A Model of Gain/Loss Asymmetry

Katsutoshi Wakai�

January 2006

Abstract

Based on a notion associated with utility smoothing, this paper pro-

poses, in an axiomatic framework, a new model of discount factors that

captures an asymmetric attitude toward gains versus losses. Its key

departure from existing models is that preferences satisfy dynamic con-

sistency. This introduces the notion of recursive gain/loss asymmetry

in discount factors: The di¤erence between future and current utility

de�nes a gain or a loss, and gains are discounted more than losses. Un-

der this mechanism, current utility becomes the endogenous reference

point. The bene�t of dynamic consistency, as well as the economic im-

plications of gain/loss asymmetry, is shown through a repeated game

example.

Keywords: discount factors, gain/loss asymmetry, recursive utility,

reference points, time-variability aversion, utility smoothing

JEL Classi�cation Numbers: D90, D91

�Department of Economics, The State University of New York at

Bu¤alo, 427 Fronczak Hall, Bu¤alo, NY 14260; kwakai@bu¤alo.edu;

http://pluto.fss.bu¤alo.edu/classes/eco/kwakai/

1

Acknowledgement

This paper is a revised version of the �rst half of Chapter II of my disserta-

tion at Yale University (Wakai (2002)). I would like to express my appreciation

to my advisor, Stephen Morris, and dissertation committee members, John

Geanakoplos and Benjamin Polak, for their invaluable advice and support. I

would also like to thank Larry Epstein and Itzhak Gilboa for valuable dis-

cussions and suggestions. I have bene�ted from comments by Richard Braun,

Takao Kobayashi, Giuseppe Moscarini, Robert Shiller, Leeat Yariv, and semi-

nar participants at the State University of New York at Bu¤alo, the University

of Iowa, the University of Rochester, The University of Texas Austin, the Uni-

versity of Tokyo, Washington University, and the RUD 2004 conference. I

am also grateful to the Editor and referees, whose comments and suggestions

greatly improved this paper.

2

1 Introduction

Intertemporal choices involve an allocation of resources over time. To analyze

this choice problem, Samuelson (1937) proposes a special form of a weighted

summation decision rule, called the discounted utility model. For example, at

each time t, the conditional utility of a consumption sequence c = (c0; :::; cT )

is expressed in an additively separable form

Vt(c) �TX�=t

���tU(c� ); (1)

where � is called a single-period discount factor and U is called an instanta-

neous utility function. Since it is analytically tractable as well as axiomati-

cally driven (Koopmans (1960)), the discounted utility model has become the

standard framework for analyzing intertemporal choices. One of the de�ning

characteristics of this model (as well as the weighted summation decision rule

in general) is that it discounts each utility sequence (U(ct); :::; U(cT )) by the

same series of discount factors (1; :::; �T�t), regardless of the distribution of

utility over time (called utility independence). However, as Frederic, Loewen-

stein, and O�Donoghue (2002) show in a recent survey article, many studies

have reported results contradicting utility independence. Among them, we

focus on the following experiment conducted by Thaler (1981).

One set of subjects were asked to imagine that they had won a monetary

prize that could be received either now or later. They were asked to state

how much money they would need to be paid if payment could be delayed (by

three months, one year, or three years). Another set of subjects were asked a

comparable set of questions, with the prize replaced by a tra¢ c ticket. Since

the immediate payment was equally preferred to the later payment, the yield

to maturity (i.e., (1=�)� 1) was computed based on (1), where instantaneous

3

utility U(:) was replaced by a monetary payment. The study found that the

yields implied by the former set of questions were much higher than the yields

implied by the latter set (we refer to this speci�c result as yield asymmetry;

we use gain/loss asymmetry to refer to the general tendency that gains are

discounted more than losses). Moreover, the di¤erence in yields increased as

the sizes of gains and losses decreased. This implies that yield asymmetry

should exist even if a monetary payment is converted to the corresponding

utility by a concave U .1

Loewenstein and Prelec (1992) explain yield asymmetry by adapting the

reference-dependent utility as developed by Kahneman and Tversky (1979,

1991): The instantaneous utility function U(ct) is replaced by the value func-

tion �(ct � qt), where qt is called a reference point. This behavioral approach

has three di¢ culties: First, there is no general guideline for how to model a

reference sequence (q0; :::; qT ) from which gains and losses are computed. Sec-

ond, the relationship between gain/loss asymmetry and notions of intertem-

poral preferences such as consumption smoothing is not well de�ned. Third,

preferences do not necessarily satisfy dynamic consistency. To address the

�rst two issues, Shalev (1997) models gain/loss asymmetry based on Gilboa�s

(1989) variation aversion, that is, the aversion to the utility variation between

adjacent periods. However, Shalev (1997) does not address the third issue of

dynamic consistency. This is problematic in application because we must also

model how the con�ict between di¤erent selves is resolved.

