decomposition of multiple attribute preference models
DESCRIPTION
This dissertation consists of three research papers on preference models of decision making, all of which adopt an axiomatic approach in which preference conditions are studied so that the models in this dissertation can be verified by checking their conditions at the behavioral level.TRANSCRIPT
Copyright
by
Ying He
2013
The Dissertation Committee for Ying He Certifies that this is the approved version
of the following dissertation:
Decomposition of Multiple Attribute Preference Models
Committee:
James S. Dyer, Supervisor
John C. Butler
Kumar Muthuraman
J. Eric Bickel
Canan Ulu
Warren J. Hahn
DECOMPOSITION OF MULTIPLE ATTRIBUTE PREFERENCE
MODELS
by
Ying He, B.Eco.; M.S.I.R.O.M.
Dissertation
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
December 2013
Dedication
To my parents and my wife.
v
Acknowledgements
I wish to take this opportunity to appreciate the people who helped me so much
during my doctoral study at The University of Texas at Austin.
I am extremely fortunate to have Professor James S. Dyer as my Ph.D advisor. I
owe my deepest gratitude to Professor Dyer for his continuous support, patient guidance,
and insightful inspiration throughout my entire Ph.D. study. He has set an example of
excellence as a researcher, mentor, instructor, and role model. I would also like to thank
Dr. John C. Butler, who always helps me think about my research from a new
perspective. I am also grateful to other member of my dissertation committee: Dr. Kumar
Muthuraman, Dr. J. Eric Bickel, Dr. Canan Ulu, and Dr. Warren J. Hahn for their helpful
comments on the dissertation and generous service on the committee.
I would like to thank my parents Yaowu He and Shulian Sun for their constant
support, continuous encouragement, and unconditional love, which makes me able to
finish my Ph.D. study in the USA.
Finally, I would like to thank my wife Ni Wang who accompanied me to the USA
and continuously provides me with her support and understanding.
vi
Decomposition of Multiple Attribute Preference Models
Ying He, Ph.D.
The University of Texas at Austin, 2013
Supervisor: James S. Dyer
This dissertation consists of three research papers on preference models of
decision making, all of which adopt an axiomatic approach in which preference
conditions are studied so that the models in this dissertation can be verified by checking
their conditions at the behavioral level.
The first paper βUtility Functions Representing Preference over Interdependent
Attributesβ studies the problem of how to assess a two attribute utility function when the
attributes are interdependent. We consider a situation where the risk aversion on one
attribute could be influenced by the level of the other attribute in a two attribute decision
making problem. In this case, the multilinear utility modelβand its special cases the
additive and multiplicative formsβcannot be applied to assess a subjectβs preference
because utility independence does not hold. We propose a family of preference
conditions called th degree discrete distribution independence that can accommodate a
variety of dependencies among two attributes. The special case of second degree discrete
distribution independence is equivalent to the utility independence condition. Third
degree discrete distribution independence leads to a decomposition formula that contains
vii
many other decomposition formulas in the existing literature as special cases. As the
decompositions proposed in this research is more general than many existing ones, the
study provides a model of preference that has potential to be used for assessing utility
functions more accurately and with relatively little additional effort.
The second paper βOn the Axiomatization of the Satiation and Habit Formation
Utility Modelsβ studies the axiomatic foundations of the discounted utility model that
incorporates both satiation and habit formation in temporal decision. We propose a
preference condition called shifted difference independence to axiomatize a general habit
formation and satiation model (GHS). This model allows for a general habit formation
and satiation function that contains many functional forms in the literature as special
cases. Since the GHS model can be reduced to either a general satiation model (GSa) or a
general habit formation model (GHa), our theory also provides approaches to axiomatize
both the GSa model and the GHa model. Furthermore, by adding extra preference
conditions into our axiomatization framework, we obtain a GHS model with a linear habit
formation function and a recursively defined linear satiation function.
In the third paper βHope, Dread, Disappointment, and Elation from Anticipation
in Decision Makingβ, we propose a model to incorporate both anticipation and
disappointment into decision making, where we define hope as anticipating a gain and
dread as anticipating a loss. In this model, the anticipation for a lottery is a subjectively
chosen outcome for a lottery that influences the decision makerβs reference point. The
decision maker experiences elation or disappointment when she compares the received
viii
outcome with the anticipated outcome. This model captures the trade-off between a
utility gain from higher anticipation and a utility loss from higher disappointment. We
show that our model contains some existing decision models as its special cases,
including disappointment models. We also use our model to explore how a personβs
attitude toward the future, either optimistic or pessimistic, could mediate the wealth effect
on her risk attitude. Finally, we show that our model can be applied to explain the
coexistence of a demand for gambling and insurance and provides unique insights into
portfolio choice and advertising decision problems.
ix
Table of Contents
Table of Contents ................................................................................................... ix
List of Tables ......................................................................................................... xi
List of Figures ....................................................................................................... xii
CHAPTER 1. INTRODUCTION .............................................................................1
1.1. Overview ................................................................................................1
1.2. Preference for risk and time ...................................................................3
CHAPTER 2. UTILITY FUNCTIONS REPRESENTING PREFERENCE OVER
INTERDEPENDENT ATTRIBUTES .....................................................................8
2.1. Introduction ............................................................................................8
2.2. Third degree discrete distribution independence .................................13
2.3. Verification and Assessment................................................................22
2.3.1. Verification of the BLII condition .....................................24
2.3.2. Assessment of the TDI decomposition ..............................28
2.4. Generality of the TDI decomposition and its relationship with some
existing decompositions .......................................................................30
2.5. A family of utility functions implying BLII condition ........................33
2.6. Nth degree discrete distribution independence ....................................41
2.7. Conclusion ...........................................................................................46
2.8. Supplemental proofs and BLII verification .........................................47
2.8.1. Proofs .................................................................................47
2.8.2. Verification of the BLII condition for utility functions
satisfying mutual risk independence ...........................................62
CHAPTER 3. ON THE AXIOMATIZATION OF THE SATIATION AND HABIT
FORMATION UTILITY MODELS ....................................................................66
3.1. Introduction ..........................................................................................66
3.2. Shifted difference independence for a measurable value function ......69
3.3. A general satiation (GSa) model ..........................................................77
x
3.4. Habit formation and satiation model with linear habit and satiation
functions ...............................................................................................84
3.4.1. A general habit formation and satiation (GHS) model ......84
3.4.2. Linear habit and satiation functions ...................................92
3.5. Axiomatization theory for risky preference .........................................98
3.6. Conclusion .........................................................................................100
3.7. Supplemental proofs ..........................................................................101
CHAPTER 4. HOPE, DREAD, DISAPPOINTMENT, AND ELATION FROM
ANTICIPATION IN DECISION MAKING .......................................................120
4.1. Introduction ........................................................................................120
4.2. The model ..........................................................................................125
4.3. The preference assumptions ...............................................................129
4.4. Risk Attitude ......................................................................................135
4.4.1. Optimism, pessimism, and risk attitude ...........................135
4.4.2. Wealth effect on risk attitude ...........................................140
4.5. Utility of Gambling ............................................................................143
4.5.1. Coexistence of gambling and purchasing of insurance....143
4.5.2. Stochastic dominance and transitivity .............................147
4.6. Decision making models ....................................................................149
4.6.1. Portfolio selection decision ..............................................149
4.6.2. Optimal advertising decision ...........................................155
4.7. Conclusion .........................................................................................159
4.8. Supplemental proofs ..........................................................................160
REFERENCES ........................................................................................................171
VITA. ....................................................................................................................180
xi
List of Tables
Table 2.1: A comparison between the TDI decomposition and some other existing
decomposition formulas. ...................................................................... 12
Table 2.2: General form of Mutual Risk-Value decomposable utility functions which
mutually satisfy BLII ............................................................................ 39
Table 2.3: Mutual Risk-Value decomposable utility functions which mutually satisfy
BLII ...................................................................................................... 65
Table 3.1: Effects from satiation and habit formation on the Delta quantity .............. 77
Table 3.2: Abbreviated notations ................................................................................118
xii
List of Figures
Figure 2.1: Third Degree Discrete Distribution Independence ................................... 15
Figure 2.2: Marginal utility functions and constants to be assessed for decomposition
(2.6) ...................................................................................................... 20
Figure 2.3: Eliciting π1, π2, and π3 ........................................................................ 25
Figure 2.4: The surface of utility function (2.8) .......................................................... 29
Figure 2.5: Relationships among different decompositions for utility function with
two attributes ........................................................................................ 33
Figure 2.6: Two indifferent lotteries with same expected values ................................ 64
Figure 3.1: Shifting value function under satiation and habit formation .................... 73
Figure 3.2: Endowed consumption and deprivation under satiation and habit
formation .............................................................................................. 75
Figure 3.3: The shifted value function in the Satiation Axiom ................................... 80
Figure 3.4: Comparison between additive independence and shifted additive
independence ........................................................................................ 99
Figure 4.1: The relationship of models (4.1) and (4.2) with some existing models . 129
Figure 4.2: Assumption 4.1: Shifted Utility Independence ....................................... 131
Figure 4.3: Assumption 4.2: Shifted Additive Independence ................................... 134
Figure 4.4: Optimistic vs. pessimistic anticipation levels ......................................... 139
1
CHAPTER 1. INTRODUCTION
1.1. OVERVIEW
This dissertation consists of three research papers on the topic of preference
models in decision making. A common topic studied in all the three papers is about the
decomposition of a multiattribute utility function representing a preference order.
However, each paper studies this topic in a different application context focusing on
different decision making problems. The subsequent three chapters of the dissertation
correspond to each paper respectively, which are all self-contained and independent. In
all three papers, when a multiattribute utility function is decomposed, we propose the
conditions on the preference which imply the utility decomposition. Therefore, the three
papers adopt an axiomatic approach to study decision models, which means that the
models in this dissertation can be verified by testing the corresponding preference
conditions proposed here.
When there are multiple criteria that concern the decision maker (DM) in a
problem context, a multiattribute utility function is used to model the DMβs preference
over alternatives. To study a multiattribute utility function, it is desirable to decompose it
into a composite function of some single attribute utility functions. The merit of such a
decomposition can be justified as follows.
2
First, in a real world decision making context with multiple attributes, a decision
analyst usually wants to assess the utility function of the DM and to base the subsequent
analysis of the problem on it. In such a case, a utility decomposition can be employed to
simplify the assessment process of the multiattribute utility function. Directly assessing a
multiattribute function is very tedious, even for a two attribute utility function. However,
if a decomposition formula can be applied to the multiattribute utility function, the
decision analyst can focus on assessing single attribute utility functions and synthesize
them into the multiattribute utility function, which makes the assessment process much
simpler. This is the problem which is discussed in the first paper.
Second, when we model some decision making phenomena by a multiattribute
utility function, a decomposition of a multiattribute utility function is associated with
some specific behavioral assumptions on preference in the decision making problem.
These behavioral assumptions usually reveal some insights about how people make
decisions. For instance, the discounted utility model (Koopmans 1960) in intertemporal
decision making assumes an additive decomposable utility function over the consumption
streams. This additive form is equivalent to the assumption that the preference over
consumption levels in one period is independent of the consumption levels in other
periods. In this case, studying the decomposition of a utility function is equivalent to
studying how people make decisions. This is exactly the approach we take in the second
and third papers in this dissertation, where utility functions are decomposed to capture
different behavioral assumptions on decision making in different contexts.
3
1.2. PREFERENCE FOR RISK AND TIME
Based on multiattribute utility decompositions, the three papers address different
decision making problems in three different contexts.
The first paper focuses on decision making under risk involved with two
attributes. In this situation, we may want to assess a von Neumann Morgenstern utility
function with two attributes to represent the preference of the DM. For this problem, the
classical approach requires the preference conditions to be verified to make sure that the
multiattribute utility function can be decomposed into some more tractable forms such as
additive or multiplicative utility functions. In the two attribute case, the additive and
multiplicative decomposition is equivalent to assuming that the two attributes are
mutually utility independent (Keeney and Raiffa 1976). However, the mutual utility
independence implies that the risk attitude on one attribute is independent of the level of
the other attribute, which may not hold in many situations.
To account for the interdependence among the attributes, we propose a family of
preference condition called th degree discrete distribution independence that can
accommodate a variety of dependencies among two attributes. The special case of second
degree discrete distribution independence is equivalent to the utility independence
condition. When the third degree discrete distribution independence holds mutually on
both attributes, the condition leads to a decomposition formula called the third degree
discrete distribution independence (TDI) decomposition, which contains many other
decomposition formulas in the existing literature as special cases. To obtain a utility
4
function by using this decomposition, the DM is required to assess the utility values of
four points obtained by arbitrarily choosing two levels on each attribute and the single
attribute utility functions on the four boundaries of the rectangle domain in the two
attributes space. Compared with the mutual interpolation independence decomposition
proposed by Bell (1979), the TDI decomposition provides a more general utility
assessment method with less effort required for the verification of the preference
condition. Furthermore, we show that for a special class of utility functions which can be
decomposed into the Risk-Value model (Jia and Dyer 1996, Dyer and Jia 1997), the
verification of the preference condition for the TDI decomposition can be simplified.
Finally, as the decomposition is more general than most existing decompositions in the
literature, it also provides a better approximate formula for two attribute utility function
when mutual utility independence condition does not hold.
The second paper focuses on the problem of decision making over time. We
consider a DM making a choice over a set of consumption streams. The classical model
in the existing literature to capture this preference is the discounted utility (DU) proposed
by Samuelson (1937) and Koopmans (1950). However, there have been many
documented experimental studies in the literature which challenge the descriptive validity
of this model (Frederick et al. 2002). It is believed that the independence axiom which
assumes that the preference over consumption levels in each period is independent of the
consumption levels in other periods is too strong to be true in many intertemporal choice
contexts. To increase the descriptive power of the DU model, this assumption is relaxed
5
to allow the past consumption levels to influence the preference over consumption in the
current period, which motivates the ideas behind the habit formation utility model (e.g.
Pollak 1970, Wathieu 1997 2004, Carroll et al. 2000), the satiation utility model
(Baucells and Sarin 2007), and a utility model with both habit formation and satiation
(Baucells and Sarin 2010). Although the habit formation utility model was axiomatized
by Rozen (2010) in a recent work, the preference conditions for both the satiation model
and the utility model with both habit formation and satiation were still unclear.
In the second paper, we axiomatize both the habit formation and satiation utility
functions by decomposing the multiattribute utility function over consumption stream
vectors, where the consumption space for each period is treated as an attribute. We
consider a case where there is no risk and assume that the DM has a strength of
preference order over consumption streams, which is represented by a utility function
called a measurable value function in the literature (Krantz 1970, Dyer and Sarin 1979).
In this case, we generalize the difference independence condition (Dyer and Sarin 1979)
to a condition called shifted difference independence to axiomatize the habit formation
and satiation model. As the difference independence implies an additive structure of the
multiattribute measurable value function (Dyer and Sarin 1979), the shifted difference
independence implies the same additive structure with a shifting quantity in the utility in
each period, which turns out to be able to capture both the satiation and habit formation
effects in the utility function. When there is risk, by following an idea similar to the one
used for strength of preference, we show that the shifted utility independence generalized
6
from utility independence (Keeney and Raiffa 1976) can be used to axiomatize satiation
and habit formation in this case. Finally, as both the satiation model and the habit
formation model are special case of the model with both satiation and habit formation
effects, our preference conditions proposed in this paper can also be used to axiomatize
these two models.
The third paper studies a decision problem where both time and risk are present.
In this problem, we consider decision making over two periods where a lottery is
resolved and paid in the second period and the DM could form an anticipation level for
the lottery in the first period. This modeling framework with two periods is consistent
with many decision problems, such as purchasing a lottery ticket or making a financial
investment. In these situations, a DM can derive utility from anticipating the payoff of the
lottery before it is resolved and paid in the second period. Forming a higher level of
anticipation for the outcome can increase the utility derived from anticipation, but it also
increases the probability of suffering from a disappointment. Therefore, the total utility a
DM experiences from this type of decision making depends on the tradeoff between
savoring a high anticipation and avoiding a high disappointment.
In the third paper, we employ the shifted utility independence condition proposed
in the second paper to decompose the total experienced utility from this two period
decision making problem into two parts. The first part captures the utility from
anticipating the payoff and the second part captures the expected utility from
experiencing both elation and disappointment based on the reference point determined by
7
the anticipation level. We show that our model contains some existing decision models as
its special cases, including the disappointment models by Bell (1985). We also use our
model to explore how a personβs attitude toward the future, either optimistic or
pessimistic, could mediate the wealth effect on her risk attitude. Finally, we show that our
model can be applied to explain the coexistence of a demand for gambling and insurance
and that it provides unique insights into portfolio choice and advertising decision
problems.
8
CHAPTER 2. UTILITY FUNCTIONS REPRESENTING
PREFERENCE OVER INTERDEPENDENT ATTRIBUTES
2.1. INTRODUCTION
Multiple attributes may need to be considered in many decision problems. The
multilinear utility function based on the utility independence condition developed by
Keeney and Raiffa (1976) has been applied widely to model preference in this situation.
For two attributes, the multilinear utility function can be reduced to either an additive or a
multiplicative utility function (Keeney and Raiffa 1976) which can be easily assessed.
However, utility independence may be a restrictive assumption for some decision
problems. For a two-attribute multilinear utility function, the mutual utility independence
condition requires that the risk aversion over one attribute is independent of the level of
the other attribute. Thus, the multilinear utility model fails to capture a decision makerβs
(DMβs) preference if her risk attitude on one attribute could be influenced by the outcome
of the other attribute.
As an example of this type of dependence, Eeckhoudt, Rey, and Schlesinger
(2007) proposed a functional form ( ) 2 ( ) [ 1] [ 1] as a
utility function over health and wealth, which exhibits increasing risk aversion on each
attribute when the other attribute is increased. In another paper on health economics, Rey
9
and Rochet (2004) discussed a two-attribute utility function over health and wealth whose
risk aversion on one attribute decreases when the other attribute increases.
Several approaches have been proposed to assess or to construct a utility function
in more general situations to allow dependence between attributes. However, not all of
these approaches are a direct generalization of the multilinear model. Fishburn and
Farquhar (1982) proposed a very general family of conditions called n-degree utility
independence conditions, which include the generalized utility independence condition
(Fishburn and Keeney 1974, 1975) as 1-degree utility independence. However, the n-
degree utility independence conditions are mutually exclusive concepts and so one
condition is not a generalization of another. Also, since Fishburn and Farquhar (1982) did
not provide assessment methods for their n-degree utility independence decompositions,
it is unclear how their formulation can be applied to assess a utility function in practice.
Another exact decomposition approach is the mixex utility function proposed by Tsetlin
and Winkler (2009), which can be reduced to either additive or multiplicative aggregation
functions as special cases. Abbasβ (2009) utility copula may be classified as an
approximation method, since the utility function depends on the type of copula used to
generate the function, but the method to choose an appropriate copula to reflect the true
preference of the DM is unclear in the paper.
We will focus on utility decomposition formulas which are generalizations of the
multilinear utility model. Both the βbilateral independenceβ decomposition, i.e.,
( ) ( ) ( ) ( ) ( ) , and the βgeneralized multiplicativeβ utility
10
function, i.e., ( ) ( ) ( ) ( ) ( ) proposed by Fishburn (1974, 1977) can
be reduced to either additive or multiplicative utility in special cases. Bell (1979) also
proposed a more general decomposition formula based on the idea of assuming mutual
interpolation independence, which contains the multilinear utility function, bilateral
independence decomposition, and general multiplicative utility as special cases. In this
approach, is interpolation independent of if the following relationship holds
( | ) ( ) ( | ) (1 ( )) ( | )
where ( | ) is called the conditional utility function and defined as ( | )
[ ( ) ( )] [ ( ) ( )] , and ( ) [
] [ ] . For any
( ) , we can always find a constant to satisfy ( | ) ( | ) (1
) ( | ), which depends on both and . However, Bellβs concept of interpolation
requires that this only depends on .
Recently, Abbas and Bell (2011) proposed a one-switch independence condition
as a generalization of the utility independence condition. They obtained a mutual one-
switch independence decomposition which is a special case of the mutual interpolation
independence model (Abbas and Bell 2011) and also contains the multilinear utility as a
special case. They also showed that the mutual one-switch independence decomposition
has overlaps with Fishburnβs generalized multiplicative decomposition and bilateral
independence decomposition (see Theorem 4 Abbas and Bell 2011), all of which are
special cases of Bellβs (1979) mutual interpolation independence decomposition.
11
In this chapter, we generalize the concept of utility independence to allow for
dependence between the risk aversion over one attribute and the level of the other
attribute to derive a new decomposition formula for a two-attribute utility function, which
we call the third degree discrete distribution independence (TDI) decomposition. The
contributions of this development include the following. First, we show that this
decomposition contains Bellβs (1979) mutual interpolation independence decomposition
as a special case, but it is even more general and therefore may be used to assess many
forms of two-attribute utility functions that cannot be assessed using Bellβs
decomposition. Further, the additional questions required to assess this more general
utility model from the DM should not be overly burdensome. Table 2.1 summarizes some
utility decomposition formulas in the literature that are generalizations of utility
independence and compares them with the TDI decomposition developed in this chapter.
Second, this utility decomposition is based on an intuitively appealing preference
independence condition that seems similar to utility independence at first glance, but is
actually more general. We believe that this new independence condition has interesting
behavioral implications that may be worthy of additional research. Third, this
independence condition can be verified much easier by a DM than Bellβs (1979)
interpolation independence decomposition, which may only be verified numerically.
Thus, the TDI decomposition discussed in this chapter provides a more practical way to
assess a more general utility function than Bellβs (1979) approach.
12
Decomposition
Name
Decomposition Formula
Multilinear
Utility (Keeney
and Raffia 1976)
( ) ( ) ( ) ( ) ( )
This decomposition contains additive decomposition and multiplicative
decomposition as special cases.
Mutual
Interpolation
Independence
Decomposition
(Bell 1979)
( ) ( | ) ( | ) ( | ) ( | ) ( ) ( | ) ( |
) ( ) ( | ) ( | ) (1 ) ( | ) ( | )
This decomposition contains multilinear utility decomposition (Keeney and
Raiffa (1976), mutual one switch independence decomposition (Abbas and
Bell 2010), generalized multiplicative decomposition (Fishburn 1977), and
bilateral independence decomposition (Fishburn 1974) as special cases.
Third Degree
Discrete
Distribution
Independence
(TDI)
Decomposition
( ) ( )(1 ( ) ( )) ( ) (1 ( ) ( ))
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
This decomposition contains Bellβs (1979) mutual interpolation
independence decomposition as a special case. The functions ( ) ( )
( ) ( ) are composite functions of
( ) ( ) ( ) (
); and and are arbitrary
values in the domain [ ] [
].
Table 2.1: A comparison between the TDI decomposition and some other existing
decomposition formulas.
The reminder of this chapter is organized as follows. In section 2.2, we introduce
the third degree discrete distribution independence decomposition and define two new
independence conditions that are necessary and sufficient for this decomposition to hold.
Binary lottery independence is one of these two conditions. In section 2.3, we use an
example to show how to verify the binary lottery independence condition in general
situations by posing a series of questions to the DM. Section 2.4 discuss the generality of
13
the TDI decomposition and its relationship with some other decomposition formulas in
the existing literature. In section 2.5, we discuss a special case which simplifies the
verification of this condition when the marginal utility functions can be decomposed into
risk-value models (Jia and Dyer 1996, Dyer and Jia 1997). In section 2.6, we generalize
the idea of the third degree discrete distribution independence condition to define a
family of th degree discrete distribution independence conditions, which implies even
more general utility functions. Section 2.7 concludes this chapter by summarizing the
relationships among the decomposition formulas developed in this chapter and some
other decomposition formulas in the literature. All the proofs are deferred to section 2.8.
2.2. THIRD DEGREE DISCRETE DISTRIBUTION INDEPENDENCE
In this chapter, we use the notation to denote both the name of an attribute and
the set of all the possible levels of this attribute. Suppose a DM has a risky preference
over two attributes and . We use the notation { ( ) (1 ) ( )} to
define a binary lottery which yields the outcome ( ) with probability and
( ) with probability (1 ). In a special case, when is fixed at the same level
for the two outcomes, we refer to this two-attribute lottery as a binary lottery
conditioned at , which is denoted by ( ) { ( ) (1 ) ( )}.
We now introduce a new decomposition of a utility function that is based on
independence conditions that are an extension of utility independence. To assess the
DMβs utility function ( ) over and using this new decomposition, we want to
14
verify whether for any outcomes on , ( ), there exist two discrete
probability distributions such that two lotteries defined on two subsets of a binary
partition of { } are always indifferent conditioned at any level of . As
there are three arbitrarily chosen levels on , we call this third degree
discrete distribution independence.
Definition 2.1. For any three outcomes on , ( ), if there exist two
discrete probability distributions such that two lotteries defined on the two subsets of a
binary partition of { } are always indifferent conditioned at any level of ,
then is third degree discrete distribution independent (TDI) of .
This condition will be met if there are two probabilities and such that the
DM is indifferent between either { ( ) (1 ) ( )} and { ( ) (1
) ( )} or { ( ) (1 ) ( )} and { ( ) (1 ) ( )} conditioned
at any level . These two cases of indifference can be visualized in the lotteries on
the left side of Figure 2.1.