Given this limitation, our objective is to provide a new model of discount

factors that not only captures an asymmetric attitude toward gains versus

losses but also satis�es dynamic consistency. Our model is based on a notion

1See Loewenstein and Prelec (1992).

4

of utility smoothing, time-variability aversion, meaning that a decision maker

(DM) is averse to utility variations de�ned over an entire time horizon. This is

second-order aversion in the sense that U already captures the notion of con-

sumption smoothing if U is concave. Formally, at each time t, the conditional

utility of a consumption sequence c is expressed in the recursive form

Vt(c) � min�t+12[�t+1;�t+1]

[(1� �t+1)U(ct) + �t+1Vt+1(c)]; (2)

where �t+1 and �t+1 describe the upper and lower bounds of single-period dis-

count factors, respectively, and satisfy 0 < �t+1 � �t+1 < 1. We provide an

axiomatization for the representation as an extension of the weighted sum-

mation decision rule. This is achieved by adapting a method developed in a

di¤erent context by Gilboa and Schmeidler (1989).

The representation (2) introduces the distinct feature called recursive gain/

loss asymmetry: The di¤erence between future utility Vt+1(c) and current

utility U(ct) de�nes a gain or a loss, and gains are discounted more than

losses. This is clearly seen by rewriting (2) as follows:

Vt(c) = U(ct) + min�t+12[�t+1;�t+1]

�t+1[Vt+1(c)� U(ct)]: (3)

More speci�cally, based on utility smoothing, (2) explains the reason for

gain/loss asymmetry shown in (3): When future utility is lower (higher) than

current utility, the DM has a stronger (weaker) incentive to increase future

utility by decreasing current utility. The DM expresses this incentive by shift-

ing weight onto (away from) the future utility. Hence, an increase from the

current utility is regarded as a gain, and it requires a lower discount factor;

a decrease from the current utility is regarded as a loss, and it requires a

higher discount factor. Under this mechanism, current utility U(ct) becomes

the reference point. Intuitively, from the reference point U(ct), the cost of

5

reducing the future utility Vt+1(c) outweighs the bene�t of increasing it. This

cost-bene�t gap is captured by a wideness in the range of discount factors

[�t+1; �t+1].

Besides being dynamically consistent, (2) addresses two other issues raised

for the behavioral approach: First, (2) provides a clear mechanism for reference-

dependence (i.e., how relative consumption matters) and a clear relationship

between the degree of gain/loss asymmetry and the size of a set of discount fac-

tors. In particular, a reference point is endogenous, meaning that it is speci�c

to each consumption sequence. Second, (2) explains the relationship between

gain/loss asymmetry and notions of intertemporal preferences, namely, time-

variability aversion and dynamic consistency.

Recursive gain/loss asymmetry also plays a crucial role in explaining yield

asymmetry, which is translated to a condition on [�t+1; �t+1]. In particular,

for a stationary and in�nite horizon version of (2), where [�t+1; �t+1] = [�; �]

for all periods, yield asymmetry exists if and only if � 6= �. Hence, recursive

gain/loss asymmetry and yield asymmetry become an equivalent notion.

Dynamic consistency makes our model tractable so that we can study the

economic implications of gain/loss asymmetry. In a simple repeated game

application, we show that as players become more gain/loss asymmetric, it

becomes easier (more di¢ cult) to �nd a subgame perfect Nash equilibrium

(SPNE) that supports constant (variable) payo¤ sequences.

The reminder of the paper proceeds as follows: Section 2 provides exam-

ples to motivate our objective. Section 3 axiomatizes the representation (2).

Section 4 characterizes the relationship of (2) to yield asymmetry and to the

discounted utility model. Section 5 provides a repeated game example. Sec-

tion 6 compares our model with other intertemporal utility functions. Section

7 concludes the paper.

6

2 Examples: Dynamic Consistency and Gain/Loss Asymmetry

Consider a three-period problem. At each t, the DM has a conditional ordering

on a set of consumption sequences c = (c0; c1; c2). Assume that orderings are

independent of the consumption history.

First, we show that if a reference point is exogenously given (i.e., the same

reference point is used to evaluate all c), preferences may not satisfy dynamic

consistency. For example, Loewenstein and Prelec (1992) propose the fol-

lowing descriptive model: Let q0 = (q00; q01; q

02) be an exogenous status quo

consumption plan. The ex-ante preference �0;q0 is represented by

V0;q0(c) =2Xt=0

v(ct � q0t )�(t); (4)

where v(ct � q0t ) is a continuous and increasing value function that evalu-

ates ct with the value v(0) = 0 and �(t) is a discount function (they adapt

the hyperbolic discount function). This model explains yield asymmetry if

v(y)=v(x) > v(�y)=v(�x) for all 0 < y < x, meaning that v(x) is less elastic

at x > 0 than at �x. Hence, (4) is based on �rst-order aversion: yield asym-

metry is de�ned as the functional form of v rather than as a characteristic of

discount factors.

Di¢ culties arise from an interpretation of the status quo (i.e., the reference

point). For simplicity, assume that the discount function is a usual geometric

series �(t) = �t, where 0 < � < 1. Also, let v(z) < �v(�z) for z > 0.2

Suppose that the conditional preference �1;q1 is represented by the same class

2The other possibility is the existence of z > 0 such that v(z) > �v(�z) for 0 < z � z.

A similar counterexample can be constructed.

7

as (4), with an exogenous status quo plan q1 = (q10; q11; q

12), i.e.,

V1;q1(c) =

2Xt=1

v(ct � q1t )�(t� 1):

Consider the following two consumption sequences:

c = (a; a+ d; a) and c0 = (a; a; a+ e),

where d; e > 0. Let fc; c0g be the DM�s choice set. Suppose that at time 0, the

DM is endowed with c so that q0 = c. Since c and c0 share the same time-0

consumption, we can investigate the consistency between the ordering at time

0 and the ordering at time 1. If c '0;q0 c0, the DM will choose either c or c0 at

time 0. For a decision problem at time 1, if c is chosen at time 0, the status

quo is q1 = c, and c '1;q1 c0. However, if c0 is chosen at time 0, the status quo is

changed to q1 = c0. Then a simple computation shows that c0 �1;q1 c. This is a

violation of dynamic consistency in the sense that the DM ex-ante knows that

c0 will be preferred at time 1 if c0 is chosen at time 0, but the DM is ex-ante

indi¤erent between c and c0. The model generates inconsistency because the

exogenous reference plans, q0 and q1, are unrelated: The conditional ordering

�1;q1 depends on the choice at time 0, but in (4), the DM fails to take into

account this relationship.

Second, we show that even if a reference point is endogenously given (i.e.,

sequence speci�c), preferences may not satisfy dynamic consistency. For exam-

ple, Shalev (1997) introduces the following model by extending Gilboa�s (1989)

representation, which is dependent on the utility variation between adjacent

periods: The ex-ante preference �0 is represented by

V0(c) � u0 +2Xt=1

��0;+t maxfut � ut�1; 0g+ �0;�t minfut � ut�1; 0g

; (5)

8

where ut is the instantaneous utility of consumption ct. When 0 < �0;+t < �0;�t ,

(5) explains gain/loss asymmetry; under appropriate values of discount factors,

(5) explains yield asymmetry. The time-t utility ut becomes an endogenous

reference point to evaluate the time-t+1 utility ut+1 but not the future utility

Vt+1(c), i.e., the utility of an entire consumption sequence evaluated at time

t + 1.3 Hence, the asymmetry introduced by (5) is not recursive. Similar to

(2), gain/loss asymmetry as shown in (5) is based on second-order aversion.

Suppose that the conditional preference �1 is represented by the same class

as (5), i.e.,

V1(c) � u1 + �1;+2 maxfu2 � u1; 0g+ �1;�2 minfu2 � u1; 0g:

Preferences are dynamically consistent if c �t+1 c0 implies c �t c0, where

c� = c0� for all � � t. Then a simple computation shows that if preferences

satisfy dynamic consistency,

�0;+1 = �0;�1 .

Hence, Shalev�s (1997) representation is dynamically inconsistent if we assume

gain/loss asymmetry (i.e., 0 < �0;+t < �0;�t ).4

The above inconsistency comes from the feature that for each �t, the dis-

count factor of ut, (1 � �t;:t+1), depends on ut+1 but not on Vt+1(c). More

precisely, utility smoothing makes discount factors dependent on the utility of

future consumption. In this case, endogenizing a reference point alone does not

3For comparison, (2) can be written as Vt(c) = U(ct) + �t+1maxfVt+1(c) � U(ct); 0g +

�t+1minfVt+1(c)� U(ct); 0g.4This result comes from the assumption used in Gilboa (1989) and Shalev (1997); their

models are based on Schmeidler�s (1989) non-additive prior model. For more details, see

Eichberger and Kelsey (1996); Sarin and Wakker (1998); Grant, Kajii, and Polak (2000).

9

necessarily resolve dynamic inconsistency; the recursive form of asymmetry is

additionally required.