15
Figure 2.1: Third Degree Discrete Distribution Independence
If the DMβs preferences satisfy the condition of being utility independent of
(Keeney and Raiffa 1976), there should be infinitely many pairs of and such that
these two binary lotteries and conditioned at on the left side of Figure 2.1
are indifferent at any level of . The condition we want to verify here is weaker than the
utility independence condition. It only requires the DM to confirm that only one such pair
of and exists. We refer to this condition as the binary lottery indifference
independence condition.
Definition 2.2. (Binary Lottery Indifference Independence) If there exist and
such that the two binary outcome lotteries ( ) and ( ) on the left side of Figure
2.1 are indifferent at any level of , we say that is binary lottery indifference
independent of , written as BLII .
(π₯π π¦) βΌ
(π₯ π¦)
(π₯ π¦)
π
1 π π
(π₯π π¦) π
β π {1 2} π {1 2}\{π}, do probabilities
π π exist such that the indifference always
holds for any level of π¦?
β π {2 3} π {2 3}\{π}, do probabilities
π π exist such that the indifference always
holds for any level of π¦?
Binary lottery indifference independence
(π₯π π¦)
(π₯ π¦)
π
1 π
(π₯π π¦)
(π₯ π¦)
π
1 π
βΌ (οΏ½οΏ½ π¦) (οΏ½οΏ½ π¦)
Trinary lottery certainty equivalent
independence
16
When the DMβs risk aversion on is influenced by the level of , her preference
does not satisfy the condition of being utility independent of ; and her certainty
equivalent for a lottery on conditioned at , defined by ( ) ( )
depends on the value of . However, it is possible that for some probabilities and ,
the DM believes that her certainty equivalents for the two binary lotteries are changed by
the same amount when the value of is varied so that she is always indifferent between
them at any level of . In section 2.3, we show how to verify this condition by eliciting
probabilities and for any ( ) conditioned at some level of
and asking the DM whether she is always indifferent between the two lotteries on the left
side of Figure 2.1 when is changed. In section 2.5, we further show that for a special
class of utility functions, this condition can be verified more easily.
The third degree discrete distribution independence condition will also be met if
one subset of a binary partition of { } contains one element and the other
subset contains three elements. In this case, the condition requires that the certainty
equivalent for a three outcome lottery on conditioned at is independent of the
level of , which is identified as the trinary lottery certainty equivalent independence.
This condition is illustrated on the right side of Figure 2.1.
We will provide some intuition for the implication of the two independence
conditions in Figure 2.1 by focusing on binary lottery indifference independence, but the
development for trinary lottery certainty equivalent independence would be similar. After
verifying the condition that BLII , we can write it as ( ) (1 ) ( )
17
( ) (1 ) ( ), where we let 3 2 in Figure 1. For 2 3, the
reasoning is the same. Then, for any { 1 2 3}, we can solve for ( ) from this
equation. Suppose we solve for
( ) ( ) ( ) (1 ) ( ) (2.1)
where and (1 ) . Since we can solve for any ( ), equation
(2.1) provides a way to assess the utility function ( ) in a bounded space
[ ] [
]. To see this, we can rewrite (2.1) by dropping the subscript of and
using the notations ( ) and ( ) to indicate that these values are functions of ,
which results in the following equation
( ) ( ) ( ) ( ) ( ) (1 ( ) ( )) ( ) (2.2)
Without loss of generality, we assume that ( ) and ( ) 1 . We
evaluate (2.2) at and respectively to determine two equations so that we can solve
for ( ) and ( ) as follows
( ) ( )( (
) ( )) ( )( ( ) (
))
( )( ( ) ( )) ( )( ( ) ( )) (2.3)
( ) ( )( (
) ( )) ( )( ( ) (
))
( )( ( ) ( )) ( )( ( ) ( )) (2.4)
Both (2.3) and (2.4) are composite functions that only depend on marginal utility
functions ( ) and ( ). Thus, we can use (2.2) to assess the utility function
( ) by assessing five marginal utility functions ( ) ( ) ( ) ( )
18
and ( ). If the DMβs preference also satisfies the condition that BLII , we can
exchange and in (2.2) and derive (2.5) for any ( )
( ) ( ) ( ) ( ) ( ) (1 ( ) ( )) ( ) (2.5)
where the equations for ( ) and ( ) can be obtained by switching and in the
formulas for ( ) given by (2.3) and for ( ) given by (2.4).
Evaluating ( ) and ( ) with (2.5) and substituting them into (2.2), we
have
( ) ( ) [ ( ) ( ) ( ) ( ) (1 ( ) ( )) ( )]
( ) [ ( ) ( ) ( ) ( )
(1 ( ) ( )) ( )] (1 ( ) ( )) ( )
(1 ( ) ( )) [ ( ) ( ) ( ) ( )]
(1 ( ) ( )) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
By using (2.3) and (2.4), we can verify that
[ ( ) ( ) ( ) ( )] ( ) . Substituting this relationship into the
above equation, we obtain the decomposition formula for the utility function ( )
below.
19
( ) ( )(1 ( ) ( )) ( ) (1 ( ) ( ))
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
(2.6)
Applying decomposition (2.6) to assess a utility function is straightforward. First,
we ask the DM to assess the marginal utility functions ( ), ( ), ( ), and
( ) on the boundaries of the attribute space [ ] [
] and the utility values
at the six black circles in Figure 2.2. These four constants ( ), ( ), ( ),
and ( ) may be assessed for any ( ) and any (
) After
we obtain these functions and values, we substitute them into the formulas for the
coefficients ( ), ( ), ( ), and ( ). Then, by substituting these coefficients
and the marginal functions into (2.6), we can obtain the assessed utility function ( ).
We showed that binary lottery indifference independence implies this
decomposition (2.6). Following a similar approach, it is easy to show (2.6) can also be
implied by the trinary lottery certainty equivalent independence. Thus, each of them is
sufficient to derive (2.6). To obtain the necessary and sufficient conditions for (2.6), we
notice that when (2.2) holds, there are two cases. If one of the three coefficients in (2.2)
(either ( ), ( ), or 1 ( ) ( )) is negative, then (2.2) is equivalent to the
binary lottery indifference independence condition. However, if all the three coefficients
( ), ( ), and 1 ( ) ( ) are positive, (2.2) is equivalent to the trinary
20
lottery certainty equivalent independence condition. Therefore, the necessary and
sufficient condition for (2.6) is that third degree discrete distribution independence
mutually holds on both attributes.
Note: the utility values at the black circles are to be assessed; and the utility values
at the white circles are to be calculated from the assessed marginal utility functions
Figure 2.2: Marginal utility functions and constants to be assessed for decomposition (2.6)
Theorem 2.1. and are mutually third degree discrete distribution independent if and
only if the utility function can be decomposed by (2.6) on the bounded attributes space
[ ] [
].
Unlike the binary lottery indifference independence condition, the trinary lottery
certainty equivalent independence condition may not be easy to verify, since it is not
apparent what kinds of preferences would satisfy this condition. Thus, verifying the
necessary and sufficient conditions for decomposition (2.6) could be a demanding task, as
π’(π₯ π¦ )
π’(π₯ π¦ ) 1
π’(π₯ π¦ ) π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
π’(π₯ π¦ )
21
it requires the DM to think about both conditions described in Figure 2.1. In the
following corollary, we specify a condition under which the mutual binary lottery
indifference independence condition is both necessary and sufficient for decomposition
(2.6).
Corollary 2.1. For any [ ] and [
], when the marginal utility
functions ( ), ( ), ( ), and ( ) are increasing functions, if both
[ ( ) ( )] [ ( ) ( )] and [ ( ) (
)] [ ( )
( )] are strictly monotonic functions of and respectively, the utility function
( ) can be decomposed by (2.6) if and only if the binary lottery indifference
independence condition holds mutually for both and .
This corollary gives us a way to simplify the verification of the necessary and
sufficient condition for decomposition (2.6) by asking the DM to focus on thinking about
the binary lottery indifference independence condition only. When assessing a utility
function, if the DMβs preference does not satisfy the mutual utility independence
condition, most existing assessment methods require the DM to assess four marginal
utility functions on the boundaries of the attributes space. Since these four marginal
utility functions will be needed to assess the utility function eventually, we may choose to
assess them first before verifying the preference condition. We can use these marginal
utility functions to calculate these ratios for both attributes in this corollary to see whether
they are strictly monotonic on each attribute space.
22
If these utility ratios are strictly monotonic, we can verify the binary lottery
indifference independence condition for both attributes as the necessary and sufficient
conditions for the decomposition (2.6). If they are not, we could make partitions on each
attribute space such that they are strictly monotonic on each subset and then focus on
verifying the binary lottery indifference independence for each subset. However, based
on our empirical analysis of utility functions over two attributes that are mutually third
degree discrete distribution independent, we would expect the monotonic condition for
the ratio in the Corollary 2.1 to be satisfied for a large majority of practical applications.
2.3. VERIFICATION AND ASSESSMENT
Equation (2.1) is a mathematical representation of the BLII condition, so if the
two functions and in equation (2.1) depend on then the condition is not satisfied.
If we rearrange the terms and divide both sides by ( ) ( ) (2.1) can be
rewritten in the following form
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
(2.7)
For any , the ratio [ ( ) ( )] [ ( ) ( )] for {1 2 3}
in (2.7) can be interpreted as a probability π ( ) that solves the equation ( )
π ( ) ( ) (1 π ( )) ( ) ; i.e., π ( )
( ) ( )
( ) ( ) for {1 2 3} .
Therefore, to verify whether and in (2.1) depend on , we can elicit π ( )
23
( ) ( )
( ) ( ) for {1 2 3} at different levels of and check whether they satisfy
(2.7) simultaneously for a unique pair of and . For each pair of and , we can
obtain a pair of and as we showed in section 2.2.
At different values of , i.e, , π ( ) π ( ), and π ( ) give three
column vectors with dimensions.
(
π ( ) π ( ) π ( )
π ( )π ( )
π ( )π ( )
π ( )π ( )
π ( )π ( )
π ( )π ( )
π ( )π ( ) )
For any , the rank of the above matrix is 3. If the rank 1, there exist
infinitely many pairs of and such that one column is a linear combination of the
other two columns. In this case, (2.7) is true for infinitely many pairs of and ; and
thus there exist infinitely many pairs of and such that the two lotteries in Figure 2.1
are indifferent, which corresponds to the case when mutual utility independence holds. If
the rank 2, there exists a unique pair of and to express one vector as the linear
combination of the other two, which implies a unique pair of and such that one of
the two lottery pairs in Figure 2.1 is indifferent conditioned on any level of . If rank
3, the three vectors are linearly independent, and there exists no such pair of and
. Therefore, the number of pairs of and that can make the two lotteries in Figure
2.1 be indifferent is either zero, or one, or infinity.
24
We will proceed as follows. First, we will assume that the DMβs preferences are
consistent with a specific form of a utility function that does satisfy the binary lottery
independence condition, and show how to verify this condition using the above idea.
Second, once the binary lottery independence condition has been verified based on
responses implied by this utility function, we will demonstrate how to recover that same
function using the assessment procedure illustrated in Figure 2.2.
2.3.1. Verification of the BLII condition
Suppose a DM needs to choose among different investment options for her
retirement pension. She believes that there are two attributes to be considered, wealth ( )
measured on the range [$0, $1] million dollars and health ( ) measured on the range [0
QALY, 80 QALY] where a QALY is defined as a quality adjusted life year. So, she
wants to assess her utility function over these two attributes to help her make a choice
over different risky plans. We assume that she can express her preferences based on the
following utility function that is otherwise unknown to her, which has been rescaled such
that ( ) [ 1] [ 1] and ( ) [ 1].
( )
( 32 11 ) ( 313 3 2 )
( 2 2 ) ( 3 3 3 )
(2.8)
25
In order to verify the BLII condition for this DM, we will elicit π for
{1 2 3}. First, we arbitrarily choose three levels of and one level of . For instance,
we can choose 2 [ ] [ 1 ],
and 32 [ ] . Then, we ask the DM to elicit the
probabilities π π π such that the indifference relationships shown in Figure 2.3 will
hold.
Figure 2.3: Eliciting π , π , and π
If the DM specifies these probabilities according to her utility function (2.8), we
will obtain the following results.
π (32 ) 2 π (32 ) 2 π (32 ) ;
( 1π 32 ππ΄πΏπ)
( π 32 ππ΄πΏπ)
π
1 π
( 2π 32 ππ΄πΏπ) βΌ
( 1π 32 ππ΄πΏπ)
( π 32 ππ΄πΏπ)
π
1 π
( π 32 ππ΄πΏπ) βΌ
( 1π 32 ππ΄πΏπ)
( π 32 ππ΄πΏπ)
π
1 π
( π 32 ππ΄πΏπ) βΌ
26
Now, we arbitrarily change the value of to 2 , and ask the DM to elicit
these probabilities again. This time, the DM will identify the following probabilities
based on her utility function (2.8)
π ( 2 ) 31 π ( 2 ) π ( 2 ) .
After we obtain these two sets of probabilities for two different health levels, we
can use the equations below corresponding to (2.7) to solve for the two variables and
.
π (32 ) π (32 ) π (32 )
π ( 2 ) π ( 2 ) π ( 2 )
Solving the above equations, we obtain 2 and 1 . Substituting
and into (2.1) and shifting the term ( ) to the left side, we have for
{32 2 }
( ) 1 ( 2 ) 2 ( ) 3 ( )
Notice that the coefficients on both sides of this equation sum to the same
constant, which in this case is 2 . So, we can divide both sides by 2 to obtain the
following equation.
1
2 ( )
1
2 ( )
2
2 ( )
3
2 ( )
This gives
1 and
for 2
[ 1 ] in (2.1). Thus, we know for
27
{32 2 } , the DM is indifferent between the two lotteries
{ 1 ( ) ( 2 )} and { ( ) 1 ( )}.
Now, we ask the DM whether she will feel indifferent between these two lotteries
if we change the health level to other values. As implied by the assumed utility
function (2.8), the DM will answer βyesβ. So, we can verify that the probabilities
1 and make the DM feel indifferent between these two lotteries for
any . Then, we should repeat this process by picking another set of , , and to
verify the existence of and again. Since we can always verify the existence of
and for different values of , , and for utility function (2.8), we are assured that
the DMβs preference represented by utility function (2.8) satisfies the BLII condition.
The preference condition verification process outlined above is simpler than that
required by Bellβs MII decomposition. Bellβs condition needs to be verified by eliciting
values of the conditional utility function ( | ) on a grid formed by choosing many
different levels on both attribute and and checking whether these values satisfy the
interpolation independence condition (see Figure 2, Bell 1979). To verify the condition
for the TDI decomposition, we still need to elicit values of the conditional utility function
at many different levels on . But, on attribute , we only need to choose two levels.
This eliminates almost half of the elicitation work required in the preference verification
process.
28
2.3.2. Assessment of the TDI decomposition
When the BLII condition has been verified as illustrated above, we can assess the
TDI decomposition for the DM by following the logic of Figure 2.2. We would assess
this DMβs single attribute utility functions on the boundaries of the two attributes domain
and six constants based on responses that would be implied by (2.8). We ask the DM to
assess the values of ( ) and ( ) first. For instance, we can assess (
)
by eliciting a probability such that ( ) βΌ { (
) (1 ) ( )} .
Suppose the DM responds according to the utility function (2.8), and we would obtain
( ) and ( ) . Then, if we assess the four single attribute utility
functions in Figure 2.2 and rescale them such that ( ) , ( ) ,
( ) 1, and ( ) , we have the following utility functions.
( ) ( )
( ) 1
1 ( )
1
1
Then, we arbitrarily choose ( 1) and ( 1). With the marginal
single attribute utility functions assessed above, we can calculate ( ), ( ),
( ), and (
). Then, to assess ( ), ( ), ( ), and ( ),
we can follow the same idea. For instance, ( ) can be assessed by eliciting
probability such that ( ) βΌ { ( ) (1 ) ( )}. Then, ( )
can be calculated by ( ) ( ) (1 ) ( ). Finally, substituting
29
these single attribute utility functions and constants into (2.6) gives (2.8), which is plotted
in Figure 2.4.
Figure 2.4: The surface of utility function (2.8)
In the above process, the assessment of the marginal utilities relies on the assessed
constants ( ) and ( ), so care should be taken to assess these values as
accurately as possible. Any errors in the assessment of these constants will be amplified
when assessing the marginal utility functions based on them.
30
2.4. GENERALITY OF THE TDI DECOMPOSITION AND ITS RELATIONSHIP WITH SOME
EXISTING DECOMPOSITIONS
To gain some insights regarding the generality of our decomposition (2.6), we
rewrite it in the form of (2.9) shown below, which is obtained by substituting the
expressions for the four functions ( ) ( ) ( ) ( ) into our decomposition
(2.6).
( ) ( ) ( ) ( ) (
) ( ) ( )
( ) ( ) (
) ( ) ( ) ( )
(2.9)
where the for {1 2 3 } and for {1 2} and {3 } are coefficients
which depend on the six constants indicated by the black circles in Figure 2.2.
The multilinear utility function for two attributes (Keeney and Raffia 1976) is of
the form ( ) ( ) ( ) ( ) ( ), which is a special case of
(2.9). The bilateral independence decomposition (Fishburn 1974) and the mutual
interpolation independence decomposition (Bell 1979) also imply utility functions of the
form of (2.9). For the bilateral independence decomposition, the coefficients
and in (2.9) only depend on ( ) and ( )
(see section 5, Fishburn 1974). For the mutual interpolation independence decomposition,
they depend on ( ) , ( ) , and ( ) for arbitrary ( ) [
]
[ ] (See theorem 1, Bell 1979). In our model, these coefficients depend on the six
31
assessed utility values indicated by the black circles in Figure 2.2. The result is that our
decomposition (2.6) is more general. In Proposition 2.1, we show that our decomposition
(2.6) contains the mutual interpolation independence decomposition developed by Bell
(1979) as a special case.
Proposition 2.1. Mutual interpolation independence decomposition implies
decomposition (2.6), but (2.6) does not imply mutual interpolation independence
decomposition.
Since the mutual interpolation independence decomposition is a very general
decomposition formula that contains many existing decompositions as its special cases
(see table 1, Bell 1979), Proposition 2.1 implies that all these decompositions identified
by Bell (1979) are also special cases of our decomposition (2.6). In the Appendix, we
show that decomposition (2.6) contains more general utility functions which cannot be
decomposed into the mutual interpolation independence decomposition, because
decomposition (2.6) allows the interpolation coefficients assumed by Bell (1979) to
depend on both and .
The generality of our decomposition (2.6) is important because it provides a
utility assessment approach that can more accurately represent the DMβs true preference
compared with the utility decomposition formulas which are special cases of this
decomposition. This point has been made by Keeney and Raiffa (see the beginning of
subsection 5.7.4, Keeney and Raiffa 1976). They also argue that a more general utility
32
function decomposition may provide a better approximation for true utility functions
because of its degrees of freedom, which they define as the number of single attribute
utility functions and constants that need to be assessed in a decomposition formula. Our
decomposition (2.6) has ten degrees of freedom (four single attribute utility functions
plus six constants in Figure 2.2) which is more than the degrees of freedom of other
existing decomposition formulas discussed by Keeney and Raiffa (Figure 5.12, Keeney
and Raiffa 1976). Bellβs (1979) mutual interpolation independence decomposition has
seven degrees of freedom. Thus, decomposition (2.6) also should provide a better
approximation of a two-attribute utility function than any of these alternative models.
Of course, the generality of a decomposition formula cannot be obtained without
some cost. The disadvantage of a more general decomposition formula is the more
complex assessment procedure associated with it. Decomposition (2.6) asks the DM to
assess three extra constants compared with Bellβs (1979) mutual interpolation
independence decomposition, so we believe this keeps the assessment task within
acceptable bounds.
In Figure 2.5, we visualize the relationships among the decompositions we
developed in this chapter with some existing decompositions including those summarized
in Table 2.1. The biggest circle in the figure represents the decomposition implied by th
degree discrete distribution independence condition as a generalization of the third degree
discrete distribution independence condition, which is discussed in section 2.6.
33
Figure 2.5: Relationships among different decompositions for utility function with two
attributes
2.5. A FAMILY OF UTILITY FUNCTIONS IMPLYING BLII CONDITION
The verification process for the BLII condition outlined in section 2.3 may be
simplified by focusing on a family of utility functions which can be decomposed into a
risk-value model (Jia and Dyer 1996, Dyer and Jia 1997) on each attribute. These risk-
value models were originally developed for single attribute utility assessments, but their
extension to the multiple attributes domain provides a variety of models that are
composed of linear and multiplicative forms that include terms commonly found in
popular multiattribute utility models, including power, exponential and logarithmic. As a
result, this development also demonstrates that there exist many different two-attribute
Mutual Interpolation Independence
Decomposition
General
Mutiplicative
Decomposition
Multi-linear Model
(Additive or Multiplicative)
(Mutual 2nd Degree Discrete
Distribution Independence)
Bilateral
Decomposition
Mutual One-Switch
Independence Decomposition
Mutual 3rd Degree Discrete Distribution Independence
Decomposition (2.6)
Mutual n-th Degree Discrete Distribution Independence
Decomposition for n>3
34
utility models that do not satisfy mutual utility independence, but that do satisfy the BLII
condition and would be practical for use in preference modeling.
In the standard risk-value model (Jia and Dyer 1996), a normalized lottery is
defined as a lottery with zero mean that is obtained by subtracting the expected value
from the lottery , namely . The set of such normalized lotteries is
denoted by { | }, where is the set of arbitrary lotteries. For a
single attribute utility function ( ), when the preference of a DM satisfies a condition
called risk independence, the expected utility of a lottery can be decomposed as
( ) ( ) ( )[ ( ) ( )] where ( ) is defined as the standard measure
of risk (see theorem 3, Jia and Dyer 1996). This risk independence condition requires that
for any , if then for any . Furthermore, if
the utility function ( ) is continuously differentiable, then it must be one of the three
functional forms: ( ) ; ( ) ; and ( ) where
, , and are constants (see theorem 4, Jia and Dyer 1996).
For a two-attribute utility function ( ), if the DMβs preference satisfies the
risk independence condition on attribute given any level of and if we combine the
utility functions ( ) and ( ) into one case by allowing the
constant to be either zero or non-zero, we know the two-attribute utility function must
be one of two forms, namely either ( ) ( ) ( ) ( ) or ( )
( ) ( ) ( ) ( ), where ( ), ( ), ( ), and ( ) are functions of .
35
These utility functions can be decomposed into a standard risk-value model on
conditioned at any level of . For example, the expected value of a lottery ( ) given
by the exponential utility function above can be written as
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) [ ( )( ) 1] ( )
( ) ( ) ( ) [ ( )( ) 1] ( )
where the standard measure of risk is ( ) ( )( ) , which depends on
.
When the standard measure of risk on is independent of , namely ( ) ,
the exponential utility function above satisfies the condition that BLII .1 The
intuition for this conclusion is that for any two binary lotteries and on the left side
of Figure 2.1, there exist probabilities and such that the two lotteries have the same
mean and standard measure of risk for either 2 3 or 3 2.
( ) ( )
(1 ) (1 )
(1 ) (1 )
1 Even though the standard measure of risk here is independent of , the utility function still allows
interdependence between attributes in the sense that the Arrow-Pratt risk measure on depends on .
36
When both the mean and the standard measure of risk are the same for the two binary
lotteries, it is easy to see from the risk-value model that the utilities for these two lotteries
are always the same conditioned at any level of .
( ) ( ) ( ) [ ( ) 1]
( ) ( ) [ ( ) 1] ( )
Therefore, when the standard measure of risk on is independent of , i.e., ( ) ,
the exponential utility function ( ) can satisfy the condition that BLII . When
the utility function on is of the quadratic form, it can be verified that the standard
measure of risk on is independent of , so, this utility function also satisfies the
condition that BLII .
The relative risk value model is established using a similar development, except
that the measure of risk is based on the ratio of (Dyer and Jia 1997). If a DMβs
preference satisfies the relative risk independence condition on at any level of , by
allowing the constants in the functional forms of the single attribute utility function that
are implied by the relative risk independence to depend on the level of (theorem 2,
Dyer and Jia 1997), we can conclude that the two-attribute utility function ( ) must
be one of the forms: (i) ( ) ( ) ( ) ; (ii)
( ) ( ) ( ) ( ); (iii) ( ) ( ) ( ) ( ); (iv)
( ) ( ) ( ) ( ) for ( ) 1; (v) ( ) ( ) ( ) ( )
for ( ) 1; (vi) ( ) ( ) ( ) ( ) ( ) for ( ) 1; or (vii)
37
( ) ( ) ( ) ( ) ( ) for ( ) 1. Following the same idea above,
if we can further verify that the relative measure of risk is independent of , we can
conclude that the utility function satisfies BLII .
Therefore, if we can verify either risk independence or relative risk independence
on , we only need to verify that the measure of risk (standard or relative) is independent
of to determine the condition that BLII . This can be accomplished by presenting
a pair of lotteries ( ) and ( ) conditioned at some level that are indifferent
for the DM and have the same mean on , i.e., ( ) βΌ ( ) at with ,
and ask the DM whether she will be indifferent between this pair of lotteries when is
changed. If the DM still feels indifferent when is changed, we can conclude that the
measure of risk in the risk-value model must be independent of . Thus, we have the
following theorem.