3 Representation

We adapt the Anscombe-Aumann (1963) framework with a temporal interpre-

tation: Preferences are de�ned on the set of sequences whose outcome at any

period is a lottery de�ned over a consumption set. This domain is rich enough

to model preference for utility smoothing independently of a functional form

of instantaneous utility. The domain of our interest, the set of consumption

sequences, is identi�ed with a subset of this larger domain, where each element

of the subset is a sequence of degenerate lotteries.

Time is discrete and varies over T = f0; 1; :::; Tg, where T > 0 (Appendix

B considers the in�nite horizon case). For a convex consumption set X � R,

let M be the set of all probability distributions over X with �nite support; M

is a mixture space under a probability operation +. An act is a function l : T

! M , where l = (l0; l1; :::; lT ). Denote by L = MT the collection of all acts;

L is a mixture space under the operation + de�ned by (l + l0)t = lt + l0t. A

constant act p is a function l : T ! M such that lt = p 2 M for all t 2 T ; p

is also identi�ed with p 2M . Let C be a collection of all constant acts.

The primitive of the model is the collection of preference orderings f�tg

� f�t j t 2 T g, whose element is de�ned on L. Each conditional ordering on

C induces a conditional ordering on M , also denoted by �t.

First, consider the following four axioms:

A1 (Conditional Preference-CP): For each t 2 T and for all l; l0 2 L,

if l� = l0� for all � � t, then l 't l0.

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A2 (Weak Order-WO): For each t 2 T and for all l; l0; l00 2 L, (i)

l �t l0 or l0 �t l and (ii) l �t l0and l0 �t l00 imply l �t l00.

A3 (Continuity-CT): For each t 2 T and for all l; l0; l00 2 L with

l �t l0 �t l00, there exist �, � 2 (0; 1) such that �l + (1 � �)l00 �t l0 and

l0 �t �l + (1� �)l00.

A4 (Non-Degeneracy-ND): For each t 2 T , there exist l; l0 2 L such

that l �t l0.

A1 states that a conditional ordering depends only on the corresponding

continuations of acts; it also implies that a preference ordering is independent

of a payo¤ history. A2-A4 are standard assumptions.

The next axiom relates the evaluation of lotteries to the evaluation of acts.

A5 (Strict Monotonicity-SMT): For each t 2 T and for all l; l0 2 L,

if l� �t l0� for all � 2 T , then l �t l0. In addition, if for some � � t, lt �t l0t,

then l �t l0.

A5 assumes that a lottery is evaluated independently at each time.

The discounted utility model fails to explain gain/loss asymmetry because

of utility independence, which is characterized by the following property:

A60 (Independence-ID): For each t 2 T , for all l; l0; l00 2 L, and for all

� 2 (0; 1), l �t l0if and only if �l + (1� �)l00 �t �l0 + (1� �)l00.

Applying A1-A3 and A60 to C, there exists an a¢ ne function Ut : M ! R

that represents �t on M . Then, by A5, a preference ordering on lottery se-

quences induces a preference ordering on utility sequences. Also, the mixture

operation + on L induces the addition between the corresponding utility se-

quences. Hence, A60 implies utility independence.

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Gain/loss asymmetry is based on utility-sequence-speci�c discounting. To

allow such discounting, we replace A60 with the following weaker axiom:

A6 (Constant-Independence-CI): For each t 2 T , for all l; l0 2 L,

p 2 C, and for all � 2 (0; 1), l �t l0if and only if �l+(1��)p �t �l0+(1��)p.

First, A6 is equivalent to A60 on C so that there exists an a¢ ne function

Ut : M ! R that represents �t on M . Second, since p does not involve

utility deviations, the pattern of utility deviations implied by l is preserved

in the pattern of utility deviations in �l + (1 � �)p. Hence, CI allows for

the possibility that the same discounting is applied only to a subset of utility

sequences sharing a similar pattern of utility deviations.

Next, consider the reasoning behind gain/loss asymmetry. If gains are

discounted more than losses, the DM tends to put greater weight on a lower

utility than on a higher utility. The motive behind this pattern of discounting

resembles utility smoothing. Hence, we introduce a notion that expresses

aversion to utility deviations:

A7 (Time-Variability Aversion-TVA): For each t 2 T , for all l; l0 2

L, and for all � 2 (0; 1), l 't l0 implies �l + (1� �)l0 �t l.

Given A1-A6, A7 is e¤ectively imposed on utility sequences. Therefore, this

axiom does not imply consumption smoothing (i.e., an instantaneous utility

function can be convex on a degenerate outcome spaceX). Intuitively, hedging

the movement of instantaneous utility over time increases overall utility.

The conditional utility based on A1-A7 is a temporal version of Gilboa-

Schmeidler�s (1989) multiple-priors utility, which does not necessarily satisfy

dynamic consistency. Hence, we restrict our attention to the family of condi-

tional orderings, for which each ordering satis�es the following crucial axiom:

12

A8 (Dynamic Consistency-DC): For each t 2 T nfTg and for all l; l0

2 L such that l� = l0� for all � � t, if l �t+1 l0, then l �t l0; the latter ranking

is strict if the former is strict.

Dynamic consistency has a normative appeal, which has been mentioned in

the introduction and discussed in detail in Section 2 through counterexam-

ples. The analytical advantage of dynamic consistency will be shown through

applications in Section 5.

The following proposition is the main result of this paper (for the proof,

see Appendix A):

Proposition 1: The following statements are equivalent:

(i) f�tg satisfy A1 to A8.

(ii) There exist an a¢ ne function U : M ! R and a collection of sets of

discount factors f[�t; �t]g1�t�T satisfying 0 < �t � �t < 1 for all t 2 [1; T ]

such that: For each t 2 T , �t is represented by Vt(:); where fVt(l)g0�t�T are

recursively de�ned by

Vt(l) � min�t+12[�t+1;�t+1]

[(1� �t+1)U(lt) + �t+1Vt+1(l)] (6)

and VT (l) � U(lT ). Moreover, [�t; �t] is uniquely de�ned for all t 2 [1; T ], and

U is unique up to a positive a¢ ne transformation.

Time-variability aversion is captured by minimization on a set of discount

factors, whereas dynamic consistency forces this operation to be applied re-

cursively. A discount factor takes on an extreme value under the recursive

application of minimization. Hence, (6) generates gain/loss asymmetry.

A special feature of this model is that the scale of future utility is compatible

with the scale of current utility. In particular, under a constant act p, U(p) =

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Vt+1(p). Hence, Vt+1(l) is greater than U(lt) if and only if l �t+1 lt, where

lt is a constant act that pays lt at each time. Intuitively, having U(lt) as

an endogenous reference point is equivalent to having a constant act lt as an

endogenous reference point.

4 Analysis

4.1 Yield Asymmetry and Conditions on Sets of Discount Factors

First, for the models based on utility smoothing, we formally de�ne yield

asymmetry as follows:

De�nition 1: For T � s > s0 � t, let u = (a; :::; a; a + bs; a; :::; a)

and u0 = (a; :::; a; a � bs; a; :::; a) be utility sequences, where bs > 0. Let v =

(a; :::; a; a + vs0 ; a; :::; a) and v0 = (a; :::; a; a � v0s0 ; a; :::; a) be utility sequences

satisfying u 't v and u0 't v0, where �t is an induced conditional ordering on

the set of utility sequences. Then the DM�s preferences show �yield asymmetry

at time t�if 0 < vs0 < v0s0 for any admissible combinations of a, bs, s, and s0.

Although highly related, recursive gain/loss asymmetry and yield asymme-

try are not equivalent notions: Recursive gain/loss asymmetry is based on a

comparison between di¤erent levels of utility in a given period, whereas yield

asymmetry is based on a comparison between two utility sequences, each of

which allocates a gain (or a loss) in a di¤erent period. Hence, to relate these

two notions, we derive necessary and su¢ cient conditions on sets of discount

factors that explain yield asymmetry (for the proof, see Appendix C).

Proposition 2: The DM�s preferences show yield asymmetry at every

14

period if and only if for each t satisfying T � t > 0,

�t(1� �t)

(1� �t+1) <�t

(1� �t)(1� �t+1); (7)

where �T+1 = �T+1 � 0.

For example, suppose that �t = � and �t = � for all t > 0. If � + � < 1

and � 6= �, yield asymmetry exists at all periods. Hence, the sets of discount

factors that satisfy the above condition are not empty.

To negate the last period e¤ect (i.e., �T+1 = �T+1 � 0), we may apply

a stationary and in�nite horizon version of (6) (see Appendix B), where our

representation becomes the following:

V ((lt; :::)) � min�2[�;�]

[(1� �)U(lt) + �V ((lt+1; :::))]: (8)

Then condition (7) becomes � < �, which is always satis�ed under � 6= �.

Hence, recursive gain/loss asymmetry and yield asymmetry become an equiv-

alent notion.