Theorem 2.2. If the preference of a DM satisfies the (relative) risk independence
condition on at any level of , and if she feels indifferent between two binary lotteries
( ) and ( ) which have the same means for any , then the preference
of this DM satisfies the condition that BLII .
To apply the Theorem 2.2, we need to find a pair of lotteries ( ) and ( )
as required by the theorem for a DM. This can be accomplished by following a similar
approach to the one we used to elicit the probabilities and in section 2.3.
38
Specifically, for the chosen ( ), we elicit π π , and π conditioned at
some level of . First, we set up one equation of and according to equation (2.7),
which requires that the two lotteries are indifferent given , i.e., ( ) βΌ ( ).
Then, by requiring the two lotteries to have the same mean on , , we can set up
another equation for and . By solving these two equations simultaneously, we can
obtain a pair of lotteries that meet the requirements of Theorem 2.2 at . Then, we
ask the DM to identify whether she will be indifferent between the two lotteries when
is changed. An example illustrating this approach is provided in the Appendix.
Compared with the verification process outlined in section 2.3, this process is
relatively easy to implement. Therefore, it is desirable to identify the utility functions that
mutually satisfy these requirements on both attributes.
From the functional forms of single attribute utility functions that can be
decomposed into either standard or relative risk-value models listed above, we can
confirm that any two-attribute utility function that satisfies the mutual BLII condition but
is not multilinear decomposable must be of the form of either ( ) ( ) ( )
( ) ( ) or ( ) ( ) ( ) ( ) ( ) with ( ) ( ) and ( )
( ) , which are the generalized multiplicative and bilateral independence
decompositions proposed by Fishburn (1977, 1974); only these two types of utility
functions can match the marginal utility functional forms that are risk-value
decomposable.
39
Now, by choosing either a linear, a logarithm, a multiplication of linear and
logarithm, an exponential, or a power function form for ( ), ( ), ( ), and ( ) in
both decompositions ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( )
( ) ( ), we can obtain all the mutual risk-value decomposable utility functions with
the measures of risk on one attribute being independent of the other attribute shown in
Table 2.2. If we eliminate the functional forms that can be obtained by simply switching
and in the functions in Table 2.2, there are fifty-six different utility functions, which
are fully listed in Table 2.3 at the end of the Appendix. For simplicity, we write power
utility functions in a general form ( ) . The constants
and take values such that all the marginal utility functions are increasing on both
attributes.
( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) { ( ) ( )
}
( ) { ( ) ( ) }
( ) ( )( ) ( ) ( )
( ) { ( ) ( )
}
( ) { ( ) ( )
}
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) { ( ) ( )
}
( ) { ( ) ( )
}
( ) ( ) ( )
( ) { ( ) ( )
}
( ) { ( ) ( )
}
( ) ( ) ( )
( ) { ( ) ( ) }
( ) { ( ) ( ) }
Table 2.2: General form of Mutual Risk-Value decomposable utility functions which
mutually satisfy BLII
40
Since the utility functions listed in Table 2.2 are special forms of these two types
of Fishburnβs decomposition, they have overlaps with the mutual one switch
independence decomposable utility functions (Abbas and Bell 2011). For these utility
functions, we may either verify the mutual one switch independence condition or mutual
BLII condition to assess them, depending on which condition is easier for the DM.
However, there are also some utility functions in Table 2.2 which do not satisfy
the mutual one switch independence condition. For example, suppose a preference is
represented by ( ) ( ) ( ) with ( )
[1 ) [1 ). To examine whether this utility function satisfies the one switch
independence condition, we rewrite it as ( ) [1 ( )
( )
], which is
the form used by Abbas and Bell (2011). To satisfy the mutual one switch independence,
both functions ( )
and
( )
must be strictly monotonic (Theorem 4 Abbas and
Bell 2011). However, it is easy to verify that these two functions are not monotonic
functions on their respective domains.
Following the similar idea, we can verify that the mutual risk-value decomposable
utility functions, which are of the general multiplicative utility functional form ( )
( ) ( ) ( ) ( ) ( ) ( ) [1 ( )
( )
( )
( )], may not satisfy mutual one switch
independence. This is because the ratios ( )
( ) and
( )
( ) may not be strictly monotonic
functions on their respective domains. There are 20 utility functions listed in the
41
Appendix which are of the functional form of ( ) ( ) ( ) ( ) ( ) that do
not satisfy the mutual one switch independence condition. For these utility functions,
verifying the BLII condition as we outlined is the only way to determine its preference
condition non-numerically.
2.6. NTH DEGREE DISCRETE DISTRIBUTION INDEPENDENCE
In section 2.2, we defined the third degree discrete distribution independence
condition based on two lotteries involving changes in one attribute only; these two
lotteries feature a total of four outcome values, three of which can be arbitrarily chosen.
In this section, we generalize this idea to consider indifference between two lotteries over
one attribute which have a total of 1 outcomes and of them are free to be chosen.
To motivate this preference condition, we consider lotteries over more than three
levels of conditioned at some level of . For outcomes ( ) and a
permutation ( ) of the index { 1 2 } with ( ) , we can choose
{ 1 2 1} so that the outcomes on form a binary partition
{{ ( ) ( ) ( )} { ( ) ( )}} of the set { } For some
permutation ( ) and , if there exist two discrete probability distributions
( ( ) ( ) ( )) and ( ( ) ( ) ( )) such that the two lotteries defined
by these distributions are always indifferent to each other at any level of , then the
following equation holds for any :
42
( ) ( ( ) ) ( ) ( ( ) ) ( ) ( ( ) )
( ) ( ( ) ) ( ) ( ( ) ) ( ) ( ( ) )
By letting an arbitrary ( ) , { 1 2 } and expressing ( ( ) )
using the other terms in the above equation, we can write ( ) as a linear combination
of these utility functions evaluated at the other points. Suppose we let ( ) , we have
( ) ( ( ) )
( )β ( ) ( ( ) )
( )β ( )
( ( ) )
Since ( ( ) ( ) ( )) and ( ( ) ( ) ( )) are two probability
distributions, we know that β ( ) 1 and β ( )
1 ( ) . So, we have
( )β ( )
( )β ( )
1. Thus, the summation of the coefficients on the
right side is one. Now, we replace the subscripts ( ) (1) ( 1) ( 1) ( )
with 1 2 1 . We have ( ( ) ) ( ) for 1 1 and
( ( ) ) ( )for 1 . If we treat as fixed values and
allow to vary, we can also redefine the coefficient of ( ) as ( ) for
{1 1}. Then, we can rewrite this equation as
( ) β ( ) ( ) (1 β ( )
) ( )
(2.10)
which is a more general form of (2.2) developed in section 2.2.
43
Definition 2.3. is said to be th degree discrete distribution independent of if for
arbitrary ( ) where 2 , there exist two discrete probability
distributions such that two lotteries defined on the two subsets of a binary partition of
{ } conditioned at some level of are always indifferent for any level of
.
This definition gives a family of preference conditions that contain the preference
condition defined in Definition 2.1 as a special case. The second degree discrete
distribution independence is equivalent to the utility independence condition (Keeney and
Raiffa 1976) as stated in the following proposition.
Proposition 2.2. is second degree discrete distribution independent of if and only if
is utility independent of .
By following the idea used in section 2.2, this th degree discrete distribution
independence condition can be used to derive a more general decomposition formula. If
we evaluate (2.10) at 1 points of , , we can solve for ( )
1 2 1 in the same way we solved for ( ) and ( ) for (2.2). So, we have
for any {1 2 1} , ( ) ( ( ) ( ) ( )) . By
substituting them into (2.10), we have the more general decomposition formula.
44
( ) β ( ( ) ( ) ( ))
( )
[1 β ( ( ) ( ) ( ))
] ( )
(2.11)
This decomposition requires the DM to assess 1 marginal utility functions
over and marginal utility functions over . Thus, we have the following theorem.
Theorem 2.3. is th degree discrete distribution independent of if and only if the
utility function can be decomposed by (2.11) on a bounded domain [ ] [
].
The verification of this condition becomes complicated since there are many
possible binary partitions for a set with a large number of outcomes. However, if a DM
can verify that the certainty equivalents for two non-degenerate lotteries defined on
1 outcomes on conditioned at some levels of would change by the same
amount when changing the level of , we can assess her utility function by using
decomposition (2.11).
This decomposition formula (2.11) was derived based on the assumption of the
preference condition on one attribute. Following the same idea as in section 2.2, we can
symmetrically define the condition of being th degree discrete distribution
independent of and derive a symmetric decomposition of (2.11). Then, by assessing
( ) in (2.11) with this symmetric decomposition, we obtain a decomposition based
45
on the mutual independence condition, which can reduce the number of marginal utility
functions that need to be assessed.
Another decomposition which also requires many single attribute utility functions
to be assessed was proposed by Tamura and Nakamura (1983). They defined a condition
called being th order convex dependent of by the following mathematical relation
(2.12) without giving a preference interpretation
( | ) β ( ) ( | )
(1 β ( )
) ( | )
(2.12)
where ( | ) [ ( ) ( )] [ ( ) ( )] is the conditional utility
function originally defined by Bell (1979) in the development of the mutual interpolation
independence decomposition.
When 1 , the above relation is reduced to the interpolation independence
condition proposed by Bell (1979). For 1, this interpolation assumption requires the
DM to assess five marginal utility functions to determine ( ). This is the same
number of marginal utility functions required to be assessed by the third degree discrete
distribution independence condition defined in section 2.2. For the th order convex
dependence assumption (2.12), the number of marginal utility functions to be assessed is
equal to that required by the ( 2) th degree discrete distribution independence
condition. Proposition 2.3 states that for the same number of marginal utility functions to
46
be assessed, our discrete distribution independence gives a more general decomposition
than the convex dependence decomposition.
Proposition 2.3. being th order convex independent of implies being (
2)th degree discrete distribution independent of .
For the preference condition on one attribute, the second degree discrete
distribution independence requires assessing three marginal utility functions (Keeney and
Raiffia 1976). The third degree discrete distribution independence requires assessing five
marginal utility functions. For nth degree discrete distribution independence condition on
one of the two attributes, there are 2 1 marginal utility functions to be assessed.
2.7. CONCLUSION
Decomposition (2.6) developed in this chapter is a general decomposition of a
two-attribute utility function that can be obtained by assessing four conditional utility
functions on the boundaries of the domain. This decomposition is equivalent to a
condition called the third degree discrete distribution independence. By generalizing the
third degree discrete distribution independence condition, we obtained a family of th
degree discrete distribution independence decompositions which contain the convex
dependence decomposition as a special case. However, with the increase of , the effort
required to verify the preference conditions and to assess the utility function increases
also. Therefore, we believe the decomposition (2.6) can be used as either an exact
47
decomposition or as an approximation for a two-attribute utility function, since, to our
knowledge, it is the most general exact decomposition that only requires assessing four
conditional utility functions. Besides the improvement on the generality of the utility
assessment method, the work in this chapter also provides a hierarchy of decomposition
formulas that include many existing decomposition formulas as their special cases, which
makes a theoretical contribution to the existing knowledge of the multiattribute utility
assessment methods.
Although the ideas developed in this chapter can be extended to decompose utility
functions with more than two attributes, the assessment would become very tedious as
more marginal utility functions would be needed. However, for a problem with more than
two attributes, we may expect that most attributes would satisfy the mutual utility
independence condition, and this more general decomposition developed in this chapter
might be applied for pairs of the attributes when mutual utility independence is not
satisfied.
2.8. SUPPLEMENTAL PROOFS AND BLII VERIFICATION
2.8.1. Proofs
Theorem 2.1. and are mutually third degree discrete distribution independent if and
only if the utility function can be decomposed by (2.6) on the bounded attributes space
[ ] [
].
48
Proof: βββ Under the assumption of mutual third degree discrete distribution
independence, we can conclude both (2.2) and (2.5) hold in section 2.2. The formulas for
( ) ( ) ( ) ( ) are easy to verify. Here we show how to derive (2.6).
Evaluating ( ) and ( ) by (2.5) and substituting them into (2.2), we
have
( ) ( ) [ ( ) ( ) ( ) ( ) (1 ( ) ( )) ( )]
( ) [ ( ) ( ) ( ) ( )
(1 ( ) ( )) ( )] (1 ( ) ( )) ( )
(1 ( ) ( )) [ ( ) ( ) ( ) ( )]
(1 ( ) ( )) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
(2.13)
Based on (2.13), if we can prove [ ( ) ( ) ( ) ( )]
( ), by substituting this relationship into (2.13), (2.13) becomes (2.6). We can verify
this is true by substituting ( ) and ( ) into ( ) ( ) ( ) ( ).
Thus, (2.6) holds.
βββ Given (2.6), from the proof above, we can conclude that the first equation in
(2.13) holds, which can be written as follows.
49
( ) ( ) ( ) ( ) ( ) (1 ( ) ( )) ( ) (2.14)
where ( ) [ ( ) ( ) ( ) ( ) (1 ( ) ( )) ( )]
and ( ) [ ( ) ( ) ( ) ( ) (1 ( ) ( )) ( )].
Given (2.14), for any ( ), we can show that there exist and
such that the following equation always holds for any .
( ) (1 ) ( ) ( ) (1 ) ( ) (2.15)
Evaluating ( ), ( ) and ( ) by (2.14) and substituting them as
well as (2.14) into (2.15), we have
[ ( ) ( ) ( ) ( ) (1 ( ) ( )) ( )]
(1 )[ ( ) ( ) ( ) ( )
(1 ( ) ( )) ( )]
[ ( ) ( ) ( ) ( ) (1 ( ) ( )) ( )]
(1 )[ ( ) ( ) ( ) ( )
(1 ( ) ( )) ( )]
By matching the coefficients for ( ), ( ), and ( ), we can see that for
any the above equation holds if the following two equations hold simultaneously for
some and .
( ) (1 ) ( ) ( ) (1 ) ( )
50
( ) (1 ) ( ) ( ) (1 ) ( )
As the above equation system has unique solutions for and , we can
conclude that there always exist and such that (2.15) holds. When , 1 , ,
and 1 are all positive or two of them on different sides are negative, (2.15) implies
the condition of being binary lottery indifference independent of . Otherwise, (2.15)
implies the trinary lottery certainty equivalent independence holds. Thus, is third
degree discrete distribution independent of .
Finally, as (2.6) is symmetric in both and , (2.6) implies a symmetric
expression of (2.14), which can be obtained by switching and in (2.14). Thus, by
following the same reasoning above, we conclude that (2.6) also implies that is third
degree discrete distribution independent of .β‘
Corollary 2.1. For any [ ] and [
], when the marginal utility
functions ( ), ( ), ( ), and ( ) are increasing functions, if both
[ ( ) ( )] [ ( ) ( )] and [ ( ) (
)] [ ( )
( )] are strictly monotonic functions of and respectively, the utility function
( ) can be decomposed by (2.6) if and only if the binary lottery indifference
independence condition holds mutually for both and .
Proof: We only prove the corollary for one attribute, the proof for the other
attribute follows the same idea.
51
Given (2.6), we have (2.2). Without loss of generality, we can assume that
and . If the three coefficients ( ) , ( ) , and (1
( ) ( )) are all positive, they can be interpreted as probabilities and the trinary
lottery certainty equivalent independence holds. Now, we show that when ( ) ( β² )
( ) ( β² )
is a strictly monotonic function of for any [ ] , two of the three
coefficients are positive and one is negative. Thus, (2.6) implies the binary lottery
indifference independence.
In section 2.2, we assessed the following coefficients, where ( ) .
( ) ( ( ) (
))( ( ) ( )) ( ( ) ( ))( ( ) (
))
( ( ) ( ))( ( ) ( )) ( ( ) ( ))( ( ) ( ))
( ) ( ( ) ( ))( (
) ( )) ( ( ) (
))(( ) ( ))
( ( ) ( ))( ( ) ( )) ( ( ) ( ))( ( ) ( ))
It is easy to verify that
1 ( ) ( )
( ( ) ( ))( ( ) (
)) ( ( ) ( ))( ( ) ( ))
( ( ) ( ))( ( ) ( )) ( ( ) ( ))( ( ) ( ))
When ( ) ( β² )
( ) ( β² ) is strictly monotonic increasing in for any , we have
( ) ( )
( ) ( )
( ) ( )
( ) ( ), which implies the denominators of the above three
coefficients are all negative. Now, we check the signs of the numerators. Since
( ) ( )
( ) ( )
( ) ( )
( ) ( ), the numerator of ( ) is negative. Thus, ( ) .
52
If , we have ( ) ( )
( ) ( )
( ) ( )
( ) ( ), the numerator of
( ) is positive. Thus ( ) . Since , we have ( ) ( )
( ) ( )
( ) ( )
( ) ( ). Since ( ) (
) the numerator of 1 ( ) ( )
is negative. So 1 ( ) ( ) .
If , we have ( ) ( )
( ) ( )
( ) ( )
( ) ( ), the numerator of
( ) is negative. Thus ( ) . Since , we have ( ) ( )
( ) ( )
( ) ( )
( ) ( ). Thus, the numerator of 1 ( ) ( ) is positive. So 1
( ) ( ) .
Thus, for , two of the three coefficients are positive and one is
negative. Similarly, we can prove this is also true when ( ) ( β² )
( ) ( β² ) is strictly
monotonic decreasing.β‘
Proposition 2.1. Mutual interpolation independence decomposition implies
decomposition (2.6), but (2.6) does not imply mutual interpolation independence
decomposition.
Proof: We first prove that the mutual interpolation independence decomposition
implies (2.6). This can be proved by showing that the assumption of being
interpolation independent of implies being third degree discrete distribution
53
independent of .
being interpolation independent of is defined by Bell (1979) as
( | ) ( ) ( | ) (1 ( )) ( | ) (2.16)
for some ( ) that only depends on and the conditional utility function is defined by
( | ) [ ( ) ( )] [ ( ) ( )].
Given (2.16), we can show that there always exist and such that (2.15)
holds for any ( ). By multiplying ( ) ( ) on both sides of
(2.16), we conclude that (2.16) is equivalent to
( ) ( ) ( | )( ( ) ( ))
( )[ ( | ) ( | )]( ( ) ( ))
(2.17)
Evaluating ( ), ( ) and ( ) by (2.17) and substituting them as
well as (2.17) into (2.15), we can prove that (2.15) always holds for some and by
following the same idea used in the proof for theorem 2.1. Then, the condition that
being third degree discrete distribution independent of follows as in the proof for
theorem 2.1.
The fact that (2.6) does not imply the mutual interpolation independence
decomposition can be shown as follows. As we shown in the proof of Theorem 2.1, (2.6)
can be written in the form of (2.14) below
54
( ) ( ) ( ) ( ) ( ) (1 ( ) ( )) ( ) (2.14)
Then, it can be shown that ( ) in the interpolation condition ( | ) ( | )
(1 ) ( | ) depends on both and when utility function is of the form (2.14).
First, we solve for from ( | ) ( | ) (1 ) ( | ) to obtain
( ) ( | ) ( | )
( | ) ( | )
Then, rewrite (2.14) as
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) (2.18)
Since ( | ) ( ) ( )
( ) ( ), using (2.18), we can obtain
( ) π ( )[
π (π¦) π’( π¦)
π’( π¦) π’( π¦)
π (π¦ ) π’( π¦ )
π’( π¦ ) π’( π¦ )] π ( )[
π (π¦) π’( π¦)
π’( π¦) π’( π¦)
π (π¦ ) π’( π¦ )
π’( π¦ ) π’( π¦ )]
π ( )[π (π¦ ) π’( π¦ )
π’( π¦ ) π’( π¦ )
π (π¦ ) π’( π¦ )
π’( π¦ ) π’( π¦ )] π ( )[
π (π¦ ) π’( π¦ )
π’( π¦ ) π’( π¦ )
π (π¦ ) π’( π¦ )
π’( π¦ ) π’( π¦ )]
Thus, the ( ) in the interpolation independence condition for utility functions of the
form (2.14) and (2.6) depends on both , which does not satisfy being II . β‘
Theorem 2.2. If the preference of a DM satisfies the (relative) risk independence
condition on at any level of , and if she feels indifferent between two binary lotteries
( ) and ( ) which have the same means for any , then the preference
of this DM satisfies the condition that BLII .
Proof: We prove the case when the risk independence condition on can be
55
verified. The case for the relative risk independence condition can be proved by
following a similar idea.
For a two-attribute utility function ( ), if we can verify the risk independence
condition on for any , the utility function ( ) can be decomposed into the
standard risk-value model on (theorem 3 Jia and Dyer 1996).
( ) ( ) ( )[ ( ) ( )]
When the utility function is continuously differentiable, we know the two-attribute utility
function ( ) is of the form either ( ) ( ) ( ) ( ) or ( )
( ) ( ) ( ) ( ), where ( ), ( ), ( ), and ( ) are functions of
(theorem 4 Jia and Dyer 1996).
For the exponential function, in section 2.5, we show that
( ) ( ) ( ) ( ) [ ( )( ) 1] ( )
If the DM feels indifferent between two lotteries ( ) { ( ) (1 ) ( )}
and ( ) { ( ) (1 ) ( )} which have the same mean for
any , by equating the expected utility of these two lotteries in term of the above risk-
value model, we have ( ) ( ) for any , which further implies that :
( ) 3 (1 ) ( ) ( ) (1 ) ( )
for any . This only happens when ( ) is a constant. To see the reason, we define
πΉ( ) ( ) 3 (1 ) ( ) ( ) (1 ) ( ) . From the equation
56
above, we know πΉ( ) for any . Thus, it must be that πΉ( ) for any ,
which implies the following identity for any
πΉ( )
( )
[
( ) 3 (1 ) ( )
( ) (1
) ( ) ]
Since we know ( ) 3 (1 ) ( ) ( ) (1 ) ( ) for
any , it is impossible that ( ) 3 (1 )
( ) ( ) (1
) ( ) for any , unless which is not true. Thus, it must
be that π ( )
π , which implies that ( ) .
In this case, the exponential function becomes ( ) ( ) ( )
( ). With Lemma 2.1 proved below, we can conclude that both the quadratic function
( ) ( ) ( ) ( ) and the exponential function ( ) ( )
( ) ( ) satisfy the condition BLII .
Lemma 2.1. If both ( ) and ( ) in ( ) ( ) ( ) ( ) ( ) ( ) are
monotonic and there exists a strictly concave transformation πΉ: β between ( )
and ( ), then the preference represented by function ( ) satisfies BLII .
Proof: The condition that BLII is equivalent to the condition that for any
( ) there exist probabilities and for either 2 3 or
3 2 such that
57
( ) (1 ) ( ) ( ) (1 ) ( )
for any holds. To prove this, we can prove that there always exits constants (may not
be probabilities) and such that the above equation holds under the conditions in
Lemma 2.1.
By substituting ( ) ( ) ( ) ( ) ( ) ( ) into the above equation,
we can verify that this equation is equivalent to the condition that there exist constants
and such that the following two equations hold simultaneously for {2 3}
{2 3}\ . This is always true as there exist unique solution to this equation system. But, we
do not know whether the constants and can be interpreted as probabilities.
( ) (1 ) ( ) ( ) (1 ) ( ) (2.19)
( ) (1 ) ( ) ( ) (1 ) ( ) (2.20)
To show they are probabilities, we consider all the possible cases, namely (1) all
the coefficients (1 ) (1 ) are positive; (2) one of them is negative; (3) two
of them are negative on each side; (4) three of them are negative. Among these cases, by
moving the negative terms to the other side of the equation, (1) is equivalent to (3), and
(2) is equivalent to (4).
In the cases (1), if we multiply (2.19) by ( ) and (2.20) by ( ) and sum
them together, we have ( ) (1 ) ( ) ( ) (1 ) ( ). As
the constants in this equation are all positive, they can be interpreted as probabilities,
58
which implies that BLII .
Now, we prove (2) and (4) are impossible under the conditions in the Lemma 2.1.
To show this, without loss of generality, we assume ( ) is more concave than ( ), so
we have πΉ( ( )) ( ) for a concave function πΉ . Denote ( ) for
1 2 3. (2.19) and (2.20) become (1 ) (1 ) and πΉ( )
(1 )πΉ( ) πΉ( ) (1 )πΉ( ). If case (2) holds (if (4) holds, we convert it to
(2)), suppose (1 ) , these two equations become
π
π
( )
; πΉ( )
π
πΉ( )
( π)
πΉ( )
( )
πΉ( )
As π
( π)
( )
1 , the above two equation imply that the certainty
equivalent of a lottery equals to its mathematical expectation under a concave utility
function πΉ( ), which is impossible. β‘
Proposition 2.2. is second degree discrete distribution independent of if and only if
is utility independent of .
Proof: By definition 2.3, the condition that being second degree discrete
distribution independent of is equivalent to the existence of ( ) such that
( ) ( ) ( ) (1 ( )) ( ) for any and any
; and the condition that being utility independent of is equivalent to
( ) ( ) ( ) ( ) f r any and .
59
βββ Given the second degree discrete distribution independence ( )
( ) ( ) (1 ( )) ( ) , we have ( ) ( ) ( ) (1
( )) ( ) . To prove ( ) ( ) ( ) ( ) holds, we can substitute
( ) and ( ) given by the discrete distribution independence condition into the
utility independence condition to get
( ) ( ) (1 ( )) ( )
( )[ ( ) ( ) (1 ( )) (
)] ( )
Which is equivalent to
( )[ ( ) ( )] ( )
( ) ( )[ ( ) (
)] ( ) ( ) ( )
If we define ( ) [ ( ) ( )] [ ( ) (
)] and ( )
( ) ( ) ( ), the above affine transformation relationship always holds.
Thus, given being second degree discrete distribution independent of , there exist
( ) and ( ) such that ( ) ( ) ( ) ( ).
β β β Given ( ) ( ) ( ) ( ) , we have ( )
( ) ( ) ( ) and ( ) ( ) (
) ( ).
Suppose for some ( ) , ( ) ( ) ( ) (1 ( )) ( ) is
equivalent to
60
( ) ( ) ( )
( )[ ( ) ( ) ( )] (1 ( ))[ ( ) (
) ( )]
[ ( ) ( ) (1 ( )) (
)] ( ) ( )
Thus, for ( ) such that ( ) ( ) (1 ( )) (
) ( ), the
relationship ( ) ( ) ( ) (1 ( )) ( ) always holds.β‘
Theorem 2.3. is th degree discrete distribution independent of if and only if the
utility function can be decomposed by (2.11) on a bounded domain [ ] [
].
Proof. This theorem can be proved by following the same reasoning in the proof
for Theorem 2.1. β‘
Proposition 2.3. being th order convex independent of implies being (
2)th degree discrete distribution independent of .
Proof: From being th order convex independent of , we know that
( | ) β ( ) ( | ) (1 β ( )
) ( | ). By choosing arbitrarily many
different values of 1 2 2, we can have a by 2 matrix shown
below.
61
(
( | ) ( | ) ( | )
( | ) ( | ) ( | )
( | ) ( | ) ( | )
( | ) ( | ) ( | ) ( | ) ( | ) ( | )
β± ( | ) ( | ) ( | )
( | ) ( | ) ( | )
( | ) ( | ) ( | ) ( | ) )
So, being th order convex independent of implies that the column vectors
are linearly dependent in the matrix. Thus, the rank of this ( 2) matrix must be
< 2. In the matrix, the column rank = row rank. Thus, the 2 rows in the matrix
must be linearly dependent, which means one row vector can be linearly expressed as a
combination of the other 1 row vectors. This is consistent with the definition of
being ( 2) th degree discrete distribution independent of , i.e., ( | )
β ( ) ( | ) . Thus, we can conclude that being th order convex independent
of implies being ( 2)th degree discrete distribution independent of . This
completes the poof.
But, the converse may not be true. If we know 2 rows are linearly dependent
from ( | ) β ( ) ( | ) , we know the 2 columns must be linearly
dependent, but we cannot prove the linear combination is a convex combination in
( | ) β ( ) ( | ) (1 β ( )
) ( | ). This gives the intuition why
our condition is more general than the convex independence condition. β‘
62
2.8.2. Verification of the BLII condition for utility functions satisfying mutual risk
independence
In this subsection, we assume a specific utility function for a DM and show how
to find a pair of lotteries required by Theorem 2.2 to verify the BLII condition.
Suppose the DM still faces the same retirement pension investment decision
problem described in section 2.3. But this time, we assume that the DM has a utility
function of the form ( ) (1 ) (2 ), which has been rescaled
such that ( ) [ 1] [ 1] and ( ) [ 1]. This function is from Table 2.3 at
the end of this section, and can be decomposed into standard risk value models on both
attributes.
After we verify the risk independence condition on , we pick four arbitrary
levels on the wealth attribute and one level on health . For example, we pick
1 , and 2 . Then, we ask
the DM to elicit three probabilities as we did when verifying the BLII in section 2.3. So,
we replace the values of and in Figure 2.3 with and
assumed here to elicit π π π .
Suppose the DM announces these probabilities according to her utility function
assumed above. We can obtain the following probabilities.
π (2 ) π (2 ) π (2 ) 3
After we elicit these probabilities, we can have an equation for and by using (2.1).
63
π (2 ) π (2 ) π (2 )
From the relationships π
and
π
in section 2.2, this equation can be written
as
π (2 )
π (2 )
1
π (2 )
Now, we can require the means of the two lotteries on in (2.1) to be equal,
and set up another equation for and . Replacing in (2.1) with
, we have
(1 ) (1 )
By substituting the values of 1
and π (2 ) π (2 ) π (2 ) 3 into the
above two equations, we can solve for
and
9
. Thus, we have the following
relationship from (2.1) for the values of and chosen here.
( )
9
( )
9
( )
8
( )
We can rearrange the above equation to obtain
( )
8
( )
9
( )
9
( )
Since
8
9
9
, we can divide both sides by
and have
8
( )
8
( )
9
( )
8
( )
64
Thus, we know the DM is indifferent between the following two lotteries, which
have the same expected value on .
Figure 2.6: Two indifferent lotteries with same expected values
After we find a pair of lottery that are indifferent at 2 with the same
expected value on , we only need to ask one further question to verify that BLII .
We present the above lotteries we found to the DM and ask βif the health level 2
is changed to any other values, will you still feel indifferent between the two lotteries?β
Under the assumed utility function, the DM can verify she is always indifferent between
these two lotteries. So, we can confirm that her utility function satisfies the BLII
condition.
( π 2 ππ΄πΏπ)
( π 2 ππ΄πΏπ)
2
11
11
(οΏ½οΏ½ 2 ππ΄πΏπ) βΌ (οΏ½οΏ½ 2 ππ΄πΏπ)
( π 2 ππ΄πΏπ)
( 1π 2 ππ΄πΏπ)
11
1
11
οΏ½οΏ½ οΏ½οΏ½
65
( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ( ) )( ( ) )
( ) ( )( ) ( ( ) )( ( ) )
( ) ( )( ) ( )( )
( ) ( )( ) ( )( )
( ) ( )( ) ( ( ) )( ( ) )
( ) ( )( ) ( )( ( ) )
( ) ( )( ) ( )( ( ) )
( ) ( )( ) ( )( ( ) )
( ) ( )( ) ( )( ( ) )
( ) ( )( ) ( )( )
( ) ( ( ) )( ) ( )( ( ) )
( ) ( ( ) )( ) ( )( ( ) )
( ) ( )( ) ( )( )
( ) ( )( ) ( )( )
( ) ( ( ) )( ) ( )( ( ) )
( ) ( )( ) ( )( ( ) )
( ) ( )( ) ( )( ( ) )
( ) ( )( ) ( )( ( ) )
( ) ( )( ) ( )( ( ) )
( ) ( )( ) ( )( )
( ) ( ) ( ) ( ) ( )
( ) ( ( ) )( ( ) )
( ) ( ( ) )( ( ) )
( ) ( )( )
( ) ( )( )
( ) ( ( ) )( ( ) )
( ) ( )( ( ) )
( ) ( )( ( ) )
( ) ( )( ( ) )
( ) ( )( ( ) )
( ) ( )( )
( ) ( ( ) ) ( ( ) )
( ) ( ( ) ) ( ( ) )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ( ) ) ( ( ) )
( ) ( ) ( ( ) )
( ) ( ) ( ( ) )
( ) ( ) ( ( ) )
( ) ( ) ( ( ) )
( ) ( ) ( )
( ) ( ( ) ) ( ( ) )
( ) ( ( ) ) ( ( ) )
( ) ( ( ) ) ( )
( ) ( ( ) ) ( )
( ) ( ) ( ( ) )
( ) ( ) ( ( ) )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ( ) ) ( ( ) )
( ) ( ( ) ) ( ( ) )
( ) ( ( ) ) ( )
( ) ( ( ) ) ( )
( ) ( ) ( ( ) )
( ) ( ) ( ( ) )
( ) ( ) ( )
( ) ( ) ( )
Table 2.3: Mutual Risk-Value decomposable utility functions which mutually satisfy
BLII
66
CHAPTER 3. ON THE AXIOMATIZATION OF THE
SATIATION AND HABIT FORMATION UTILITY MODELS
3.1. INTRODUCTION
The discounted utility (DU) model proposed by Samuelson (1937) has been a
dominant model of intertemporal choice for about half a century. One of the main reasons
can be attributed to the seminal work by Koopmans (1960) who showed that the model
could be derived from a set of appealing axioms. However, even Samuelson and
Koopmans had some reservations about the descriptive validity of the model (Frederick
et al. 2002). There have been a number of documented experiments challenging the
validity of the DU model as a descriptive model in the last two decades (Frederick et al.
2002). One of the primary mechanisms to improve the descriptive validity of the DU
model is to relax the independence axiom to allow past consumption to influence the
experienced utility derived from the current consumption. Prior consumption could
influence preference over current and future consumption in two distinct ways: habit
formation and satiation (Read et al. 1999).
During the last few decades, several models have been proposed to capture the
effect of habit formation on the utility derived from consumption (experienced utility) in
each period (e.g. Pollak 1970, Wathieu 1997 2004, Carroll et al. 2000). Lichtendahl et al.
(2011) showed that a correlation averse DM will exhibit habit-like consumption behavior
where the optimal consumption in each period depends on the consumption levels in past
67
periods. Rozen (2010) axiomatized a habit formation model with linear habit functions
over an infinite time horizon.
The satiation effect on intertemporal utility functions has been modeled by
relatively few studies compared to habit formation (Bell 1974, Baucells and Sarin 2007).
Satiation captures the psychological phenomenon that people may feel satisfied by
previous consumption, so additional consumption of the commodity in the current period
provides less incremental utility. In short, habit formation may cause people to become
addicted to a previously consumed commodity, while satiation may lead to boredom due
to previous consumption. Each effect results from different influences of the past
consumption experience on current and future consumption.
Synthesizing ideas from the habit formation model (Wathieu 1997) and the
satiation model (Baucells and Sarin 2007), Baucells and Sarin (2010) proposed a hybrid
model of habit formation and satiation (HS) that combines the influence of both effects of
past consumption on the experienced utility in each period. This model assumes the
overall utility from a consumption stream can be represented by the following function.
( ) β [ ( ) ( )]
In the above utility function, ( ) , and ( ) and
( ) represent the satiation level and habit level in period respectively, both
of which depend on the past consumptions stream . The utility difference
68
term ( ) ( ) is defined as experienced utility of period . We chose to
modify the original notation of Baucells & Sarin (2010) so that it is easier to recall each
symbolβs role in the model: we use for consumption rather than , for satiation
rather than , and for habit formation rather than . In their concluding remarks,
Baucells & Sarin (2010) identified the need for future research efforts to axiomatize the
habit formation (Ha) and satiation (Sa) model.
In this chapter, we propose a hierarchy of axioms to develop a General Habit
Formation and Satiation (GHS) model that can be reduced to either a General Satiation
(GSa) model or a General Habit Formation (GHa) model. These general models allow
more flexible functional forms for the satiation function and the habit formation
function . By assuming further restrictive axioms, we obtain models with a recursively
defined linear satiation function and a linear habit function .
The mathematical form of our habit formation model is similar to the model
axiomatized by Rozen (2010), but the behavioral assumptions underlying the preference
conditions are different. In Rozenβs theory, the main axiom is based on the concept of
compensation for the consumption under the assumption of ordinal preference, while the
main axiom in our paper is based on shifting the measurable value function that is used to
compare the strength of preference over consumption streams under the assumption of
cardinal preference. Our GHS model is axiomatized over a finite horizon which is
consistent with Wathieuβs (1997) assertion that it is essential for the habit formation
model to be finite (see section 3 Wathieu 1997).
69
The reminder of the chapter is organized as follows. In section 3.2, we formally
define the main preference condition used to axiomatize the models in a measurable
preference context, which we refer to as shifted difference independence, and discuss the
implication of this condition. In section 3.3, we axiomatize a General Satiation (GSa)
model, which contains Baucells and Sarinβs satiation model (2007) and Bellβs
intertemporal preference model (1974) as special cases. Section 3.4 focuses on the
axiomatization of the General Habit Formation and Satiation (GHS) model including a
discussion of more restrictive axioms that support the linear satiation and habit functions.
In section 3.5, we discuss how our theory can be applied to axiomatize these models in a
risky preference context. Section 3.6 concludes the paper. All the proofs are provided in
section 3.7.
3.2. SHIFTED DIFFERENCE INDEPENDENCE FOR A MEASURABLE VALUE FUNCTION
We first motivate the main preference condition used to axiomatize habit
formation and satiation with a two day consumption example. For this purpose, we
denote a two period consumption stream by ( ) , where the consumption
space in each period is the real set, i.e. : β and : β. Levels of consumption in
each period are denoted by lower case letters, and the date of consumption is denoted by
the subscript; e.g., denote four different consumption levels at time
{1 2}. We assume a measurable preference order (Fishburn 1970, Krantz et al. 1971,
Dyer and Sarin 1979) over the set of consumption streams . The theory of
measurable preference assumes that a DM can not only compare two alternatives, but
70
also can compare her strength of preference over exchanges of alternatives. This
measurable preference is essentially a binary relation on ( ) ( ). The
expression ( )( ) ( )( ) means that the strength of preference for
the exchange from ( ) to ( ) is greater than or equal to that from ( ) to
( ).
We assume the existence of a measurable value function ( ) to represent
in the sense that ( ) ( ) ( ) ( ) if and only if
( )( ) ( )( ). Given this measurable preference order , we can
define the ordinal preference order on by requiring that ( ) ( ) if
and only if ( )( ) ( )( ) for any ( ) ( ) ( )
(see Dyer and Sarin 1979). Then, this ordinal preference order is also represented by
in that ( ) ( ) if and only if ( ) ( ) . Finally, we also
define the strict and indifference preference relations, , βΌ , and βΌ, in the standard
way in the literature (Fishburn 1970, Krantz et al. 1971). There are different methods in
the literature to elicit the strength of preference and we refer readers to Farquhar and
Keller (1989) for a detailed discussion on this topic.
To motivate our new preference condition, let us first consider the following
example for consumption over two time periods. Assume a new cupcake vendor has
opened near your office. You are asked to evaluate streams of cupcake consumption over
two days. You may feel that your preference for consumption on day 2 may be influenced
by your consumption on day 1. To obtain more insight into your preference for cupcake
71
consumption, you can compare a pair of consumption exchanges: from ( 2) to ( 3)
and from (1 2) to (1 3). In the first exchange, you plan to consume ( 2) but you have
an opportunity to exchange it for ( 3); in the second exchange, you plan to consume
(1 2) but you have an opportunity to exchange it for (1 3). After considering the two
situations, you ask yourself whether the value increase from the first exchange is larger or
smaller than that from the second exchange. The answer to this question may be different
in the following three cases.
First, it is possible that you feel the exchange from ( 2) to ( 3) produces a
larger satisfaction increase than the exchange from (1 2) to (1 3). In both
exchanges, the number of cupcakes on day 2 increases from 2 to 3, but consuming
1 cupcake on day 1 causes you to assign less value to the additional cupcake on
day 2 due to satiation. If we assume decreasing marginal utility for additional
cupcakes, we can create a new exchange from ( 2 π₯ π ) to ( 3 π₯
π ) that
produces the same satisfaction as the increase from (1 2) to (1 3) for some
π₯ π . This quantity π₯
π shifts the measurable value function to the left
(see Figure 3.1 A and B).
Second, if you develop a habit of consuming cupcakes after your first day of
consumption, you might feel that the exchange from ( 2) to ( 3) produces a
smaller satisfaction increase than the exchange from (1 2) to (1 3). In this case,
the higher consumption level on day 1 leads to the formation of a higher habit
level. A higher habit level causes a stronger craving for consumption on day 2,
72
which makes the same increase on day 2 produce a larger value given the higher
consumption level of day 1. Following the analogy from the satiation case, we
may create a new consumption exchange from ( 2 π₯ ) to ( 3 π₯
) such
that the exchange from (1 2) to (1 3) is equivalent to an exchange from
( 2 π₯ ) to ( 3 π₯
) for some π₯ . This quantity π₯
also shifts
the measurable value function to the right (See Figure 3.1 C and D).
Third, it is also possible that both the satiation and the habit formation from the
day 1 consumption affect your preference over day 2 consumption. In this case, a
higher consumption on day 1 will have two opposing influences on your
preference for consumption on day 2, and the net effect is determined by the
relative strength of satiation compared to habit formation. If the effect of satiation
is stronger, more consumption on day 1 can decrease your satisfaction derived
from the same increase of day 2 consumption; otherwise increased consumption
may increase your satisfaction. In other words, you may feel an exchange from
(1 2) to (1 3) is equivalent to an exchange from ( 2 π₯ ) to ( 3 π₯ ) for
either π₯ or π₯ . In a special case, when the effect of satiation equals to
that of the habit formation, or if neither effect is present, π₯ and the
measurable value function is not shifted.
73
Figure 3.1: Shifting value function under satiation and habit formation
After verifying the existence of the shifting quantity π₯ in the above example, if
you can further determine that the π₯ produced by satiation and habit formation only
depends on the change of the consumption levels on day 1, i.e. from 0 to 1, and is
π(1 3) π(1 2) π( 3 π₯
π ) π( 2 π₯
π )
π( 3) π( 2)
π( π )
π
2
3
π( π )
π
2
3
π₯ π
π(1 π ) A B
Satiation
Shifting quantity
π₯ π
π( 3) π( 2)
π( π )
π
2
3
π(1 3) π(1 2)
π( 3 π₯ )
π( 2 π₯ )
π( π )
π
2
3
π(1 π ) C D
Habit Formation
π₯
Shifting quantity
π₯ <0
74
independent of the increase in consumption on day 2, your measurable preference
satisfies a more general condition called shifted difference independence which we
formalize as follows.
Definition 3.1. is said to be shifted difference independent of if for any
, there exists π₯ ( ) β such that for any , ( π₯ ( ))
( π₯ ( )) βΌ ( )( ).
This condition captures all three of the cases discussed in the example above.
When π₯ ( ) is equal to zero, shifted difference independence is reduced to the
difference independence condition (Dyer and Sarin 1979), which implies an additive
multiattribute value function. The shifted difference independence condition also implies
that the value function ( ) has an additive structure, i.e., it is time separable. To
see this, we can write the preference relation assumed in the condition above in terms of
the measurable value function as ( ) ( ) ( π₯ ( ))
( π₯ ( )). This relation implies additive separability of the two attribute
measurable value function ( ) ( ) ( π₯ ( ))
( π₯ ( )) when and in the above relation. This is used as the main
axiom to obtain the additive structure of our satiation and habit formation model.
As we discussed in the cupcake example, under satiation, the effects of past
consumption may spill over into the current and future periods, and the decision maker
75
may experience endowed consumption at a zero consumption level consistent with a left
shift of the value function. This endowed consumption can make the DM feel as if she
has already consumed an amount of the commodity due to her positive past consumption.
Because of this endowed consumption, the DM may become satiated leading to the
experienced utility on day 2 ( π₯ π ) (π₯
π ) smaller than the experienced utility on
day 1 ( ) ( ) for the same consumption, namely ( π₯ π ) (π₯
π )
( ) when . Under habit formation, the DM has a desire to maintain a certain
amount of consumption due to past consumption. In this case, zero consumption is
perceived as a deprivation by the DM consistent with a right shift of the value function.
Deprivation produced by shifting the value function to right has also been discussed by
Hoch and Loewenstein (1991). Endowed consumption and deprivation are illustrated in
Figure 3.2.
Figure 3.2: Endowed consumption and deprivation under satiation and habit formation
π
π£(π π₯ π )
π£(π π₯ )
π£(π₯ π ) π£( )
Endowed consumption
π£( ) π£(π₯ )
Deprivation
Right shifted value function under
habit formation Left shifted value function
under satiation
( )
76
When both satiation and habit formation are present, the shifting quantity π₯
reflects the net effect of both satiation and habit formation resulting from first period
consumption. This quantity assumed in the shifted difference independence condition can
be elicited by asking a DM to compare the strength of preference over the same exchange
on day 2 consumption conditioned on different levels of day 1 consumption. To
determine whether there is a habit formation effect on consumption in the second period,
the DM can be asked to identify a neutral consumption level in this period. The βneutralβ
level of consumption in each period can be elicited by asking the DM to identify a
consumption level which makes her feel neither satisfied nor dissatisfied (Baucells and
Sarin 2010, Baucells et al. 2011). This is the consumption level that makes the
experienced utility in each period equal to zero, i.e., which is obtained by
setting ( ) ( ) . In our two period context, this βneutralβ
consumption in the second period is denoted by ( ). The satiation level in the second
period is denoted by ( ) and π₯ ( ) can be simply denoted by π₯ ( ).
In the first case of the cupcake example, we considered a situation where there is
no habit formation effect, so π₯ ( ) totally reflects the satiation effect, thus ( )
and π₯ ( ) ( ). In the second case, we considered a situation where there is no
satiation effect ( ) so the shifting quantity only reflects the habit formation
effect, i.e. π₯ ( ) ( ); the negative sign on ( ) reflects the opposite effects
of satiation and habit formation on measurable preference. In the third case, there are
both satiation and habit formation effects and π₯ ( ) ( ) ( ) . Thus,
77
increasing satiation ( ) can increase π₯ ( ) and increasing habit formation ( )
can decrease π₯ ( ).
In Table 3.1, we summarize different combinations of habit formation level and
satiation level corresponding to the GSa, GHa, and GHS models axiomatized in this
paper. The habit level ( ) can be either positive or zero corresponding to the case
where habit formation is present or not. For satiation, the function ( ) is also allowed
to be negative. Baucells and Sarin (2010) identified this case as craving caused by
accumulated unmet need.
( )
(No habit formation is present)
( )
(Habit formation is present )
( ) (No satiation is present)
No satiation and habit formation
(DU model)
π₯ ( )
Habit formation only
(GHa model)
π₯ ( ) ( )
( )
(Positive satiation is present)
Satiation only
(GSa model)
π₯ ( ) ( )
Both satiation and habit
formation
(GHS model)
π₯ ( ) ( ) ( ) ( ) if ( ) ( ) ( )
( ) craving
(Negative satiation is present
when there is accumulated unmet
need)
Satiation only
(GSa model)
π₯ ( ) ( )
Both satiation and habit
formation
(GHS model)
π₯ ( ) ( ) ( )
Table 3.1: Effects from satiation and habit formation on the Delta quantity
3.3. A GENERAL SATIATION (GSA) MODEL
To axiomatize a general satiation (GSa) model, we consider the measurable
preference of a DM on a set of multiple period consumption streams on a finite discrete
time horizon {1 2 }. We denote the set of all the possible consumption levels in
period by β, the set of consumption streams that last from period to period
78
as π π , the set of consumption streams from 1 to period as
, and the set of consumption streams from 1 to as
. The vectors
in these consumption stream sets are denoted by π : ( π ) π ,
( ) , and ( ) respectively. We use different lower
case letters to denote different realizations of . So, π : ( π )
π and π : ( π ) π are two different consumption streams over the same
time horizon. The same lower letter with different subscripts, e.g., and π , should be
interpreted as realizations of consumption in different periods, which may or may not be
equal to each other. We use π ( π ) to denote zero consumption streams and
π .
The measurable preference order on the consumption set is assumed to
be represented by a continuously differentiable measurable value function
( ). To state our axioms, we extend the shifted difference independence
condition defined for a two period consumption space in the previous section to
conditional shifted difference independence for a multiple period consumption space,
where the future consumption is conditioned at a specific level when we shift the value
function.
Definition 3.2. is said to be conditional shifted difference independent of given
, if for any there exists shifting quantity π₯ ( )
79
which is independent of the future consumption after period such that β
, ( )( ) βΌ ( π₯ ( ) )(
π₯ ( ) ).
Now, we present our first axiom for the GSa model.
Axiom 3.1. (Satiation) For any {2 } , is conditional shifted difference
independent of with shifting quantity π₯ ( ) defined in Definition 3.2,
given that .
This axiom says that given zero consumption levels in the future, the past
consumption levels produce an effect on the strength of preference at time only
through shifting the value function of the preference at period . Conditioning the
strength of preference comparison on zero future consumption levels allows the
possibility of non-negative consumption streams of different length. For a cupcake
consumption problem similar to the one in section 3.2 with more than two consumption
periods, this axiom assumes that when any period is evaluated as the last period of a
consumption stream, the changes in the previous consumption levels affect the preference
over the last period consumption by shifting its value function according to the
magnitude of the previous changes. For a DM whose preference satisfies the two period
condition described in the cupcake example in section 3.2, it is likely that her preference
may also satisfy the condition assumed in this axiom if she believes that extending a two
80
period choice problem to a three (or more) period horizon would not change the way she
compares the consumption streams.
By shifting the value function to the left, the DM evaluates the consumption in the
last period on a flatter part of the value function. The shifting quantity in this axiom
works as a satiation level which depends on the past consumption: the more you feel
satiated from past consumption, the less you value the same consumption in the last
period of your consumption stream. In the Appendix, we show that by assuming the
existence of satiation for the last consumption period we can recursively prove the
existence of satiation for all previous periods.
Figure 3.3 depicts Axiom 3.1 by assuming that the past consumption stream is
changed from a zero vector to a non-negative vector. In the GSa model, the shifting
quantity π₯ ( ) only reflects the satiation level, so we define a satiation function
as ( ) π₯ ( ).