4.2 Relationship between the Discounted Utility Model and Time-

Variability Aversion

First, we investigate a binary relation of �more time-variability averse� (or

equivalently, �more gain/loss asymmetric�). From the representation of (6), it

is clear that the instantaneous utility function U in�uences e¤ective intertem-

poral substitution. Thus, we can only permit a binary relation of �more time-

variability averse�for pairs of preference relations that embody the same rank-

ing of a single-period consumption lottery. Then Vt is more time-variability

averse than V 0t if and only if U = U0, and

[�� ; �� ] � [�� ; �� ]0 for all � 2 ft+ 1; :::; Tg:

15

where [�� ; �� ] and [�� ; �� ]0 represent sets of discount factors in (6) for Vt and

V 0t , respectively.

The relationship between the discounted utility model and (6) is most

clearly shown by (8): If � = � = �, (8) is equivalent to the discounted utility

model with the discount factor 0 < � < 1, i.e.,

V ((lt; :::)) � (1� �)1X�=t

���tU(l� ): (9)

(9) is the least time-variability averse in the class of (8), i.e., there exists

no other Vt that is strictly less time-variability averse than (9). Hence, the

discounted utility model is a special case of our model, in which the DM is

time-variability neutral (i.e., in A7, l 't l0 implies �l + (1� �)l0 't l).

5 Application

To study the economic implications of gain/loss asymmetry, we present a sim-

ple repeated game application. The bene�t of dynamic consistency becomes

evident through the construction of an SPNE. Consider the following version

of the prisoner�s dilemma (the numbers are instantaneous utilities). The re-

sults in this example will not change under a positive a¢ ne transformation of

payo¤s. Hence, they are independent of the level of utility but dependent on

the pattern of utility deviations involved in a payo¤ sequence.

Player 1

Player 2

A B

A (5,5) (10,1)

B (1,10) (4,4)

16

We focus on two di¤erent payo¤ sequences. The �rst sequence is an in�nite

repetition of (5,5). To generate this sequence under an SPNE, consider the

following strategy.

(a) Play A until the opponent plays B.

(b) If the opponent plays B, then play B for the next N > 0 periods.

(c) If (b) happens, after N periods of punishment, play A until the opponent

plays B.

(d) If (c) happens, and if the opponent plays B, play B forever.

This strategy involves a combination of Nash reversion with a preceding

punishment phase. The intention of this strategy is to give the opponent an

opportunity to learn the bene�t of cooperation.

First, consider discounted-utility players with the same discount factor

0 < � < 1. For this strategy to be an SPNE,

5 � 10(1� �) + 4�, and (10)

5 � 10(1� �) + 4�(1� �N) + 5�N+1 = 10(1� �) + 4� + �N+1. (11)

For example, let � = 0:9, which satis�es (10). Then by (11), N � 8; since the

payment at a single-period derivation is high, it requires a long punishment

phase.

Second, suppose that players�preferences are represented by (8) with the

same set of discount factors [�; �], where 0 < � < � < 1. We assume that

� 2 [�; �] so that the players here are more gain/loss asymmetric than the

discounted-utility players above. For the above strategy to be an SPNE,

5 � 10(1� �) + 4�, and (12)

5 � 10(1� �) + 4� (1� �)(1� �)(1� �

N) + 5��N = 10(1� �) + 4� + ��N . (13)

17

By assumption, � � 0:9 � �, which satis�es (12). By (13), N � 1 if � �

(6� � 5)=�; the right hand side is increasing in � (it is 49at � = 0:9). This

shows that if players are su¢ ciently gain/loss asymmetric, the subgame perfect

strategy requires only a minimum punishment phase. Hence, as players become

more gain/loss asymmetric, it becomes easier to achieve cooperation if payo¤s

are constant.

The second sequence is (10,1) at an even period and (1,10) at an odd

period. To generate this sequence under an SPNE, consider the following

strategy using Nash reversion.

(a) Player 1 plays B if t is even and plays A if t is odd until Player 2 plays B

at an even t.

(b) If Player 2 plays B at an even t, Player 1 plays B forever.

(c) Player 2 plays A if t is even and plays B if t is odd until Player 1 plays B

at an odd t.

(d) If Player 1 plays B at an odd t, Player 2 plays B forever.

Consider the discounted-utility players. For this strategy to be an SPNE,

1 + 10� � 4(1� �2):

This condition is satis�ed under � = 0:9. Similarly for the gain/loss asymmet-

ric players,

V ((1; 10; 1; 10; :::)) � 4, and (14)

V ((1; 10; 1; 10; :::)) =1

1� ���(1� �) + �(1� �)10

= 10� 9(1� �)

1� ��: (15)

Again, � � 0:9 � �. By (14) and (15), � � 3=(9 � 6�); the right hand

side is increasing in � (it is 56at � = 0:9). For example, let � > 51

54. Then

3=(9�6�) > 0:9 so that the �rst strategy is an SPNE for someN but the second

18

strategy is not (if N � 8, the �rst strategy is an SPNE at any � � 0:9 � �).