Figure 3.3: The shifted value function in the Satiation Axiom
π( π‘ ππ‘ π‘ )
πΆπ‘ πΆπ‘
π(ππ‘ ππ‘ π‘ )
πΆπ‘
πΆπ‘
π(ππ‘ π₯π‘ π‘ )
π(ππ‘ π‘ )
πΆπ‘ π₯π‘
π( π‘ π₯π‘ π π‘(ππ‘ ) π‘ )
π( π‘ π π‘(ππ‘ ) π‘ )
π π‘(ππ‘ ) π π‘(ππ‘ ) π₯π‘ π π‘(ππ‘ )
Shifted value function
81
This axiom allows the satiation level of the last period of a consumption stream to
depend on all the previous consumption levels, or to depend only on a subset of the
previous periodsβ consumption levels. Therefore, this axiom allows a situation where the
satiation may decay very quickly with the passage of time, with ( ) ( )
being the extreme case.
The second axiom for the GSa model is presented as follows.
Axiom 3.2. (Time Invariance) β {1 } , for any π π π π π ,
( π π π )( π π π ) ( π π π ) ( π π π ) if and only if
( ) ( ) ( )( ) when π , π
, π , and π for any {1 }. Moreover, β {1 } and ,
if ( ) βΌ ( ) for some , then ( ) βΌ
( ) when and .
The first part of this axiom states that, for a one period consumption stream, if the
strength of preference for π over π is greater or equal to that for π over π in period
when the consumption levels in other periods are zeros, the strength of preference
should be unchanged if the same consumption levels are compared in any other time
period {1 } given that you consume nothing at time periods other than . The
second part of the time invariance axiom simply assumes that if two levels of
consumption at two different periods are indifferent, they should be perceived to be
82
indifferent when the consumption in both periods is advanced by the same amount of
time.
The spirit of this time invariance axiom has been assumed in other time
preference models. For instance, the first part of our axiom is a version of an
independence condition which says that the strength of preference comparison between
two exchanges is independent of the time index. In the time preference literature,
outcome monotonicity is usually assumed to axiomatize preference models (Fishburn and
Rubinstein 1982, Baucells and Heukamp 2012). Outcome monotonicity implies that the
outcome attribute is preferentially independent of the time attribute, but not vice versa.
So, outcome monotonicity is a special case of the more general independence condition.
The second part is similar to the assumption of stationarity of preference (Fishburn and
Rubinstein 1982, Baucells and Heukamp 2012).
Finally, the third axiom is about the impatience of preferences.
Axiom 3.3. (Impatience) For any {1 1} when for some
and , ( ) ( ).
This impatience axiom is also called time monotonicity and allows the
discounting of the value function in each period in the model. Fishburn and Rubinstein
(1982) and Baucells and Heukamp (2012) also assume impatience.
83
Based on the Axioms 3.1, 3.2, and 3.3, we can show that there exists a general
satiation (GSa) model for the value function ( ) over all possible
consumption profiles.
Theorem 3.1. Axioms 3.1, 3.2, and 3.3 hold if and only if the measurable preference
over the consumption streams can be represented by the following model,
with [ 1]
( ) β [ ( ( )) ( ( ))]
where ( ): β is called the satiation function with ( ) and
( ) .
Other models in the literature are special cases of this more general satiation
model (GSa). When the satiation function ( ) is always zero, i.e., ( ) , the
above model is equivalent to the DU model proposed by Koopmans (1960). When
( ) ( ( ) ), the GSa model is reduced to the Satiation Model (Sa)
proposed by Baucells and Sarin (2007). When ( ) ( ) , the GSa is
equivalent to the model proposed by Bell (1974).
In this section, we axiomatized the GSa model by assuming shifted difference
independence on consumption streams. In the next section, we show that the idea used in
this section can be extended to axiomatize a model with both satiation and habit
84
formation. We also show that stronger forms of the shifted difference independence
imply linear satiation and habit formation functions in our framework.
3.4. HABIT FORMATION AND SATIATION MODEL WITH LINEAR HABIT AND SATIATION
FUNCTIONS
3.4.1. A general habit formation and satiation (GHS) model
Baucells and Sarin (2010) proposed a hybrid model of habit formation and
satiation (HS) which inherits the characteristics from both the satiation model and the
habit formation model. In this subsection, we propose axioms that are necessary and
sufficient for a general habit formation and satiation (GHS) model which admits general
functional forms for both satiation and habit formation. In the next subsection, we
provide stronger preference conditions to axiomatize linear satiation and habit formation
functions.
We consider a horizon with periods as the life time of a DM. In this horizon,
past consumption experience can develop into a consumption habit which has an
influence on the experienced utility. However, the habit developed from past
consumption may not last to the end of the life horizon. It is possible that the habit can be
βresetβ (Wathieu 1997) or changed for some reason, either subjective or objective, such
that it only takes effect on finite periods less than . For example, a consumer may
develop a habit of drinking a glass of iced tea during lunch in a hot summer season, but
she may reset this habit and switch to drinking a glass of coffee during lunch when the
85
cold winter comes. Here the habit developed from consuming iced tea during lunch in a
hot summer does not last until the end of , and it is terminated due to the change of the
season. In the context of habit formation, when the habit can only last for a number of
periods less than , this number of periods can influence the preference of a DM.
For simplicity, we consider a DM who consumes iced tea on a four period life
horizon , where each period is one day. Suppose the DM is evaluating a
consumption stream (2 2 2 ), where she consumes two glasses of iced tea on each of the
first three days but zero glasses of iced tea on the fourth day. Her satisfaction derived
from this consumption stream depends on how long the habit lasts. If all four days occur
during the warmer fall season, zero consumption on the fourth day will be perceived as
deprivation since the DM has established a habit for consumption of iced tea. However, if
the habit only lasts to the third day because the fourth day marks the start of winter, the
DM may reset her habit of consuming iced tea and initiate a habit for consumption of
coffee. In this case, the consumption of zero glasses of iced tea on the fourth day will be
perceived as neutral. We distinguish the number of periods a habit can last from the
number of periods in the life horizon by calling the former concept a habit horizon.
We need a richer set of consumption streams to accommodate the impact of the
different habit horizons on preference. Specifically, we use vectors with different lengths
to denote different habit horizons. We assume that the consumption levels are zeros after
the habit horizon is over until . For instance, ( ) denotes the consumption stream
on a two period habit horizon with zero consumption in future periods until the end of the
86
life horizon ; while ( ) denotes a consumption stream with the same
consumption levels in each period but with a three period habit horizon. These two
consumption streams could be perceived differently in the habit formation context.
Formally, on a life horizon with periods, we denote a consumption stream with
a period habit horizon with non-negative consumption levels up to period and zero
levels after period by ( ). Comparatively, a consumption stream with
a 1 period habit horizon with non-negative consumption levels up to period and
zero levels after period is denoted by ( ) ( ). The set of all
consumption streams with habit horizon and 1 are denoted by and
respectively. Now, we define a set of consumption streams with all possible habit
horizons {1 2 } by β . We assume that there exists a measurable
preference order on this set which is represented by a continuously differentiable
measurable value function : β β . Thus, unlike the traditional theory where
preference is assumed on a set of vectors with the same dimension, our theory assumes a
preference order on a set of vectors with dimensions varying from 1 to , which
contains all vectors of the form {( ) ( ) ( ) ( )}. This setup,
where a preference order is assumed on a set of vectors with different lengths, is also
employed by Gilboa and Schmeidler (2001) to axiomatize a habit formation model with
linear utility functions in each period.
87
As we argued above in the iced tea example, in the context of habit formation, a
DM will perceive the iced tea consumption stream (2 2 2) to be different from the
consumption stream (2 2 2 ). Since the DM will perceive deprivation from the zero
consumption in (2 2 2 ), she will prefer the first consumption stream to the second one,
i.e., (2 2 2) (2 2 2 ). Thus, by increasing the consumption level in the fourth period
of the second stream, we can find a quantity defined as the neutral consumption level
such that the DM feels (2 2 2) βΌ (2 2 2 ) . This quantity is the habit level that
makes the DM feel neutral under habit formation for consumption on a four period habit
horizon that is otherwise identical to the corresponding consumption stream on a three
period habit horizon.
We present our first axiom about the neutral consumption level for the GHS
model below.
Axiom 3.4. (Neutral Consumption) For any {1 1} and any , there exists
( ) β such that βΌ ( ( )).
Axiom 3.4 assumes the existence of the neutral consumption level in the habit
formation context, which is also assumed by Gilboa and Schmeidler (2001). In a special
case, when there is no habit formation effect, ( ) for any and , βΌ
( ), which is equivalent to the case of satiation only discussed in the previous
section.
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Axiom 3.5. (Satiation and Habit Formation) For any {2 }, the last period
consumption of is shifted difference independent of the past consumption.
Axiom 3.5 says that the strength of preference for the last period of any habit
horizon is not altered by exchanging the past consumption levels from to if
the value function in the last period of a habit horizon is shifted by π₯ ( ).
Consider a multiple period cupcake consumption example where both satiation and habit
formation may impact preferences. The Satiation and Habit Formation axiom assumes
that no matter how long the habit horizon may last, the cupcakes consumed in previous
periods will produce an effect on the preference over the cupcakes consumed in the last
period of a habit horizon by shifting the value function. Although the changes in the past
consumption levels shift the value function in multiple future consumption periods up
to the end of the habit horizon, the Satiation and Habit Formation axiom only dictates
how the changes in past consumption influence the value function in the last period of a
habit horizon.
In the appendix, we prove that this condition is sufficient to show that all the
previous periods of a habit horizon are also subject to this shifting of the value function
induced by the changes of the past consumption levels. Again, a DM may think about a
simple two-period version of the cupcake example where the habit formed from the first
period consumption has an effect on her preference over the second period through
shifting the value function. If she can verify that the shifting quantity exists for the two
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period case, she may also have reason to believe that her preference should satisfy the
multiple period extension implied by the Satiation and Habit Formation axiom. In this
axiom, the shifting quantity π₯ reflects the net effect on the value function produced by
both satiation and habit formation from the past consumption, so we refer to π₯ as the
adjustment function in the rest of this paper.
Two additional axioms are based on the same motivation as the time invariance
and impatience axioms assumed in the previous section. The only difference is that in this
section we focus on non-zero consumption in the final period of a habit horizon.
Axiom 3.6. (Time Invariance) β {1 } , for any π π π π π ,
( π π )( π π ) ( π π ) ( π π ) if and only if
( )( ) ( ) ( ) when π , π , π , and
π for any {1 } . Moreover, β {1 }and if ( ) βΌ
( ) for some , then ( ) βΌ ( ) when
and .
Axiom 3.7. (Impatience) For any {1 1} when for some
and , ( ) ( ).
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The following theorem states that the above four axioms are necessary and
sufficient to derive a GHS model for the measurable preference represented by
( ).
Theorem 3.2. Axioms 3.4, 3.5, 3.6, and 3.7 hold if and only if the measurable preference
on can be represented by the following model, β {1 } for any ,
( ) β π [ ( π π ( π ) π ( π )) ( π ( π ))]
π
and π ( π ): π β β , π ( π ): π β β , with ( ) , ( ) ,
π ( π ) , π ( π ) and [ 1].
As a general model that accounts for both satiation and habit formation, the GHS
model can be reduced to either the GHa or GSa model in our framework by requiring
more restrictive preference conditions. From the relation π₯ ( ) ( )
( ), when π₯ ( ) ( ) the satiation ( ) , and the GHS model is
reduced to GHa model.
To see how our GHS model can be reduced to a GSa model, we notice that the
main difference in the assumptions for the two models is that the consumption set
used to axiomatize GHS is a larger set that contains the consumption set used to
axiomatize GSa. The measurable preference on this larger set can be reduced to
the measurable preference on in section 3.3 when βΌ ( ) for any . If this
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relationship holds, different habit horizons do not influence preferences. The axioms in
this section are reduced to the corresponding axioms of the GSa model in section 3.3, and
the GHS model is reduced to the GSa model.
Finally, we compare the preference conditions we used in our paper to axiomatize
the GHS model with the preference conditions employed by Gilboa and Schmeidler
(2001) and by Rozen (2010). Our Axiom 3.4 is a condition similar to the neutral
continuation axiom proposed by Gilboa and Schmeidler (2001), which is used to
axiomatize a well-being model with a linear utility function in each period. Under some
conditions, their well-being model can also be interpreted as a consumption model with
habit formation. This neutral continuation condition is defined in the context of
consumption streams with different lengths, which provides a way to axiomatize the habit
level in each period.
In Rozenβs (2010) work on habit formation, each of the evaluated consumption
streams has the same infinite length. Thus, neutral continuation does not work in this
setup. Rozen (2010) introduced a condition called habit compensation for ordinal
preference. Our satiation and habit formation axiom is motivated by a similar idea
applied to the measurable strength of preference. In contrast to Rozen (2010), we also
permit the adjustment function π₯ to be negative.
Shifted difference independence is defined for a measurable preference and
allows the adjustment function π₯ to be either positive or negative. This is because we
also need to axiomatize a utility difference structure ( ) ( ) in order to
92
model satiation in addition to habit formation. This utility difference is naturally
interpreted as a measure of strength of preference. Moreover, as both satiation and habit
formation are two independent effects, we employ both the neutral consumption (Axiom
3.4) and shifted difference independence (Axiom 3.5). Axiom 3.4 produces the habit
effect , and Axiom 3.5 produces the net shifting quantity π₯ .
3.4.2. Linear habit and satiation functions
Linear habit functions have been widely assumed in different habit formation
utility models in the literature (Pollak 1970, Wathieu 1997, Carroll et al. 2000). Baucells
and Sarin (2007, 2010) also assume linear functions to model the habit formation and
satiation in their models. In the context of habit formation on an infinite horizon, Rozen
(2010) proposed a set of axioms that guarantee the existence of a linear functional form
for the habit function. However, we are not aware of any work on this topic in the context
of both habit formation and satiation. In this subsection, we propose a set of stronger
axioms that specify a linear habit formation and satiation model.
When axiomatizing linear satiation function in the GSa model, we assume a
preference order on the set as we did in section 3.3. Axioms 3.8 and 3.9 are
necessary and sufficient for a linear satiation function given by
( ) ( ( ) ) for any .
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Axiom 3.8. (Strong Shifted Difference Independence)
For any {1 }, β , β , and β , there
exists a unique ( ) such that ( )( )
βΌ ( ( ) )( ( ) ).
This axiom is stronger than the conditional shifted difference independence used
in Axiom 3.1, which only compares strength of preference over the last period of non-
zero consumption levels and assumes that the past consumption experience can only shift
the value function in the last period. In Axiom 3.8, the strength of preference is compared
in multiple periods. To illustrate, we return to the cupcake example in section 3.2 with
consumption streams of more than two periods; e.g., the DM may have a four period
consumption stream of cupcakes ( ) . For this four period consumption
problem, Axiom 3.8 states that if two consumption streams have the same level in period
1 but differ from period 2 on, the strength of preference over the two different streams is
unchanged under different period 1 consumption if the value function in period 2 is
shifted appropriately. In other words, it assumes that the past consumption levels before
period affect the future preference only through shifting the value function in period .
It is easy to verify that this is a necessary condition for the recursively defined linear
satiation function ( ) ( ( ) ) . But, to obtain the sufficient
condition for this satiation model, we also need the following assumption.
94
Axiom 3.9. (Independence of Irrelevant Past Consumption)
For any and any 1 , if π π π , then
(( π π π ) ( π π π )) (( π π π ) ( π π π )) for
any π π .
This axiom says that if the two past consumption streams have the same
consumption level in some period , then the common consumption level in this period
does not affect how the value function is shifted in Axiom 3.8. In other words, the DM
only shifts the value function according to the changes in the past consumption levels; the
unchanged past consumption is irrelevant to the shifting of the value function. This is an
axiom similar to the one used by Rozen (2010) to axiomatize a linear habit formation
function. In our context, we show that the combination of Axiom 3.8 and Axiom 3.9 is
necessary and sufficient for the existence of a linear recursively defined satiation function
as in our GSa model.
Theorem 3.3. Under the assumption of the GSa model, Axioms 3.8 and 3.9 hold if and
only if the satiation function ( ) in the GSa model is recursively defined by
( ) ( ( ) ) for some β and any {2 }.
This recursive satiation function reduces to the satiation function proposed by
Baucells and Sarin (2007, see equation (4)) when for any . With a given initial
95
satiation level , the recursive relation implies a linear satiation function given by
the following formula.
For the GHS model, the preference order is assumed on the set as we did in
subsection 3.4.1. We need another version of the conditional shifted difference
independence condition to axiomatize linear habit formation and satiation.
Axiom 3.10. (Multiple Period Shifted Difference Independence) For any
{2 }, any , and any π π π , there exists a unique vector
π ( ): ( ( ) ( ) π ( )) π such that
( π )( π ) βΌ ( π π ( )) ( π π ( )).
Axiom 3.10 is stronger than Axiom 3.5 assumed in subsection 3.4.1; when ,
Axiom 3.10 reduces to Axiom 3.5. Unlike Axiom 3.5 which assumes that the changes in
the past consumption levels only shift the value function in the last period of a habit
horizon, Axiom 3.10 assumes that these changes can shift the value functions in multiple
future periods. In the cupcake example, if there are four periods in a habit horizon,
Axiom 3.10 assumes that the changes in consumption levels on days 1 and 2 can cause
the DM to shift her value functions on both days 3 and 4 such that her strength of
preference over the consumption levels on days 3 and 4 is unchanged. Furthermore, if the
habit horizon is extended, the following axiom assumes that this extension of the habit
96
horizon will not change the π ( ) in Axiom 3.10 on the shorter habit horizon
given that all consumption levels on the shorter habit horizon are unchanged.
Axiom 3.11. (Consistent Shifting) For any and
, π ( ) is equal to ( ) from to { }.
Following the cupcake example, for a five day habit horizon, the changes in the
consumption levels on days 1 and 2 will cause the DM to shift her value functions for
days 3, 4, and 5. Axiom 3.11 says that if identical changes are made on days 1 and 2 for
both four day and five day habit horizons, the on days 3 and 4 should be equal for
both habit horizons.
Finally, we also need Axiom 3.12 assumed below, which is analogous to Axiom
3.9.
Axiom 3.12. (Independence of Irrelevant Past Consumption) For any and
and any 1 , if , then for any
π (( ) ( ))
π (( ) ( )) .
97
Based on Axioms 3.10, 3.11, and 3.12, the habit function and satiation functions
in the GHS model become linear functions of past consumption, and the satiation
function is recursively defined.
Theorem 3.4. Under the assumption of the GHS model, Axioms 3.10, 3.11 and 3.12 hold
if and only if the satiation function and the adjustment function are given by the
following formulas
( ) (π₯ ( ) ) for some β;
π₯ ( ) for some β;
The habit function ( ) is linearly defined by the formula
( ) ( ) ( ) ( )
( ) ( ) .
When for any , the satiation function reduces to ( )
( ( ) ( ) ) as assumed by Baucells and Sarin (2010). For the
habit formation, our linear functional form given in this theorem also contains the
functional form ( ) (1 ) ( ) for any as a special case.
This form for the habit formation function is assumed by Wathieu (1997, 2004) and
Baucells and Sarin (2010). To see this, setting in the habit formation function
given above, we have the following series of habit functions when π₯ and .
98
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
β¦β¦β¦.
Let , we can write ( ) (1 ) . Then, substituting
and into ( ) ( ) ( ) , we have
. When (1 ) or ( ) (1 ), we
have ( ) (1 ) ( ) . We can continue this process to show that
( ) (1 ) ( ), for any , is a special case of our model.
3.5. AXIOMATIZATION THEORY FOR RISKY PREFERENCE
All the models we have developed in the previous sections are based on the
measurable preference order represented by a measurable value function .
However, we can axiomatize the same GSa, GHa, and GHS models for a von Neumann-
Morgenstern utility function representing a risky preference order over the
consumption profiles by following the same ideas.
First, we define shifted additive independence and conditional shifted additive
independence by following the same logic of Definitions 3.1 and 3.2. We apply the same
notation used for the value function development, except that we replace the measurable
99
value function by . Also, we use {( ) ( )} to denote an even chance binary
outcome lottery on the two attribute space .
Definition 3.3. is said to be shifted additive independent of if for any ,
there exists π₯ ( ) β such that , {( ) ( π₯ ( ))} βΌ {(
π₯ ( )) ( )}.
We can compare this shifted additive independence condition with the additive
independence condition (Fishburn 1965) in the Figure 3.4.
Note: the left graph shows additive independence; the right one shows shifted additive
independence
Figure 3.4: Comparison between additive independence and shifted additive
independence
From Figure 3.4 above, we can see that shifted additive independence assumes the
two even chance binary lotteries with outcomes on the opposite angles of a parallelogram
πΆ
πΆ
{(π₯ π¦) (π§ π€)} βΌπ {(π§ π¦) (π₯ π€)}
π§
π₯
π€ π¦
πΆ
πΆ
π₯
π§
π€ π€ Ξ (π₯ π§) π¦ π¦ Ξ (π₯ π§)
{(π₯ π¦) (π§ π€ Ξ (π₯ π§))} βΌπ {(π§ π¦
Ξ (π₯ π§)) (π₯ π€)}
100
are indifferent to each other. When the parallelogram is a rectangle, this condition is
reduced to additive independence (Fishburn 1965).
This condition has the same implication for the utility function as the shifted
difference independence condition has for the measurable value function. Therefore, we
can assume similar axioms for a utility function over consumption steams with more than
two periods and duplicate all the theorems developed in this paper for a utility function
.
3.6. CONCLUSION
In this chapter, we present a framework to axiomatize a general habit formation
and satiation utility model, which contains many existing models of satiation and habit
formation as special cases. The main axiom used in our framework is motivated by the
concept of shifting the measurable value function, which captures how the strength of
preference over current period consumption can be influenced by the past consumption.
Although we also axiomatize the linear satiation and habit formation functions in this
chapter, the GHS model admits more general forms of the satiation and habit formation
functions. Finally, the framework in this paper provides theoretical foundations for a
GHS model in both the measurable preference context and the risky preference context.
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3.7. SUPPLEMENTAL PROOFS
Theorem 3.1. Axioms 3.1, 3.2, and 3.3 hold if and only if the measurable preference
over the consumption streams can be represented by the following model,
with [ 1]
( ) β [ ( ( )) ( ( ))]
where ( ): β β is called the satiation function with ( ) and
( ) .
Proof: It is easy to verify that Axioms 3.1, 3.2, and 3.3 are all necessary
conditions of the GSa model. We only show they are also sufficient here.
To obtain the additive structure in the model, we consider the value increase from
zero consumption level to a positive consumption level in period , given that the
future consumption levels are zeros and past consumption levels are equal to . By
applying Axiom 3.1, we have
( ) ( ) ( π₯ ( ) )
( π₯ ( ) )
Define ( ) π₯ ( ), we have
( ) ( ) ( ( ) ) ( ( ) )
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Denote ( ) by ( ) for any {1 }, where ( ) should
be understood as ( ) and ( ) should be understood as ( ).
The above equation can be written as
( ) ( ) [ ( ( )) ( ( ))] (3.1)
By assuming that the initial satiation is zero, i.e. ( ) , and setting
( ) , we can write ( ) as ( ( )) ( ( )). By noticing that the
first term on right side of (3.1) is of the same form as the left side of (3.1) with a different
time index, we can sum (3.1) for {2 } and simplify to obtain the following
equation.
( ) β [ ( ( )) ( ( ))] (3.2)
Now, we derive the relationship between the value functions in each period. From
the first part of Axiom 3.2, we know for any two periods {1 }, the value
functions and π are strategically equivalent with each other. So, for any we have
( ) ( ) and ( ) ( ) for some β . ( )
( ) for all implies that . Then, from the second part of Axiom
3.2 and the affine transformation relationship shown above, we can conclude that for
some there exists such that ( ) ( ) ( ) and ( )
( ) ( ) for any . Thus, we can conclude that ( ) ( )
( ), which implies . So, we know for any , ( ) ( ).
103
Denote ( ) by ( ) . By applying the relationship ( ) ( ) for
{2 } in (3.2), we obtain
( ) β [ ( ( )) ( ( ))]
Finally, by using Axiom 3.3, we have ( ) ( ) ( ) ( ),
which implies , so ( 1). β‘
Theorem 3.2. Axioms 3.4, 3.5, 3.6, and 3.7 hold if and only if the measurable preference
on can be represented by the following model, β {1 } for any ,
( ) β π [ ( π π ( π ) π ( π )) ( π ( π ))]
π
and π ( π ): π β β , π ( π ): π β β , with ( ) , ( ) ,
π ( π ) , π ( π ) and [ 1].
Proof: It is easy to verify that Axioms 3.4, 3.5, 3.6 and 3.7 are necessary
conditions of the GHS model, so we only verify that they are also sufficient here. The
idea of the proof is similar to that used in the proof of Theorem 3.1, except that we prove
the existence of the habit formation function by using Axiom 3.4.
Consider the value increase from zero consumption level to a positive
consumption level in period , given that after there is no habit formation effect and
104
past consumption levels are equal to . By applying Axiom 3.5, for any
{2 3 } we have
( ) ( ) ( π₯ ( )) ( π₯ ( ))
Define π₯ ( ): π₯ ( ) and ( ) for {2 } , we
have
( ) ( ) ( π₯ ( )) (π₯ ( )) (3.3)
By adding and subtracting ( ( )) in (3.3), we obtain
( ) ( ) ( ( )) (π₯ ( ))
[ ( π₯ ( )) ( ( ))]
(3.4)
Assuming ( ) in (3.3) and defining ( ): ( ) π₯ ( ),
(3.3) becomes
( ( )) ( ) ( ( )) (π₯ ( )) (3.5)
Using Axiom 3.4, ( ( )) βΌ , we have ( ( ))
( ). Replacing ( ( )) by ( ) in (3.5) and substituting it into (3.4),
we have
( ) ( ) [ ( ( ) ( )) ( ( ))] (3.6)
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The first term on right side of (3.6) is of the same form as the left side of (3.6)
with a different time index (shorter time horizon). Following the same reasoning used in
the proof of Theorem 3.1, we can obtain the following additive value function for any
habit horizon , where ( ) and ( ) .
( ) β[ π ( π π ( π ) π ( π )) π ( π ( π ))]
π
(3.7)
From Axiom 3.6, using the same reasoning in the proof of Theorem 3.1, we can
conclude that for any , π ( ) π ( ) and π ( ) π ( ) . Thus, for any
{1 }, (3.7) can be written as
( ) β π [ ( π π ( π ) π ( π )) ( π ( π ))]
π
Finally, from Axiom 3.7, we conclude that ( 1). β‘
Theorem 3.3. Under the assumption of the GSa model, Axioms 3.8 and 3.9 hold if and
only if the satiation function ( ) in the GSa model is recursively defined by
( ) ( ( ) ) for some β and any {2 }.
Proof: To prove Theorem 3.3, we need Lemmas 3.1 and 3.2, which are proved by
adapting ideas from Rozen (2010) to our satiation and habit formation context.
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Lemma 3.1 says that the shifting effect in Axiom 3.8 produced by changing
the past consumption from to is equal to the cumulative effect of first
changing past consumption from to and then from to .
Lemma 3.1. (Triangle Equality) For any , , and ,
we have ( ) ( ) ( ) for any .
Proof: Applying Axiom 3. 8, we have
( )( ) βΌ ( ( ) )(
( ) ) βΌ ( ( )
( ) )( ( )
( ) ) βΌ ( ( ) )(
( ) )
From the last indifference relation and the uniqueness of the shift quantity
assumed in Axiom 3.8, we can conclude that ( ) ( )
( ). β‘
Lemma 3.2 says that the shifting effect in Axiom 3.8 produced by a vector of
past consumption is the summation of the effects produced by the individual consumption
levels in each period.
Lemma 3.2. (Additive Separability) There exists functions : β such
that ( ) ( ) ( ) ( ) ( ).
107
Proof: By iteratively using the triangle equality, we have
( ) ( ( )) (( ) ( ))
(( ) ( ))
(( ) ( ))
(3.8)
Define ( ) ( ( )). Starting from the second term on the
right side of the above equation, we iteratively apply Axiom 3.9 to replace the common
past consumption levels with zero consumption levels. Then, define ( )
(( ) ( )) for 3 . Substituting ( ) into (3.8), we
obtain the additive separable expression for ( ). β‘
Now, we prove Theorem 3.3. The necessary part is easy to verify, so we only
show the sufficient part here.
We first prove ( ) ( ) . For this purpose, we consider a strength
of preference relation where only ( ) and ( ) appear in the GSa model in
period t. Specifically, we consider the relation
( )( ) βΌ ( ( ) )(
( ) )
assumed by Axiom 3.8. Under the assumption of the GSa model, this relation can be
written as
108
( ( )) ( ( )) ( ( ) ( ))
( ( ) ( )) (3.9)
The value difference on the left side of (3.9) reflects a consumption increase from
( ) to ( ) while the right hand side reflects an increase from
( ) ( ) to ( ) ( ) , both of which are
increased by the same amount . Under the assumption of a concave increasing
value function ( ) , (3.9) holds if and only if the following equation (3.10) holds.
Mathematically, this can be verified by taking derivatives with respect to (or ) on
both sides of (3.9) and then applying the monotonicity of ( ).
( ) ( ) ( ) (3.10)
When , (3.10) is reduced to ( ) ( ) . Thus, we
have ( ) ( ) ( ).
Now, to derive the relationship between ( ) and ( ) , we consider
another strength of preference relation assumed by Axiom 3.8,
( )( ) βΌ (
( ) )( ( ) )
As was true in equation (3.9) the value differences in period are equal on both sides of
the relation, and so the representation of this strength of preference relation in the GSa
model can be reduced to
109
[ ( ( )) ( ( ))]
[ ( ( )) ( ( ))]
[ ( ( ( )))
( ( ( )))]
[ ( ( ( )))
( ( ( )))]
By the same logic used above, taking derivatives with respect to on both
sides of the above equation and using the monotonicity of ( ), we obtain the following
equation.
( ) ( ( )) (3.11)
From (3.10), we have ( ) ( ) ( ) . Substituting this
equation into (3.11), we obtain
( ) ( ( ) ( )) (3.12)
If we set , (3.12) becomes ( ) (
( )) since ( ) . Now, define πΉ( ) ( ) , we have
( ) πΉ( ( )).
Finally, from the fact that ( ) ( ) proved above and Lemma 3.2,
we know that ( ) is also additively separable, and ( ) ( )
110
[ ( ) ( ) ( )]. Therefore, ( ) should be independent
of . Then, from ( ) πΉ( ( )), we know that πΉ( ) must be a
linear function. Otherwise, ( ) will depend on . Thus, for some
, ( ) ( ( )).
The above reasoning works for any , so the in the GSa model is a recursively
defined linear function. β‘
Theorem 3.4. Under the assumption of the GHS model, Axioms 3.10, 3.11 and 3.12 hold
if and only if the satiation function and the adjustment function are given by the
following formulas
( ) (π₯ ( ) ) for some β;
π₯ ( ) for some β;
The habit function ( ) is linearly defined by the formula
( ) ( ) ( ) ( )
( ) ( ) .
Proof: To prove Theorem 3.4, we need the following Lemmas 3.3, 3.4, 3.5, and
3.6. Lemmas 3.3 and 3.4 can be proved by the same idea used to prove Lemmas 3.1 and
3.2. We only prove Lemmas 3.5 and 3.6 here, which are proved by adapting the logic
from Rozen (2010) to accommodate both satiation and habit formation.
111
Lemma 3.3. (Triangle Equality)
π ( ) π ( ) π ( ).
Lemma 3.4. (Additive Separability)
( ) ( ) ( ) ( ) .
Lemma 3.5 says that the shifting effect produced by changing consumption in
one period from some level by some non-zero amount is independent of the level of
the starting point of consumption , when the consumption in the other periods are at
zero levels.
To simplify the discussion, we define the notation :
( ) for {1 1} . By this notation, ( )
denotes ( ).
Lemma 3.5. (Weak Invariance)
For any β β {2 1} ( ( )
) (
).
To prove the result, we consider a strength of preference relation where the same
shifting effect in period can be produced either by changing the past consumption
levels from periods 1 to 1 or by changing the past consumption levels from periods
1 to 1. This implies that the shifting effect in period in this relation only
depends on the consumption change before period . By Axiom 3.10, this strength of
preference relation is of the following form, β {2 1},
112
( )( ) βΌ ( π
( π ) (
π )
( π )) (
π ( π )
( π ) (
π ))
for some 1, where we assume the habit horizon is . The length of the habit
horizon does not matter here, as long as we consider a habit horizon with more than
periods.
Now, we treat the first 1 periods as past, which implies that the past
consumption levels are changed from ( ) to (
π
( π ) (
π )) in the strength of preference relation. This
implies
( π ) (
( π (
π ) ( π )))
This equation holds for any , since the left side of the equation is the vector of shifting
effect produced by the changes of the consumption levels in the first 1 periods,
which should be independent of . Therefore, we have:
113
( (
π ( π ) (
π )))
( ( π (
π ) ( π )))
( ( π (
π )))
(3.13)
Equation (3.13) says that the vector of shifting effect for period to only
depends on the marginal change ( π ) for the past consumption change in
period , namely the change from to ( π ), and is independent of the
base consumption level . To obtain the desired result, we apply the triangle equality
and Axiom 3.12 on both sides of (3.13) to replace the nonzero consumption levels in
periods other than , which results in the equality stated in the lemma.
By the triangle equality, the left side of (3.13) can be written as:
( (
π ( π ) (
π )))
( ( (
π ))
)
(( ( π ))
( π (
π ) ( π )))
(3.14)
114
By applying Axiom 3.12 to the second term on the right side of (3.14), we can
replace ( π ) by (
π ) in period . Then, (3.14) becomes
(3.15) as below
( (
π ( π ) (
π )))
( ( (
π ))
)
( ( π )
( π (
π ) ( π )))
(3.15)
By applying the triangle equality again to the right side of (3.13), we have:
( ( π (
π )))
( ( π )
)
( ( π )
( π (
π )))
(3.16)
Substituting (3.15) and (3.16) into (3.13), we can conclude that the right side of
(3.15) equals to the right side of (3.16). Then, by cancelling the like terms on both sides,
we have:
115
( ( π )
) ( ( (
π ))
).
Denote ( π ) and . The equal relationship of the first
elements of the two vectors above leads to the desired result ( ( )
)
(
). β‘
Lemma 3.6 says that the shifting effect ( ) not only additively
depends on as stated by Lemma 4 but also linearly depends on .
Lemma 3.6. (Linearity) For some β {1 1} , ( )
β .
By Lemma 3.4, we have: ( ) ( ) ( ) ( ),
where ( ) is defined to be ( ) in the same way that we define ( ) in
the proof of Lemma 3.2. Then, by the triangle equality, we have β {1
1} β
( ) ( ( ) ) (
) (
( ) )
By applying Lemma 3.5, we have (
( ) ) (
) .
Thus, we conclude β {2 1} ( ) ( ) ( ), which is a Cauchy
equation (AczΓ©l 2006). The solution to this equation is ( ) for some β.
116
Because we only prove the weak invariance for consumption level changes taking
place from period 2 to period 1 in Lemma 3.5, we can only obtain the linearity of
( ) for {2 1}. To obtain a linear function of ( ), we can consider our
model on a horizon starting from period . In this case, we obtain ( )
( ) β . If we take period as exogenous input in our model, we can take
( ) as the initial shifting effect for the value function, which depends on the previous
consumption experience before period 1. This is consistent with the assumption of the
existence of initial satiation and habit formation in the model by Baucells and
Sarin (2010). Therefore, if we absorb the initial shifting effect ( ) into the value
function, we have ( ) β . β‘
Now, we prove the Theorem 3.4. The necessary part is easy to verify, we only
show the sufficient part here.
First, we define ( ) ( ) and prove π₯ ( ) ( )
by following similar ideas used to prove ( ) ( ) in Theorem 3.3. We
consider the following strength of preference indifference relation assumed by Axiom
3.10.
( )( ) βΌ ( ( ))( ( ))
By the same logic used in the proof of Theorem 3.3, representing the above
relation by the GHS model and taking derivatives with respect to , we conclude that
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( ) ( ) ( ) ( ) ( ) . From the relationship
π₯ ( ) ( ) ( ), we have
π₯ ( ) ( ) π₯ ( ) (3.17)
When , we obtain π₯ ( ) ( ) from (3.17). Since
( ) is a linear function by Lemma 3.6, π₯ ( ) is also linear. Therefore, there
exist β such that π₯ ( ) .
Since satiation ( ) and habit formation ( ) are two independent
effects in our framework, given certain past consumption levels, the variation of one
effect does not influence the other effect. Therefore, for fixed , when there is no
satiation effect, π₯ ( ) ( ) implies that ( ) is a linear function. Since
π₯ ( ) is always linear, when there exists a satiation effect which implies ( ) is
nonzero, both ( ) and ( ) must be linear functions as well.
Now, to prove ( ) is a recursively defined function of π₯ ( ),
we consider another type of strength of preference relation assumed by Axiom 3.10.
( )( ) βΌ ( ( )
( ))( ( ) ( ))
To express the above relations in a compact form of the GHS model, we use the
abbreviated notations shown in Table 3.2.
118
Full Abbreviated Full Abbreviated
( ( )) ( )
( ( )) ( )
( ( ))
( )
( ( ))
( )
Table 3.2: Abbreviated notations
Since Axiom 3.11 implies that ( ) in the first strength of preference
indifference relation is equal to the ( ) in the second strength of preference
indifference relation, the first relation can be used to cancel the equal utility in period
in the second indifference relation. Thus, we can write the second strength of preference
relation as follows.
[ (
) ( )] [ (
) (
)]
[ ( ( )
) ( )]
[ ( ( )
) (
)]
(3.18)
Again, taking the derivative with respect to and respectively on both
sides of (3.18), we can conclude that
( )
and
( )
. These results reduce (3.18) to the
following equation.
(
) ( ) (
) (
) (3.19)
119
Taking the derivative of both sides of (3.19) with respect to , we obtain
( )
( )
Since ( ( )) and
( ) are
both linear functions that have the same functional forms and differ only in the values of
the arguments, we conclude that
. This implies ( )
(
) , so we have
from the monotonicity of ( ) ,
which is
( ) ( ( )) (3.20)
By the reasoning similar to that used in the proof for Theorem 3.3, (3.17) and
(3.20) imply that there exists a function :β β β such that ( ) (
π₯ ( )). By the linearity of ( ), we conclude that there exists such that
( ) ( π₯ ( )) . Finally, with the linear π₯ ( ) and ( )
proved above, we can derive the expression for the linear ( ) given in Theorem 3.4
from the relationship ( ) ( ) π₯ ( ). β‘
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CHAPTER 4. HOPE, DREAD, DISAPPOINTMENT, AND
ELATION FROM ANTICIPATION IN DECISION MAKING
Hope itself is a species of happiness, and, perhaps, the chief happiness which this world
affords; but, like all other pleasures immoderately enjoyed, the excesses of hope must be
expiated by pain.
β Samuel Johnson
4.1. INTRODUCTION
When the mega millions jackpot prize reached its highest level of $656 million on
March 30 2012, the public experienced lottery fever. The topic
β#IfIWonTheMegaMillionsβ was trending on Twitter during that week as people
anticipated how their life would change in the event that they won the jackpot2. Even
though the chance of winning the prize was minuscule β 1 in 175,711,536 as listed on the
website of www.megamillions.com βmany people were still willing to pay a few dollars
to play it. With a few dollars, they bought hope, which allowed them to dream about what
they would do with hundreds of millions of dollars. Dreaming about winning in the days
between buying a ticket and learning the outcome of the lottery drawing may have
brought more pleasure to the players than using a few dollars to buy a snack or a cup of
coffee.
Lottery buyers in the Mega Millions lottery experience more utility by
anticipating a higher expected payoff from the lottery, because anticipating a favorable
2 See Yahoo news: http://news.yahoo.com/blogs/sideshow/mega-millions-hits-record-640-million-jackpot-
160916556.html
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result is in itself a pleasurable process. This type of behavior is consistent with the theory
of utility from anticipation, which is based on the assumption that people not only derive
utility when experiencing an outcome but also from anticipating the outcome (Akerlof
and Dickens 1982, Loewenstein 1987, Elster and Lowenstein 1992).
However, anticipating a higher expected payoff may also result in more
disappointment when a player does not win the lottery. Adopting the old saying, βBlessed
is he who expects nothing, for he shall never be disappointedβ is consistent with lowering
anticipated expected payoff. The notion that a DM can subjectively change her
anticipation level for an uncertain payoff has been studied extensively in psychology and
behavioral science (Taylor and Shepperd 1998, Van Dijk et al. 2003, Carroll et al 2006).
In all of these studies, scholars confirmed that people tend to lower their expectations or
predictions for a self-relevant event as the event draws near. Van Dijk et al (2003)
hypothesize that people lower their expectations to protect themselves from suffering a
major disappointment when the uncertainty of a proximate self-relevant event is resolved.
Thus there are two competing cognitive strategies that a decision maker (DM)
might employ to increase her experienced utility: savoring a higher anticipated payoff
before the uncertainty of the payoff is resolved or anticipating a less desirable payoff to
avoid disappointment when the lottery is resolved. These two competing strategies have
been verified in experimental studies by Loewenstein and Linville (1986).
In this paper, we propose a decision making model to capture the tradeoff between
these two conflicting strategies that influence the DMβs total experienced utility from an
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uncertain outcome paid in the future. Besides the behavioral findings reviewed above, our
research is also closely related to the concept of disappointment (Bell 1985, Loomes and
Sugden 1986, Jia et al. 2001). In these disappointment models, a DM anticipates that she
will experience either elation or disappointment when the lottery is resolved and paid,
depending on whether the realized outcome is superior or inferior to her reference point.
The reference point against which the outcome is compared to form elation and
disappointment is assumed to be either the mathematical expectation of the lottery (Bell
1985, Jia et al. 2001) or the expected utility of the lottery (Loomes and Sugden 1986).
However, these models do not apply to a decision maker who subjectively chooses to
lower her expectation to avoid disappointment, as the expectations in these
disappointment models are based on objective probabilities.
Our model is a special case of a model proposed by Gollier and Muermann
(2010), hereafter the GM model, where a DM forms her expectation of the anticipated
outcome based on her subjective probabilities. Before the uncertainty of the outcome is
resolved, she can savor the anticipation; after the uncertainty is resolved, she experiences
either elation or disappointment by comparing the realized payoff with a reference point
determined by her subjective expectation. The GM model assumes that the DM chooses
an optimal subjective belief to balance the tradeoff between savoring higher expectation
and avoiding higher disappointment. This assumption in the GM model is related to the
line of research on optimal beliefs in expected utility introduced by Brunnermeier and
Parker (2005) and Brunnermeier et al. (2007). We allow for any possible anticipated
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payoff level in the decision making process rather than assume the DM is capable of
determining the optimal anticipation to maximize her utility. Savoring anticipation and
avoiding disappointment may be only two of many considerations that influence how
people form their beliefs about the future.
Our model asserts a different process of forming an anticipated outcome to savor
than GM, which results in different implications. For example, we show that in a
portfolio choice problem our model is consistent with the empirical finding that optimism
will lead to more investment in the risky asset relative to the risk free asset (Manju and
Robinson 2007, Balasuriya 2010, Nosic and Weber 2010). In contrast, the GM model
conflicts with these empirical findings. This conflict is addressed by an extension of the
GM model proposed by Jouini et al (2013). However, neither GM nor Jouini et al (2013)
propose preference conditions for their models. In contrast, we also develop an axiomatic
basis for our model with preference assumptions that can be evaluated for their
reasonableness.
We refer to the anticipated expected payoff based on a DMβs subjective
probabilities as the anticipation level. By changing her subjective probabilities over
outcomes, the DM could change her anticipation level for a lottery. This anticipation level
influences two types of utility derived from a lottery: utility of anticipation and
anticipated experienced utility. Utility of anticipation is the pleasure or pain that the DM
βconsumesβ before the lottery is resolved, where anticipation can be interpreted as a
psychological state (Caplin and Leahy, 2001). Anticipated experienced utility is
124
determined based on the DMβs prediction of how disappointed or elated she will feel
when the lottery is resolved. By incorporating these two types of utility into a unified
framework, our model captures four different emotions we may observe in a risky
decision context: hope, dread, elation, and disappointment.
While elation and disappointment have been modeled in disappointment theory
(Bell 1985, Loomes and Sugden 1986, Jia et al 2001, DelquiΓ© and Cillo 2006), there are
few studies that embed hope and dread in a decision model. One exception is Chew and
Ho (1994) who did model hope as the preference for the late resolution of the uncertainty
in a recursive utility framework. Caplin and Leahy (2001) proposed a very general model
that incorporates the utility derived from anticipatory feelings β such as anxiety, hope,
and suspense β in the decision making process. However, they did not allow the
anticipatory feelings to influence the decision makerβs reference point, thus emotions of
disappointment and elation are not captured by their model. We model hope as the
anticipation of a gain and dread as the anticipation of a loss consistent with Lowenstein
(1987).
The reminder of the chapter is organized as follows. In section 4.2, we introduce a
general model and show that this model contains the Risk-Value model (Jia and Dyer
1996) as a special case. We then make additional assumptions about the components of
this general model and obtain a model similar to GM, which also contains Bellβs (1985a)
disappointment model as a special case. In section 4.3, we propose preference conditions
to axiomatize the models discussed in section 4.2 while section 4.4 explains how the
125
DMβs optimistic or pessimistic attitude toward the future may influence the risk attitude
of the DM in a manner consistent with empirical findings. Section 4.5 utilizes our model
to explain the coexistence of gambling and purchasing insurance, which provides an
intuitive interpretation to this widely recognized puzzle in decision theory. In section 4.6,
we apply our model to portfolio choice and the selection of the optimal advertising level
to demonstrate the variety of factors that might affect preference that our model can
accommodate. Section 4.7 concludes the paper. All the proofs are provided in section 4.8.
4.2. THE MODEL
In this paper, we use to denote a lottery of payoffs and to denote an
anticipation level. The bounded sets of payoffs and anticipation are denoted by β
and β respectively. In general, the anticipation level depends on the lottery, which
can be denoted by . Thus, is a function of consistent with Caplin and Leahy
(2001). However, when it is clear which lottery is associated with the anticipation level
we will drop the subscript and simply use .
We consider two periods in our model. In the first period, the DM chooses the
anticipation level of the lottery over monetary outcomes under consideration. She
derives utility from before the lottery is resolved by savoring it. In the second period,
the lottery is resolved and she experiences either elation or disappointment induced by
comparing the received outcome of the lottery with a reference point determined by .
Thus, the DMβs evaluation of a lottery in the first period is based on a two attribute
126
vector ( ). The total ex ante utility derived from this lottery with an associated
anticipation is evaluated by the DM according to the following model with ( )
( ) ( ) ( ) ( ( )) (4.1)
The total ex ante utility ( ) in this model is decomposed into two parts, the
utility of anticipation ( ) and the anticipated experienced utility ( ( )) .
Since the utility function is unique up to an affine transformation, we can rescale it such
that ( ) , ( ) , and ( ) . This rescaling leads to zero total ex ante
utility when she both anticipates and receives a zero outcome. The function ( ) is a
trade-off factor between the two components of the total ex ante utility. For a lottery , if
the DM anticipates , this positive anticipation creates hope for the DM; if the DM
anticipates , this negative anticipation creates dread. Since this anticipation is
the outcome the DM anticipates before the lottery is resolved, the reference point used
by the DM to form elation and disappointment should be influenced by this anticipation
level. Specifically, we assume the reference point depends on the anticipation level
through a function ( ). For any realized outcome , the DM experiences ( ) and
will be elated when ( ) and disappointed when ( ).
We do not address the psychological mechanisms that may form the anticipation.
Instead, we allow the DM to form the anticipation in many possible ways. If the DM
forms her anticipation level by using her subjective probability over the possible future
outcomes, then the anticipation level can be interpreted as the certainty equivalent of
127
the lottery in a way consistent with the interpretation of anticipation in the GM
model. We also assume that this anticipation is bounded by the minimum possible
outcome and the maximum possible outcome of a lottery, namely min( ) and max( )
respectively, which is consistent with the argument by Jouini et al. (2013). If the DMβs
anticipation level for is the mathematical expectation of the lottery , ; and
she also chooses the anticipation as the reference point when determining the elation and
disappointment, i.e., ( ) , our model is reduced to ( ) ( )
( ) ( ). If we also assume ( ) ( ), our model (4.1) is reduced to a
Risk-Value model (Jia and Dyer 1996). In this sense, our model (4.1) can be considered a
General Risk-Value model where the risk is measured by the anticipated experienced
utility from elation and disappointment and value is measured by the utility of
anticipation.
Although model (4.1) can be obtained by assuming some weak preference
conditions as we show in the next section, it is not an easy model to study and it is more
general than other models considered in the literature. A more parsimonious model that
captures the tradeoff between anticipation and disappointment can be obtained by
assuming a constant tradeoff factor ( ) 1 and a linear reference point function
( ) for some constant [ 1].
( ) ( ) ( ) (4.2)
In Figure 4.1, we show that both of model (1) and (2) are special case of the GM
128
model. The GM model and its extension proposed by Jouini et al. (2013) assumes that the
DM always adopts her optimal belief: the anticipation level that maximizes the total ex
ante utility derived from the lottery . In contrast, by adopting a descriptive perspective,
we do not assume that the DM is capable of optimizing her anticipation when facing a
lottery In this way, the anticipation level in our model reflects the DMβs optimistic or
pessimistic attitude toward the future as we discuss in subsequent sections.
For model (4.2), if both and are linear and the DMβs anticipation equals the
mathematical expectation of the lottery , this model reduces to the
disappointment model proposed by Bell (1985a). In another special case, if the DMβs
preferences are not affected by anticipation, elation, or disappointment, we have
and ( ) becomes a constant. In this case, model (4.2) reduces to the expected utility
model. In Figure 4.1, we illustrate the relationships between our models (4.1) and (4.2)
and other preference models in the literature.