Hence, as players become more gain/loss asymmetric, it becomes harder to

�nd an SPNE that supports variable payo¤ sequences.

6 Other Related Literature

Koopmans (1960) proposes the aggregator function that represents dynami-

cally consistent preferences. Concave, nonlinear, and smooth aggregator func-

tions can exhibit e¤ects similar to gain/loss asymmetry. However, the de-

gree of asymmetry converges to zero as the sizes of gains and losses decrease,

which contradicts the results shown by Thaler�s (1981) experiment. On the

contrary, our aim is to build a speci�c aggregator function under which the

degree of asymmetry does not diminish as the sizes of gains and losses de-

crease. As applications show, our approach provides a clear parametrization

of gain/loss asymmetry independently of a utility level, which is not possible

under a smooth aggregator function. Also, given the Euler inequality derived

in a di¤erent context by Epstein and Wang (1994), the analytical cost of (6)

caused by non-di¤erentiability is relatively minor.

In terms of a model of discount factors, Uzawa (1968), Epstein (1983), and

Shi and Epstein (1993) propose a model under which the discount factor at

any future date depends on historical consumption. However, their models are

not based on utility smoothing and do not show gain/loss asymmetry. On the

other hand, similarly to our model, Geo¤ard (1996) proposes, in a continuous-

time setting, the following non-additively-separable model under which the

discount rates depend on an entire sequence of consumption:

V (c) � minrt2R

Z 1

0

f(ct; Bt; rt)dt; (16)

19

where R is a set of admissible paths of discount rates, and f gives current

felicity as a function of the current values of consumption ct, discount factor

Bt, and discount rate rt. Although (16) does not satisfy dynamic consistency

in general, Geo¤ard (1996) shows that the recursive utility of Epstein (1987)

belongs to this class of the representation if f(ct; Bt; rt) = BtF (ct; rt). Indeed,

our model (8) can be written as follows:

V (c) � minB2B

1Xt=0

BtF (ct; �t+1);

where F (ct; �t+1) � (1� �t+1)u(ct) and B is a collection of admissible normal-

ized discount-factor processes B de�ned by B0 � 1 and by Bt+1 � �t+1Bt with

�t+1 2 [�; �]. Hence, in a discrete time setting, our model provides axiomati-

zation for a particular form of (16).

Finally, Epstein and Schneider (2003) propose recursive multiple-priors

utility, where a set of priors is recursively constructed. In (6), a set of dis-

count factors is similarly constructed. To exploit this similarity, we adapt

their proof and suitably modify it to �t our framework.

7 Conclusion

This paper makes the following contributions: (i) Based on time-variability

aversion, we develop a new model of discount factors that not only captures

gain/loss asymmetry but also satis�es dynamic consistency; (ii) Dynamic con-

sistency introduces the notion of recursive gain/loss asymmetry, where current

utility becomes an endogenous reference point to evaluate future utility. The

model provides a clear mechanism for reference dependence and explains the

relationship among gain/loss asymmetry, time-variability aversion, and dy-

namic consistency; and (iii) Under a stationary and in�nite horizon version of

20

the model, recursive gain/loss asymmetry and yield asymmetry are equivalent.

Also, through an example, we show the applicability of our model originated

by dynamic consistency.

Appendix A: Proof of Proposition 1

If T = 1, Proposition 1 is equivalent to Theorem 1 of Gilboa and Schmeidler

(1989) (GS). Hence, assume that T > 1. We prove (i) ) (ii), which is based

on Lemmas A.1-A.4 (given Theorem 1 of GS and Lemmas A.1-A.4, (ii)) (i)

is trivial). We adapt the arguments in Epstein and Schneider (2003).

Lemma A.1: There exists an a¢ ne function U : M ! R, unique up to a

positive a¢ ne transformation, such that for all t 2 T , U represents �t on M .

Proof. By CP, WO, CT, and CI, there exists an a¢ ne Ut : M ! R,

unique up to a positive a¢ ne transformation, that represents �t on M .

For a given t < T , consider l; l0 2 L such that l� = p 2M and l0� = q 2M

for all � � t+ 1 and l� = l0� for all � < t+ 1. By CP, WO, and SMT, l �t+1 l0

if and only if p �t+1 q. By WO and DC, l �t+1 l0 if and only if l �t l. By

CP, WO, and SMT, l �t l0 if and only if p �t q. Hence, p �t+1 q if and only if

p �t q. The conclusion follows by induction applied from t = 0 to T � 1. �

For the remaining lemmas, we use U in place of Ut. By ND, assume that

[�1; 1] � U(M). In particular, let p[0] 2 M satisfy U(p[0]) = 0. Assume a

standard topology on Rt.