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Figure 4.1: The relationship of models (4.1) and (4.2) with some existing models
4.3. THE PREFERENCE ASSUMPTIONS
In this section, we discuss the preference conditions that imply models (4.1) and
(4.2) in section 4.2. We assume that there is a risky preference over the two attribute
space , which is represented by a von Neumann and Morgenstern utility function
( ). Since the anticipation level can be interpreted as a psychological state which
reflects DMβs beliefs, this setup is consistent with the premise that people not only have
preferences over payoffs but also over their beliefs about payoffs as proposed by Akerlof
and Dickens (1982) and with the assumption that the DM could have a preference order
π(οΏ½οΏ½ π) π£(π) π½(π)πΈπ’ (οΏ½οΏ½ πΎ(π))
General Risk Value Model (4.1)
π(οΏ½οΏ½ π) π£(π) πΈπ’(οΏ½οΏ½ πΎπ)
Anticipation Disappointment Tradeoff Model (4.2)
π(οΏ½οΏ½ πΈοΏ½οΏ½) π’(πΈοΏ½οΏ½) π½(πΈοΏ½οΏ½)πΈπ’(οΏ½οΏ½ πΈοΏ½οΏ½)
Risk Value Model (Jia and Dyer 1996)
π(οΏ½οΏ½ π) πΈπ’(οΏ½οΏ½)
Expected Utility Model
π(οΏ½οΏ½ οΏ½οΏ½) πΈοΏ½οΏ½ πΈπ’(οΏ½οΏ½ πΈοΏ½οΏ½)
Disappointment Model (Bell 1985)
Gollier and Muermannβs model (2010)
π(οΏ½οΏ½) maxπ π£(π) πΈπ’(οΏ½οΏ½ π)
When π πΈοΏ½οΏ½, π£ π’,
and πΎ(π) π
When πΎ(π) πΎπ,
π½(π) 1
When πΈπ’(π π) π½(π)πΈπ’(π πΎ(π))
and no max operation is applied to
determine a
When πΎ ,
π£ (π)
When π πΈοΏ½οΏ½ πΎ 1 π£(π₯) π₯ and
π’(π₯) ππ₯ π₯ ππ₯ π₯
130
over the psychological states as proposed by Caplin and Leahy (2001). The set of simple
lotteries defined over is denoted by and different lotteries on the payoff space are
denoted by , , and so on. Given these definitions, the preference condition leading to
model (4.1) can be stated as follows.
Assumption 4.1. (Shifted Utility Independence) For any and any ,
( ) ( ) implies that there exists quantity π₯( ) β such that (
π₯( ) ) ( π₯( ) ).
This assumption states that for lotteries resolved and paid in the second period, a
DMβs preference order over these lotteries is the same under different levels of
anticipation if the lotteriesβ payoffs are adjusted by a constant amount that depends on the
two distinct anticipation levels. For instance, consider a gambler choosing between
betting on a pair of horse races where she anticipates winning $100 for each bet. She may
have the same risky preference over the two races if instead she anticipates winning $150
if all the possible payoffs are increased by an amount that depends on both $150 and
$100. In a simple special case, for example, this increase could be $50=$150-$100 if
preferences are linear in dollars. When the outcomes are dollars, the higher anticipation
may be completely compensated by the increased payoff levels in the lotteries, and any
possible disappointment and elation from each original lottery is kept the same in the
transformed lottery.
131
In general, there may also exist situations where the required adjustment quantity
for the lotteries does not match the exact difference between the two levels of
anticipation. However since the adjustment is affected by the two different anticipation
levels, we expect it to be a function of both and , i.e., π₯( ) in Assumption 4.1.
Figure 4.2 provides a graphical depiction of Assumption 4.1.
Figure 4.2: Assumption 4.1: Shifted Utility Independence
When π₯( ) for any , Assumption 4.1 is equivalent to the
assumption that is utility independent of (Keeney and Raiffa, 1976). Utility
independence implies that, for example, the utility function over when anticipation is
is an affine transformation of the utility of when anticipation is , e.g. ( )
( ) ( ) ( )(Keeney and Raiffa 1976). Similarly, Assumption 4.1 implies that
( ) ( ) ( ) ( Ξ( ) ) , since the preference order over ( ) is
strategically equivalent to the preference order over ( Ξ( ) ). Assumption 4.1
leads to the additive representation of model (4.1) when and ( ) is defined to be
(οΏ½οΏ½ π₯(π π) π)
(οΏ½οΏ½ π)
(π₯ π)
(π₯ π)
π
1 π
(οΏ½οΏ½ π)
(π¦ π)
(π¦ π)
π
1 π
(οΏ½οΏ½ π₯(π π) π)
(π¦ π₯(π π) π)
(π¦ π₯(π π) π)
π
1 π
(π₯ π₯(π π) π)
(π₯ π₯(π π) π)
π
1 π
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( ) [π₯( ) π₯( )] . Therefore, we conclude that a utility function ( )
representing risky preference over can be decomposed into model (4.1) under
Assumption 4.1.
Theorem 4.1. Assumption 4.1 holds if and only if the utility function ( ) can be
decomposed into (4.1) with ( ) , ( ) , and ( ) .
As discussed in section 4.2, model (4.2) can be obtained as a special case of
model (4.1) by assuming ( ) 1 and ( ) for some [ 1]. To state the
preference assumptions for model (4.2), we denote {( ) ( )} as a binary lottery that
results in either ( ) or ( ) with even chances.
Assumption 4.2. (Shifted Additive Independence) For any and ,
there exists π₯( ) such that {( ) ( )} βΌ {( π₯( ) ) (
π₯( ) )}.
This assumption describes a situation that may happen if a DM is uncertain about
her anticipation level. Caplin and Leahy (2001) adopted a similar assumption in their
anticipatory feeling model. In our paper, we can consider a DM who has an even chance
to obtain lottery or on day 2 and the lottery she receives will be resolved and paid
two weeks later. In this case, the DM will form her anticipation level for each lottery and
begin to savor it when she learns which lottery she will receive on day 2. But, on day 1,
133
the DM is uncertain about her anticipation level. If the DM also forecasts that her
anticipation levels will be and for and respectively, the lottery she evaluates
on day 1 is {( ) ( )} . If we assume there exist and such that
( ) βΌ ( ) and ( ) βΌ ( ) , the lottery faced by the DM can be written as
{( ) ( )}, which is the lottery discussed in Assumption 4.2.
Specifically, Assumption 4.2 assumes that the DM faces two such options
{( ) ( )} and {( ) ( )} . Since and , ( ) is a lower payoff
associated with a higher level of anticipation and ( ) is a higher payoff associated
with a lower level of anticipation. Thus, the option produces either a large disappointment
or a large elation. The second option {( ) ( )} yields either a lower payoff
associated with lower anticipation or higher payoff associated with higher anticipation,
which produces neither high disappointment nor high elation. Put another way, the level
of anticipation and the outcome received are negatively correlated for option 1 and
positively correlated for option 2.
If the DM is correlation seeking in the sense defined by Eeckhoudt et al. (2007) in
the payoff-anticipation space, she may feel like playing it safe leading to the preference
relation {( ) ( )} βΎ {( ) ( )} . This situation happens when the utility
function has the property of ( ) , the condition for disappointment
aversion (Gollier and Muermann, 2010). As a result, if the attractiveness of the second
option can be reduced by some amount, it is possible that the DM is indifferent between
the two lotteries. This can be achieved by spreading out the outcomes of the preferred
134
lottery on the payoff attribute while holding the mean constant (mean preserving spread).
More formally, there may exist π₯( ) such that {( ) ( )} βΌ
{( π₯( ) ) ( π₯( ) )} as illustrated in Figure 4.3. This preference condition
was proposed by He et al. (2013) to axiomatize a habit formation and satiation utility
function for intertemporal choice.
Figure 4.3: Assumption 4.2: Shifted Additive Independence
Using Assumption 4.2, we conclude that the trade-off factor in model (4.1) is
equal to 1. To obtain a linear reference point function ( ) so that model (4.1)
reduces to model (4.2), we also need the following technical assumption.
Assumption 4.3. (Linear Shifting Quantity) For any , there is a unique
π₯( ) ( ) [ ] satisfying the condition in Assumption 4.2.
Under Assumptions 4.2 and 4.3, we can conclude that the utility function ( )
can be decomposed into model (4.2) as formally stated in Theorem 4.2.
π π¦ π₯
π
π
Ξ(π π)
Ξ(π π)
π¦ Ξ(π π)
π΄
π₯ Ξ(π π)
135
Theorem 4.2. Assumptions 4.2 and 4.3 hold if and only if the utility function ( )
can be decomposed into (4.2) with [ 1] ( ) ( ) .
4.4. RISK ATTITUDE
4.4.1. Optimism, pessimism, and risk attitude
When facing an uncertain outcome, a DMβs attitude toward the future outcome
may be classified into two categories, optimistic and pessimistic. When a DMβs
anticipation level increases, we say that she become more optimistic which also means
that she become less pessimistic, and vice versa. Intuitively an optimistic DM believes
better outcomes are more likely to occur and therefore will take more risks than a DM
who is pessimistic. This positive relationship between optimism and risk seeking
behavior has been modeled and tested in the literature (Misina 2005, Anderson and
Galinsky 2006, Dillenberger and Rozen 2011). However, there may be situations where
pessimistic people are more risk seeking; for instance, desperate people may take more
risky actions (Lybbert and Barrett, 2011). In another study, Mansour et al. (2008) found
that pessimism is positively correlated with the risk tolerance, implying that more
pessimistic people are more risk seeking. In this paper, we call these two types of
interaction between anticipation and risk attitude increased risk seeking behavior due to
optimism (pessism), respectively, and show that our model (4.2) can be used to describe
both.
Following convention, we define the risk attitude by comparing the expected
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utility of a lottery with the utility of its expectation. From this section on we denote the
anticipation level as to emphasize that the anticipation level is associated with a
particular lottery, . If ( ) ( ) ( πΈ ), we say the DM is risk seeking
(neutral, averse). For a certain outcome, we assume that a DM will anticipate the
outcome itself as it is the only feasible anticipation level, so that πΈ . The
certainty equivalent for a lottery when the DM anticipates receiving should clearly
depend on the anticipation level, which is denoted by ππ( | ) and solved from
(ππ( | ) ππ( | )) ( ) . The risk premium for lottery under
anticipation is defined as ( ) ππ( | ) , which also depends on the
anticipation level .
We say that a DM becomes more optimistic if the DMβs anticipation level
increases; and we say a DM becomes more pessimistic if the DMβs anticipation level
decreases. In the economics literature (BΓ©nabou and Tirole 2002, Epstein and Kopylov,
2007), optimism (pessimism) is defined by assigning higher subjective probabilities over
better (worse) outcomes. If the DMβs anticipation is interpreted as the certainty
equivalent for the lottery based on her subjective probabilities as in GM, the optimism
and pessimism defined here is consistent with the concepts commonly used in the
literature.
In Proposition 4.1, we use , , and to denote the derivatives of , ,
and , respectively. The proposition states that, in our model, more optimism about a
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lottery (a higher level of anticipation) could either increase or decrease the risk premium
of a lottery. So, our model is consistent with increased risk seeking behavior due to
optimism (pessimism).
Proposition 4.1. Under the assumptions of model (4.2), for a given lottery when the
DM anticipates , the risk premium ( ) for depends on in the following
ways:
i. If ( ) ( ) , then ( ) The DM exhibits
increased risk seeking behavior due to optimism .
ii. If ( ) ( ) , then ( ) The DM exhibits
increased risk seeking behavior due to pessimism.
When risk premium ( ) is positive the DM is risk averse and case i describes
a situation where more optimism leads to less risk aversion. Since being less risk averse
implies that the DM is getting closer to risk seeking behavior, we refer to this increased
risk seeking behavior due to optimism in our paper. When ( ) is negative the DM is
risk seeking and case i describes a situation where more optimism leads to more risk
seeking as ππ( | ) increases. The results of case ii can be interpreted in the similar
way.
This proposition states that whether a DM exhibits increased risk seeking
behavior due to optimism or pessimism is determined by the comparison between the
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marginal utility of anticipation and the marginal anticipated experienced utility. When
( ) ( ), the utility from higher anticipation derived by the DM is
larger than the utility lost from the increase in disappointment, and thus the DMβs
increased risk seeking behavior is due to optimism.
Case i in Proposition 4.1 may be common for the Mega-millions lottery players
discussed in the introduction. The lottery ticket buyers derive more utility from a higher
anticipation than they lose from disutility due to the potential disappointment. This is
consistent with the observation that many people purchase a lottery ticket as a way to
acquire hope. Proposition 4.1 predicts that for DMs that gamble and buy lottery tickets,
high levels of optimism are associated with more risk seeking behavior. This also
explains why lottery companies spend money on advertising that depict people winning
the lottery to increase the anticipation level of the public such that they might become
more risk seeking and buy more tickets.
Similarly, in case ii of Proposition 1 a DM worries more about the possible
disappointment. If she is more pessimistic, her anticipation will be lower. Therefore, she
will be less worried about the possible utility loss from a larger disappointment
associated with higher anticipation. This is consistent with the empirical finding that a
negative emotional state may cause people to become more risk seeking (Zhao 2006,
Chuang and Lin 2007), because they value the chance of elation from receiving a better
than anticipated lottery outcome that would improve their negative emotional state.
Case ii cannot be explained by the EU model because shifting probability mass from the
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bad outcome to the good outcome will increase the expected utility of a lottery. Thus,
more optimistic always implies less risk averse behavior in the EU model.
The evaluations of a lottery in these two cases with different attitudes towards the
future can be illustrated in Figure 4.4.
Figure 4.4: Optimistic vs. pessimistic anticipation levels
For simplicity, we assume the utility model for a DM is of the form ( )
( ) ( ) where 1 in model (4.2). If this DM is optimistic about the future
(left side of Figure 4.4), she evaluates the lottery above by (1 ) (1 1 )
( 1 ) (1 ) ( ) ( 1 ). If she is pessimistic about the future,
she evaluates the lottery by ( ) (1 ) ( ) . Thus, risk seeking due to
optimism (pessimism) occurs when (1 ) ( ) ( 1 ) ( ) ( )
(1 ) ( ), which is equivalent to (1 ) ( ) ( ) [ (1 ) ( 1 )].
This relationship will hold if the DM is more (less) sensitive to anticipation than to the
anticipated elation and disappointment. Whether a DM exhibits risk seeking behavior due
to optimism or pessimism can be explained by the tradeoff between the two sources of
πΏ
1 1
2
1
2
πΏ
1
1
2
1
2
Optimistic Pessimistic
Anticipation
Anticipatio
n
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utility in our model.
4.4.2. Wealth effect on risk attitude
By allowing the DM to choose the level of anticipation, our model can also
capture how the DMβs anticipation mediates the wealth effect on her risk attitude. For any
anticipation level chosen by the DM for lottery at wealth level , there is a unique
certainty equivalent ππ that solves the equation ( ππ ππ) (
). For a given , this certainty equivalent is a function of . In this subsection, we
use ππ( ) to denote the derivative of ππ to emphasize this point. However, we
also use the notation ππ to indicate this function for simplicity when no derivative of the
function is taken. Under model (4.2), the equation that defines the certainty equivalent
above can be written as
( ππ) ( ππ ( ππ)) ( ) ( ( ))
(4.3)
We can investigate how the certainty equivalent is affected by the wealth level
at different levels of anticipation by taking the derivative with respect to on both sides
of (4.3), and solving for ππ( ) .
ππ( ) π£β²( οΏ½οΏ½ ) π£β²( ππ) ( πΎ)[πΈ β²( πΎ( οΏ½οΏ½)) β²( ππ πΎ( ππ))]
π£β²( ππ) ( πΎ) β²( ππ πΎ( ππ))
Under the standard assumptions that and , the sign of ππ( )
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is determined by the numerator. Thus, we have the following proposition about the
sign of ππ( ) .
Proposition 4.2. When 1, ( ) , we have: ππ( ) ( ) if and only
if ( )ππ; when [ 1), we have:
i. If , ππ and (1 )ππ
implies ππ( )
ii. If , ππ and (1 )ππ
implies ππ( ) .
iii. If , ππ and (1 )ππ
implies πππ( )
π
iv. If , ππ and (1 )ππ
implies ππ( )
Moreover, if we replace with and ( )ππ with (
)ππ, the sign of ππ( ) is unchanged.
When 1, the DM uses the anticipation as the reference point to predict the
level of disappointment and the sign of ππ( ) is determined by the sign of
ππ. If we assume , a relatively optimistic DM who anticipates ππ
becomes more risk averse with an increase in wealth, ππ( ) ; and a relatively
pessimistic DM who anticipates ππ becomes less risk averse with an increase in
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wealth, ππ( ) .
When 1, the sign of ππ( ) not only depends on the sign of ππ
but also depends on the sign of [(1 )ππ ] , as summarized by
Proposition 4.2. These four cases demonstrate that our model has the descriptive power to
capture many different ways that optimism and pessimism can mediate the wealth effect
on risk aversion. For instance, in case ii of Proposition 4.2, ππ and
(1 )ππ implies ππ (risk seeking). So, in this case, a DM who is
relatively pessimistic ( ππ) and risk seeking ( ππ) will become more risk
seeking at a higher level of wealth ππ( ) . Among these four cases, cases i
and iii are of special interest, as they describe two seemingly conflicting empirical
observations that people with lower levels of wealth can be either more risk averse or
more risk seeking (Caballero 2010, Vieider et al. 2012).
For case i, it is straightforward to show that ππ implies both
ππ ππ ππ and ππ . By combining these two
inequalities, we obtain (1 )ππ. Recognizing that case i can be obtained
from ππ , we can conclude that a DM with an anticipation level lower than
the certainty equivalent of the lottery ππ and exhibiting risk averse behavior
ππ will become more risk averse when her wealth level decreases, i.e., ππ( )
. This is consistent with the observation that people with lower levels of wealth
are often more risk averse than people with higher levels of wealth and are therefore less
143
likely to participate in high return investment activities which are usually associated with
high risks, resulting in the βpoverty trapβ (Mosley and Verschoor, 2005, Yesuf and
Bluffstone 2009).
Similarly, if the condition ππ is satisfied then using the result of case
iii, we can conclude that a DM with anticipation level higher than the certainty equivalent
ππ and exhibiting risk seeking behavior ππ will become more risk seeking
when her wealth level is decreased, i.e., ππ( ) . The result of this case
matches the observation that DMs with lower levels of wealth may be more involved in
gambling than DMs with higher wealth levels (Lesieur 1992). When gambling, people
may anticipate favorable results; in our terms, gamblers are optimistic about the payoff of
the lottery, i.e., ππ. Thus, the certainty equivalent of the lottery for a high wealth
gambler is smaller than that for a low wealth gambler, which results in relatively less
gambling for high wealth DMs. Bosch-Domenech and Benach (2005) found that people
with lower levels of wealth are more risk seeking than people with higher levels of
wealth when facing lotteries with large absolute payoffs. This empirical finding may also
be explained by case iii, since a lottery with large payoffs is more likely to induce a high
anticipation leading to more risk seeking behavior for people with lower levels of wealth.
4.5. UTILITY OF GAMBLING
4.5.1. Coexistence of gambling and purchasing of insurance
A widely recognized puzzle that cannot be explained by standard utility theory is
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the coexistence of gambling and insurance purchasing, implying that people are
simultaneously risk seeking and risk averse (Friedman and Savage 1948). This puzzling
problem can be traced back to the work of von Neumann and Morgenstern who believed
that gambling behavior is inconsistent with expected utility theory (von Neumann and
Morgenstern 1944, p. 28 and Bleichrodt and Schmidt 2002). Only a few studies have
axiomatized the utility of gambling (e.g. Diecidue, et al. 2004) and typically, the utility of
gambling is modeled by appending an extra utility term to the standard expected utility
model or applying different utility functions to non-degenerate and degenerate lotteries
(Fishburn 1980, Conlisk 1993 Schmidt 1998, Diecidue et al. 2004). A common weakness
of these studies is that they do not provide a psychological explanation for why people
would use different utility functions to evaluate risky lotteries and certain outcomes.
In this subsection, we show that our model not only explains the coexistence of
these two seemingly conflicting behaviors, but also provides intuitive psychological
motivations for the choices: in different choice contexts, a DM might form anticipation in
different ways. When purchasing a lottery ticket, a DM may focus on imagining a future
based on winning the prize of the lottery after hearing stories of the lucky players who
have won large prizes. This leads to anticipating a good outcome from the lottery even if
she knows that the chance of winning is very small. However, when facing a possible
loss, e.g., the destruction of her house by a tornado, a DM is influenced by the horrible
images of a tornado from the media, which leads to focusing on imagining the large loss
she may suffer if a tornado hits her house. In this case, a DM may anticipate an extremely
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bad situation when the loss occurs. This explanation is consistent with the research on
affect in decision making which shows that positive affect usually induces optimistic
beliefs and less perceived risk, and negative affect usually induces pessimistic beliefs and
more perceived risk (Johnson and Tversky 1983, Slovic et al. 2005, VΓ€stfjΓ€lln et al.
2008).
To simplify the illustration, we consider a special case of model (4.2) with ,
which implies a DM who does not anticipate elation or disappointment, but a similar
analysis can be obtained for . First, we consider a DM with a wealth level of
who is facing a loss of π β with probability . She faces a lottery that yields π
with probability and with probability 1 . We denote the premium for a full
coverage insurance policy by . According to our model, the utility from purchasing the
insurance is given by ( ) ( ) and the utility from not insuring is given by
( ) ( π) (1 ) ( ). Then, the condition of purchasing the insurance is
( ) ( π) (1 ) ( ) ( ) ( )
which is equivalent to
( ) ( ( π) (1 ) ( )) ( ) ( ) (4.4)
Under the assumption , the left hand side of (4.4) is positive as long as the
insurance premium is not greater than the price of fair insurance π. Therefore, if the
anticipation level for the lottery is not too high, (4.4) always holds and the DM
prefers to buy insurance. Even when the anticipation reaches its highest level and she
146
anticipates losing nothing, i.e., , the DM may still choose to buy insurance when
the utility difference on the right hand side of (4.4) is smaller than the utility difference
on the left side. But, if the DM always anticipates , which seems plausible,
the right hand side of (4.4) is always negative. Thus, when the DM will
always buy insurance under the standard assumptions that , , and .
Now, we consider the utility from gambling for a DM. Suppose that a lottery with
a large payoff β with small probability and a zero payoff with probability
1 is available for purchase at its expected payoff . According to our model, a DM
that anticipates the nonzero payoff will buy the lottery when the following condition
holds
( ) ( ) (1 ) ( ) ( ) ( )
which is equivalent to
( ) ( ) ( ) ( ) (4.5)
The left hand side of (4.5) is the utility difference from anticipation and the right
hand side is the utility difference from anticipated experienced utility. The DM may buy
the lottery because the utility difference ( ) ( ) may not be very large.
However, when is large and is small, the difference ( ) ( ) could be very
large. In other words, anticipating a large prize from a lottery may produce much more
marginal utility than anticipating the certain payoff of the expectation of the lottery;
( ) ( ) β« ( ) ( ) ( ) [ ( ) (1 ) ( )].
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4.5.2. Stochastic dominance and transitivity
A widely recognized problem in modeling the utility of gambling is that stochastic
dominance is usually violated by the proposed models (Fishburn 1980, Schmidt 1998,
and Diecidue et al. 2004). Stochastic dominance is a desirable rationality requirement
that should not be violated by a preference model from both normative and prescriptive
perspectives. To remedy this problem, Bleichrodt and Schmidt (2002) propose a context
dependent model that does not violate stochastic dominance, but does violate another
desirable property: transitivity of preference (Luce 2000, MacCrimmon 1968). Stochastic
dominance or transitivity is violated by these models in part because they apply different
utility functions to represent the unique preference order on a set including both risky and
riskless alternatives (see Bleichrodt and Schmidt 2002, Table 1). In our model, under the
appropriate assumptions, the violation of both stochastic dominance and transitivity can
be avoided.
By definition (Bleichrodt and Schmidt 2002, Diecidue et al. 2004), a preference
order satisfies stochastic dominance if for any degenerate or non-degenerate lottery
, any two certain outcomes , and any ( 1] , if , then
(1 ) (1 ) . Under model (4.2), the preference relation is
represented by ( ) ( ) ( ) ( ). Since both functions ( ) and
( ) are monotonically increasing, we know . Compounding lottery with
will have a total ex ante utility greater than or equal to that from compounding lottery
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with in many cases, but it will depend on how a DM forms her anticipation for the
compound lotteries. In Assumption 4.4, we describe a situation where the larger the
payoff compounded with a lottery, the higher the anticipation formed by the DM. This
assumption implies that when an outcome of a lottery is improved the anticipation level
should not decrease, which seems to be reasonable.
Assumption 4.4. (Consistent Compounding) A DM is said to be consistently
compounding in anticipation if for any and , her anticipation levels for the
compound lotteries (1 ) and (1 ) satisfy the condition
( ) ( ) .
Proposition 4.3 states that under Assumption 4.4, the preference in our model
satisfies stochastic dominance when is small enough, i.e., the DM is not very sensitive
to the potential disappointment.
Proposition 4.3. Under Assumption 4.3, there exists π β, such that when [ π],
( (1 ) ( ) ) ( (1 ) ( ) ) for any and any
.
A smaller indicates that the gambler is not sensitive to disappointment, which
may be true in practice. A gambler may be driven by the hope created by a large
anticipated outcome. If the effect of disappointment is also strong (large ), the utility of
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anticipation could be reduced by the disappointment, which would further reduce the
motivation for gambling. Since we observe many people repeating gambling activities,
we may infer that is small for these DMs.
Finally, by introducing the anticipation level in the choice set, our model can
avoid the problem of intransitivity encountered by Bleichrodt and Schmidt (2002). This is
apparent since the total ex ante utility ( ) is a representation of a transitive
preference order defined on the two attribute space .
4.6. DECISION MAKING MODELS
4.6.1. Portfolio selection decision
As previously discussed, a major difference between our model and the GM
model is that we do not assume that the DM optimizes her anticipation level to maximize
the total ex ante utility in decision making. Instead, we allow the anticipation level to be a
parameter that can be influenced by both exogenous and endogenous factors. This leads
to different implications for optimal decision making in the context of the portfolio
choice problem.
The portfolio choice problem involves the following choices. A decision maker
has initial wealth denoted by β. She selects β to invest in the risky asset which
has a random gross return . Her remaining wealth is invested in a risk free asset
which has a gross return . The objective is to select an optimal to maximize her
utility from holding both risky and risk free assets.
150
In the GM model, for any given level of the allocation to the risky asset, the DM
selects her anticipation level so that her total ex ante utility is maximized. Thus, this
optimal anticipation level is a function of the allocation to the risky asset. Then, under the
optimal anticipation, the DM optimizes the allocation to the risky asset to maximize her
total ex ante utility. By solving a problem set up in this circular way, GM obtained the
result that optimism is negatively related to allocation to the risky asset, which seems to
be counterintuitive. They acknowledged that their result is somewhat surprising and that
it conflicts with the results predicted by optimal expectations models (Brumnermeier and
Parker 2005, Gollier 2005). Empirical studies have also confirmed that more optimistic
investors tend to hold more risky assets (Manju and Robinson 2007, Balasuriya 2010,
Nosic and Weber 2010). This shortcoming of the GM model is also addressed by Jouini
et al. (2013) in an extension of the GM model.
In the extended GM model (Jouini et al. 2013), the feasible domain of the
anticipation level is modified to show that the GM model can be consistent with the result
of the empirical studies on the relationship between optimism and investment in risky
asset. In this paper, we provide an alternative explanation and propose that the surprising
result in the GM model can be induced by the optimal anticipation assumption. It may be
true that in some cases a DM will intentionally adjust her anticipation level for the lottery
she chooses when she tries to increase her total ex ante utility, but this may not be a
general rule that applies to all situations. The belief of a DM, which we model as the
anticipation level, may be influenced by the context of the decision. For instance, in a
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bear market, no matter how optimistic an investor used to be, she may not be able to form
an optimistic anticipation level for money invested in stocks. Moreover, as we discussed
above, the circular set up of GM assumes that a DM can forecast how the allocation
decision will influence the total ex ante utility through the optimal anticipation, which
seems challenging in practice. In our development we relax this demanding requirement
that the DM will be able to optimize her anticipation level intentionally when facing such
a portfolio selection problem.
We allow the anticipation level to be influenced by contextual factors and set up
the portfolio decision model as follows. Influenced by the economic environment, the
DM forms anticipation for the random risky return, which is bounded by the
minimum and maximum possible outcomes of , i.e, [min max ]. Then, her
anticipated total wealth is given by ( ) ( ) . The utility of
anticipation is ( ( ) ) and the anticipated experienced utility is
(( ) (( ) ))
( (1 ) [( ) ( )] )
Utilizing these components in (4.2), the DM optimizes the allocation of her wealth to
risky asset by solving the following problem
max
( ) ( ( ) ) ( (1 ) [( ) ( )] )
(4.6)
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If we assume that both and are concave functions, ( ) is also a concave
function of as the sum of concave functions is still concave. Then, it is easy to obtain
the following result.
Proposition 4.4. The optimal investment in the risky asset in (4.6) is ( ) if
and only if π£β²( π)( οΏ½οΏ½ π)
β²( π( πΎ)) ( )[ ( ) ( )] .
We expect a DM who invests in the financial market to anticipate that the risky
asset return exceeds the risk free return ; otherwise the investor would not
choose to invest in the risky asset. In this case, we can rewrite the optimal investment
condition in Proposition 4.4 as if and only if ( ) ( (1 ))
( ) ( ). Since the risk premium of the risky asset, , is positive,
when the anticipated return is low , ( ) ( ) is always negative
because [ 1] . In this case ( ) ( (1 )) ( ) ( )
always holds as and are both positive. Recall that the anticipation level can be
interpreted as a certainty equivalent of the lottery from anticipation based on subjective
probabilities. Thus, under the assumption of concave , when the subjective
probabilities become closer to the objective probabilities. So, the above results implies
that if the DM holds more rational beliefs that are closer to the objective probabilities, she
will always invest in the risky asset.
If the anticipated return is relatively high, e.g. , ( ) ( )
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is always smaller than one. If the marginal utility ratio ( ) ( (1 )) is
always larger than one, then and the DM always invests in the risky asset. In this
case, the inequality can be explained as the result of optimistic beliefs that
deviate significantly from objective probabilities. Thus, this result implies that when a
DM is very optimistic, she will invest in the risky asset only if the marginal utility she
derives from anticipation is large enough to counter the potential disappointment.
Now, we treat the anticipated return as a choice parameter and analyze how it
influences the optimal investment level which we will denote as ( ).
Proposition 4.5. There exists a β such that π ( οΏ½οΏ½)
π οΏ½οΏ½
if and only if (
( ) ) .
This proposition states that when the marginal utility from anticipation is large
enough at the optimal investment level, i.e., ( ( ) ) , the DM will
invest more given a higher anticipation level. This is a very intuitive result. The DM
will only increase the investment in a risky asset when the marginal utility she derives
from anticipation is large enough to offset the utility loss from the potential
disappointment.
Besides being consistent with the empirical finding that more optimistic DMs will
invest more in risky assets, our model can also be employed to explain the equity
premium puzzle, which can be described as follows: In order to explain the much higher
154
available returns of risky assets (stocks) compared to riskless assets (bonds), investors
must have extremely high levels of risk aversion. GM noticed that the literature on
optimal expectations (Brunnermeier and Parker 2005, Gollier 2005) assumes the DM
always optimizes her beliefs and selects a risker portfolio, reinforcing the equity premium
puzzle, while their model implies that optimism of a DM is negatively related to the
investment in a risky asset, reducing the equity premium puzzle. However, empirical
studies suggest that a more optimistic DM will invest more in the risky asset (Manju and
Robinson 2007, Balasuriya 2010, Nosic and Weber 2010). Thus, although the GM model
is consistent with the equity premium puzzle, it conflicts with both our intuition and the
empirical finding that optimism should induce more risk taking behavior and more
investment in risky asset.
Our model can explain the equity premium puzzle and accommodate behavior
consistent with the notion that optimistic investors invest more in the risky asset. As
shown by Proposition 4.4, our model can be used to represent preferences with either
( ) or ( ) depending on functional form used to
model utility from anticipation. To be consistent with the empirical finding on the
relationship between optimism and risk taking, we should assume preferences exhibit
( ) , implying that a more optimistic DM will invest more in risky asset.
To explain the equity premium puzzle, we propose that if DMs in the financial market are
generally pessimistic, i.e., anticipate a lower level of , the model proposed here
implies a decrease in the demand for the risky asset, which increases the equity premium.
155
Finally, we should emphasize that this descriptive flexibility comes from the relaxation of
the optimal anticipation (belief) assumed by models in the literature (Brunnermeier and
Parker 2005, Gollier 2005, Gollier and Muermann 2010).
4.6.2. Optimal advertising decision
In this section, we explore the optimal advertising level for a marketer facing a
consumer who trades off the utility of anticipation and the utility from anticipated
disappointment consistent with model (4.2). We will model the consumerβs decision to
purchase or not to purchase a single unit of a product. Further, we assume that the
customer will not know the quality of the product until after it is purchased and will
model the predicted quality as a simple lottery defined on a bounded payoff set
β. However, we assume that the consumer has some knowledge about the probability
distribution of this uncertain quality level. Before purchasing the product, the consumer
anticipates the quality of the product [min max ] . Under the assumption of
model (4.2), the total ex ante utility derived from purchasing one unit of this product is
given by ( ) ( ) while the total ex ante utility from not purchasing the
product is .
Following the convention in the economics literature (e.g. Shogren 1994), we
assume the consumer has additive utility over wealth and her consumption of the product,
i.e., ( ) ( ) ( ) . The consumerβs willingness to pay ( ( )) is
determined by solving equation (4.7)
156
( ) ( ) ( ( ) ) ( ) (4.7)
where ( ) is the utility function over her wealth.
For this problem, we can show that the maximum willingness to pay is obtained at
an interior level of the anticipation in the domain [min max ] under some standard
assumptions.
Proposition 4.6. Under some standard assumptions,
,
(min ) ( min ) and (max ) ( max ) , there
exists an interior optimal anticipation (min max ) such that ( ) is
maximized.
This proposition states that if a consumer derives utility from both anticipation
and the anticipated experienced utility, the optimal level to anticipate should be neither
too high nor too low.
Now, we consider a seller who is attempting to sell a new product to a collection
of consumers, each with a concave willingness to pay function ( ) due to the
tradeoff between high anticipation and high disappointment. To model the heterogeneity
of the consumers in the market, we assume the willingness to pay of each customer is
given by π π( ) ( ) π , where π is a mean zero random variable with
cumulative density function πΉ that captures the uniqueness of a consumerβs preferences.
If the seller sets the price of the product at π, the consumer will buy the product if
( ) π π . Therefore, the response function is given by (π ) 1
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πΉ(π ( )), which depends on both the price π and anticipation . Further, the
seller can influence the anticipated quality of the product through her advertising effort,
, measured in dollars. We assume that the consumerβs anticipated quality of the product
is positively related to the advertising effort and is given by the linear relationship
π π, where [ ] with (max π) π, which is the effort level
that will cause the consumer to anticipate the highest possible quality. In this linear
relationship, π is the base anticipation level of the consumer when no advertising effort
is exerted; and π is the anticipation increase produced by one marginal unit of
advertising effort. This assumption of a positive relationship between the anticipated
quality of the product and the advertising effort has been documented by Deighton
(1984). Kirmani and Wright (1989) also verified that the perceived advertising expense
has a positive relationship with consumersβ expectation of product quality in a laboratory
setting.
It has been argued that increasing the expected quality of a product can increase
the demand for the product (Goering 1985) and that advertising is a likely mechanism to
increases the consumersβ quality expectation and therefore product sales (Simon and
Arndt 1980, Bagwell 2005, Erdem et al. 2008). However, as we show in Proposition 4.7,
the response function in our context is maximized at an βappropriateβ level of advertising
effort , because a high anticipation level of product quality produced by the
advertising can also induce high anticipated disappointment, decreasing the consumerβs
willingness to pay. In other words, advertising can raise a consumerβs expectation so high
158
that she would prefer not to purchase the product for fear of being disappointed with its
actual quality.
Proposition 4.7. For fixed price π, the response function is maximized at an advertising
effort level οΏ½οΏ½ π
π , when ( ) , ( ) ( ) .
Finally, we consider a problem where a seller determines the optimal advertising
effort for a given price π to maximize her profit π( ) . Each unit of product is
assumed to have a constant cost of production .
max
π( ) (π ) (1 πΉ(π (π π)))
The first order condition of the above problem is (π )πΉ (π (π
π)) (π π)π 1 3. Since attains its maximum at οΏ½οΏ½ π
π, we know
that the optimal advertising effort to maximize the total profit is π , so that
(π π π) [ (π )πΉ ( (π π π))]
>0. Therefore, we have the
following proposition.
Proposition 4.8. For fixed price π, the profit is maximized at an advertising effort level
that is lower than the effort level maximizing the willingness to pay, π .
This result implies that sellers of a product should not always seek to increase
3 We also assume the second order condition is satisfied:
π π( )
π .
159
consumersβ willingness to pay. When willingness to pay is above (π π π), the
marginal cost of the advertising effect β the unit cost in our model β outweighs the
marginal contribution to the profit produced by the increase in willingness to pay, which
further reduces the total profit. Again, increasing the anticipated quality level of a product
via advertising can reduce sales when customers grow concerned that their high
expectations cannot be satisfied and choose to abstain from a purchase.
4.7. CONCLUSION
In this chapter, we propose preference conditions for a decision making model
which incorporates both the utility of anticipation β hope and dread β and the anticipated
experienced utility β elation and disappointment β in a decision making process. This
model captures optimism and pessimism by allowing the DM to choose to anticipate a
high or low outcome of a lottery. The level of anticipation serves two roles in our model:
it is the source of the utility of anticipation in the period before the lottery is resolved as
well as the reference point used to form elation and disappointment after the lottery is
resolved.
We show that our model can account for how optimism could influence both the
DMβs risk attitude as well as the wealth effect on that risk attitude. This optimism can
explain the coexistence of gambling and purchasing of insurance without violating
stochastic dominance and transitivity. Finally, we discuss the applications of this model in
both finance and marketing contexts. In a simple setting with one risky and one risk-free
asset, we show that our model can capture the widely observed behavior that investor
160
optimism is positively correlated with investment level in the risky asset. It also provides
an explanation for the equity premium puzzle without conflicting with the empirical
finding that optimism leads to more investment in a risky asset. In a marketing context,
we show that using advertising to increase the customerβs anticipation level of product
quality with the intent to increase her willingness to pay does not always increase the
demand for a product. This result conflicts with the intuition that product demand is
increasing with advertising and it should be studied in more detail with controlled
experiments.
4.8. SUPPLEMENTAL PROOFS
Theorem 4.1. Assumptions 4.1 holds if and only if the utility function ( ) can be
decomposed into
( ) ( ) ( ) ( ( )) (4.1)
with ( ) , ( ) , and ( ) .
Proof: Sufficiency: by Assumption 1, we have ( ) ( ) ( ) (
π₯( ) ) ( ) ( ) ( [π₯( ) π₯( )] π₯( ) ) , since ( ) and
( π₯( ) ) are strategically equivalent to each other. Let and define
( ) [π₯( ) π₯( )] , ( ): ( ) , and ( ): ( π₯( ) ) in
( ) ( ) ( ) ( [π₯( ) π₯( )] π₯( ) ) , we have (1). By
definition of ( ) , we have ( ) . Since utility function is unique up to affine
161
transformation, we can rescale the utility function ( ) such that (π₯( ) )
and ( ) . Thus, we have ( ) and ( ) .
Necessity: for any and any , if ( ) ( ) , from model (4.1)
we have,
( ) ( ) ( ) ( ( )) ( ) ( ) ( ( )) ( )
which implies ( ( )) ( ( )) . For any , since β ,
( ) , this inequality is equivalent to
( ) ( ) ( ( ) ( ) ( ))
( ) ( ) ( ( ) ( ) ( ))
Define Ξ( ) ( ) ( ), so we have ( Ξ( ) ) ( Ξ( ) ). β‘
Theorem 4.2. Assumptions 4.2 and 4.3 hold if and only if the utility function ( )
can be decomposed into
( ) ( ) ( ) (4.2)
with [ 1] ( ) ( ) .
Proof: Sufficiency: Assumption 4.2 implies ( ) ( ) (
π₯( ) ) ( π₯( ) ) . Let and rescale ( ) such that
( ) , we have ( ) ( π₯( ) ) (π₯( ) ) . Define ( )
162
Ξ( ) , ( ) (π₯( ) ) , and ( ( )) ( ( ) ) , we have
( ) ( ) ( ( )).
Now, we prove ( ) is linear. According to Assumption 4.2, for any
and , we have {( ) ( )} βΌ {( π₯( ) ) ( π₯( ) )} .
Expressing this condition in term of ( ), we have
( ) ( ) ( π₯( ) ) ( π₯( ) )
Let π₯( ), the above equation is equivalent to
( ) ( ) ( π₯( ) ) ( π₯( ) )
Similarly, we have for
( ) ( ) ( π₯( ) ) ( π₯( ) )
( π₯( ) ) ( π₯( ) )
( π₯( ) π₯( ) ) ( π₯( ) π₯( ) )
Thus, we have
( ) ( ) ( π₯( ) ) ( π₯( ) )
( π₯( ) π₯( ) ) ( π₯( ) π₯( ) )
According to Assumption 4.3, this π₯( ) is unique which is a function depends
on the difference between and , namely π₯( ) ( ) is unique. Thus, from
the uniqueness of this π₯( ), we have
163
( ) ( ) ( )
By setting in above equation, we have ( ) ( ) ( ) . Let
, we have ( ) ( ) ( ), which is a Cauchy functional equation
(AczΓ©l 2006). The solution to this equation is ( ) for β . Because
Assumption 4.3 states that ( ) [ ] for , we have [ 1]. Since
we defined ( ) Ξ( ) ( ) , we have ( ) . Finally, from π₯( ) ,
( ) (π₯( ) ), and ( ( )) ( ( ) ), it is easy to conclude that
( ) and ( ) .
Necessity: Given ( ) ( ) ( ), we have
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ( ) ) ( ) ( ( ) )
( ( ) ) ( ( ) )
Define π₯( ): ( ) , since [ 1] , we know there exits π₯( )
( ) [ ] such that Assumption 4.2 holds. This also proves π₯( )
[ ] in Assumption 4.3.
Finally, to prove the uniqueness of π₯( ) stated in Assumption 4.3, suppose
there exists another π₯ ( ) π₯( ) such that ( ) ( ) (
π₯( ) ) ( π₯( ) ) ( π₯( ) ) ( π₯( ) )
Let π₯( ) , we have
164
( π₯( ) ) ( π₯( ) )
( π₯( ) ) ( π₯( ) )
( π₯( ) π₯( ) ) ( π₯( ) π₯( ) )
( ) ( ) ( ) ( )
Since are arbitrary, are also arbitrary. Denote utility function
( ) by ( ). The last equation above is equivalent to ( ) ( ) ( )
( ) for any . Taking derivative with respect to , we have ( ) (
) . Then, taking derivative with respect to , we have ( ) , which
implies ( ) ( ) is a linear function in . This violates the law of diminishing
marginal utility. Thus, the π₯( ) is unique. β‘
Proposition 4.1. Under the assumptions of model (4.2), for a given lottery when the
DM anticipates , the risk premium ( ) for depends on in the following
ways:
i. If ( ) ( ) , then ( ) The DM exhibits
increased risk seeking behavior due to optimism .
ii. If ( ) ( ) , then ( ) The DM exhibits
increased risk seeking behavior due to pessimism.
Proof: According to the definition ( ) ππ( | ), ( ) ( )
if and only if πππ( | οΏ½οΏ½)
π οΏ½οΏ½
( ). By definition (ππ( | ) ππ( | )) ( )
165
and model (4.2) ( ) ( ) ( ) , we have (ππ( | ))
((1 )ππ( | )) ( ) ( ). Thus, we have:
[ (ππ( | )) (1 ) ((1 )ππ( | ))] ππ( | )
( ) ( )
It is easy to verify that πππ( | οΏ½οΏ½)
π οΏ½οΏ½
( ) if and only if ( ) (
) ( ). β‘
Proposition 4.2. When 1, ( ) , we have: ππ( ) ( ) if and only
if ( )ππ; when [ 1), we have:
i. If , ππ and (1 )ππ
implies ππ( )
ii. If , ππ and (1 )ππ
implies ππ( ) .
iii. If , ππ and (1 )ππ
implies πππ( )
π
iv. If , ππ and (1 )ππ
implies ππ( )
Moreover, if we replace with and ( )ππ with (
)ππ, the sign of ππ( ) is unchanged.
166
Proof: From (4.3) in the text, we know
πππ( )
π
π£β²( οΏ½οΏ½ ) π£β²(ππ ) ( πΎ)[πΈ β²( πΎ( οΏ½οΏ½)) β²( ππ πΎ( ππ))]
π£β²(ππ ) ( πΎ) β²( ππ πΎ( ππ))
When 1, since , the sign of πππ( )
π is determined by the comparison
between and ππ. When [ 1). We only show case i here. The other cases can be
obtained by following the same idea. When , from , by
Jensenβs inequality, we can conclude ( ( )) (
( )) . From , (1 )ππ implies (
(1 ) ) ((1 )ππ (1 ) ). Thus, we have
( ( )) ( ππ ( ππ))
( ( )) ( ππ ( ππ))
Moreover, from and ππ , we have ( ) (ππ ) .
Therefore, we can conclude the numerator of the above equation is positive. Since we
also assume and , we conclude that ππ( ) in this case.β‘
Proposition 4.3. Under Assumption 4.4, there exists π β, such that when [ π],
( (1 ) ( ) ) ( (1 ) ( ) ) for any and any
.
Proof: Since the lottery X is the common part for both compounding lotteries
167
considered here, we simply denote the anticipation by ( ) indicating that the
anticipation depends on , which is the certain payoff compounded with . Stochastic
dominance requires that when :
( (1 ) ( )) ( ( )) ( ( )) (1 ) ( ( ))
( ( )) ( ( )) (1 ) ( ( )) ( (1 ) ( ))
Under the assumption of consistent compounding in anticipation, we have ( )
( ) for any , namely ( ) ( ) , stochastic dominance is
satisfied by our model when ( (1 ) ( )) , which is equivalent to
the condition
( ( )) ( ) ( ( ))(1 ( )) (1 ) ( ( )) ( )
Without loss of generality, we assume that marginal utility is bounded, i.e.,
[ β² β²]. Then, let π solves the following equation
( ( )) ( ) β²(1 π ( )) (1 ) β²π ( )
π ( ( )) ( ) β²
[ β² (1 ) β²] ( )
when ( ) , ( ) , and ( ) .
Then, we have for any [ π],
168
( ( )) ( ) ( ( ))(1 ( ))
( ( )) ( ) β²(1 π ( )) (1 ) β²π ( )
(1 ) ( ( )) ( )
which implies that the stochastic dominance holds. β‘
Proposition 4.4. The optimal investment in the risky asset in (4.6) is ( ) if
and only if π£β²( π)( οΏ½οΏ½ π)
β²( π( πΎ)) ( )[ ( ) ( )] .
Proof: Taking derivative with respect to in (4.6), we have:
( )
( ( ( ) )( )
[( ) ( )] ( (1 ) [( ) ( )] )
Since we assume and are concave, ( ) is also concave. Thus, ππ( )
π |
( ) is equivalent to ( ), which leads to the result. β‘
Proposition 4.5. There exists a β such that π ( οΏ½οΏ½)
π οΏ½οΏ½
if and only if (
( ) ) .
Proof : When no derivative of ( ) is taken, we keep using for simplicity.
By differentiating the first order condition with respect to for (4.7), we can solve for
169
π ( οΏ½οΏ½)
π οΏ½οΏ½
as follows
π ( οΏ½οΏ½)
π οΏ½οΏ½
πΎπΈ[( π) πΎ( οΏ½οΏ½ π)] β²β²(π( )) πΎπΈ β²(π( )) π£β²(π( )) ( οΏ½οΏ½ π)π£β²β²(π( ))
( οΏ½οΏ½ π) π£β²β²(π( )) πΈ[( π) πΎ( οΏ½οΏ½ π)]
β²β²(π( ))
where we define two functions ( ) (1 ) [( ) ( )] and
π( ) ( ) to simplify the expression above. Under the assumption that
and , the denominator of the right hand side is negative. Therefore, a
negative numerator is equivalent to a positive π ( οΏ½οΏ½)
π οΏ½οΏ½
. A negative numerator is
equivalent to the condition: [( ) ( )] ( ( )) ( ( ))
(π( )) ( ) (π( )) . Finally, define [( ) (
)] ( ( )) ( ( )) ( ) (π( )), we can get the result in the
proposition. β‘
Proposition 4.6. Under some standard assumptions,
,
(min ) ( min ) and (max ) ( max ) ,
there exists an interior optimal anticipation (min max ) such that ( ) is
maximized.
Proof: Note that the consumerβs willingness to pay is a function of her
anticipation for this one unit of the product. Differentiating both sides of (4.7) with
respect to aX,
( ) ( ) ( ( ) ) ( ) (4.7)
170
we find ( ) π£β²( οΏ½οΏ½) πΎπΈ β²( πΎ οΏ½οΏ½)
π€β² ( ( οΏ½οΏ½))
. When , ( ) (
) , and ( ) ( ) , we have ( )
and ( ) . If we take the derivative of (4.7) with respect to twice, we
can solve for
( ) ( ) ( ) ( ( ))
( ( ) )
( ( ))
Under the assumption of , and , we can verify that ( )
. Thus, we can conclude the result stated in the proposition. β‘
Proposition 4.7. For fixed price π, the response function is maximized at an advertising
effort level οΏ½οΏ½ π
π , when ( ) , ( ) ( ) .
Proof: By taking the derivative of (π ) 1 πΉ(π (π π)) with
respect to , we have (π ) πΉ (π (π π)) (π π)π. Since πΉ ,
we know from Proposition 4.6 that when π π , , and (π ) ,
sales are increasing with ; when π π , and (π ) , sales are
decreasing in . β‘
171
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VITA
Ying He was born in Xianyang, Shaanxi Province, People Republic of China, on
March 31th, 1982, the son of Yaowu He and Shulian Sun. After graduating from
Xianyang Shi-Yan High School in July 2000, he entered Xian Jiaotong University, where
he spent eight years before came to the USA. He received his bachelor degree in
economics from School of Economics and Finance at Xiβan Jiaotong University and
entered a Ph.D. program in School of Management to study management science in July
2004. In Aug 2008, he entered the Ph.D. program in the department of Information, Risk,
and Operations Management at The University of Texas at Austin to pursue his Ph.D.
degree. In Dec 2010, he received a M.S. degree in Information, Risk, and Operations
Management from The University of Texas at Austin.
Permanent Address: IROM department, 2110 Speedway Stop B6500, Austin, TX 78712
This manuscript was typed by the author