Lemma A.2: For each t 2 T , there exists a non-empty, closed, and

convex set of strictly positive discount factors, Dt � RT�t+1; each of whose

21

elements, bt 2 Dt, satis�esPT

�=t bt� = 1 such that �t is represented by Vt(:);

where

Vt(l) � minbt2Dt

TX�=t

bt�U(l� ) (A.1)

Moreover, Dt is unique.

Proof. This follows from Theorem 1 of GS and Lemma A.1. �

For each t < T , let Ut � f(U(lt); Vt+1(l))j l 2 Lg. By (A.1), [�1; 1] �

[�1; 1] � Ut. De�ne the following function It : Ut ! R by

It(U(lt); Vt+1(l)) � Vt(l): (A.2)

By SMT and DC, (A.2) is well de�ned and increasing on Ut. Also, It(1; 1) = 1.

Lemma A.3: For a given t < T , there exists a unique [�t+1; �t+1] � R

satisfying 1 < �t+1 � �t+1 < 1 such that for all l 2 L,

It(U(lt); Vt+1(l)) = min�t+12[�t+1;�t+1]

[(1� �t+1)U(lt) + �t+1Vt+1(l)]: (A.3)

Proof. We adapt the arguments in Lemma 3.3 of GS.

(i) It is homogeneous on Ut: For any x; x0 2 Ut with x0 = �x for 0 < � � 1,

It(x0) = �It(x). Let l; l0 2 L satisfy l0 = �l+(1��)p[0] and (U(lt); Vt+1(l)) = x.

Since U is a¢ ne, U(l0t) = �U(lt). By (A.1), Vt+1(l0) = �Vt+1(l). By (A.2),

�It(x) = �Vt(l) and

It(x0) = It(�U(lt); �Vt+1(l)) = It(U(l

0t); Vt+1(l

0)) = Vt(l0):

By (A.1), Vt(l0) = �Vt(l) so that It(x0) = �It(x).

(ii) Extend It by homogeneity to R� R.

(iii) It is monotone on R� R (by construction).

22

(iv) It satis�es constant additivity on R� R: For any x; x0 2 R� R with

x0 = (x1; x2) and x1 = x2, It(x + x0) = It(x) + It(x0). By homogeneity, it is

enough to consider 2x; 2x0 2 Ut. Let l; p 2 L satisfy (U(lt); Vt+1(l)) = 2x and

(U(p); Vt+1(p)) = 2x0. Since U is a¢ ne,

1

2U(lt) +

1

2U(p) = U(

1

2lt +

1

2p). By

(A.1),1

2Vt+1(l) +

1

2Vt+1(p) = Vt+1(

1

2l +

1

2p). Hence, by (A.1) and (A.2),

It(x+ x0) = Vt(

1

2l +

1

2p) = Vt(

1

2l) + Vt(

1

2p) = It(x) + It(x

0):

(v) It is superadditive on R� R: For any x; x0 2 R� R, It(x + x0) �

It(x) + It(x0). By homogeneity, it is enough to consider 2x; 2x0 2 Ut. Let

l; l0 2 L satisfy (U(lt); Vt+1(l)) = 2x and (U(l0t); Vt+1(l0)) = 2x0, where l0� = l0� 0for all �; � 0 > t. Since U is a¢ ne,

1

2U(lt) +

1

2U(l0t) = U(

1

2lt +

1

2l0t). By (A.1),

1

2Vt+1(l) +

1

2Vt+1(l

0) = Vt+1(1

2l +

1

2l0). Hence, by (A.1) and (A.2),

It(x+ x0) = Vt(

1

2l +

1

2l0) � Vt(

1

2l) + Vt(

1

2l0) = It(x) + It(x

0):

The conclusion follows from Lemma 3.5 and Theorem 1 of GS. �

Let �t+1 � [�t+1; �t+1]. De�ne a collection of discount factors, D0t, by

D0t � fbt 2 RT�t+1++ jbt = (1��t+1; �t+1bt+1); where �t+1 2 �t+1 and bt+1 2 Dt+1g:

(A.4)

Each element bt 2 D0t satis�esPT

�=t bt� = 1. Since �t+1 andDt+1 are non-empty,

closed, and convex, so is D0t; D0t is also compact.

Lemma A.4: D0t = Dt.

23

Proof. For all l 2 L,

Vt(l) = minbt2Dt

TX�=t

bt�U(l� )

= min�t+12[�t+1;�t+1]

f(1� �t+1)U(lt) + �t+1Vt+1(l)g (by (A.3))

= min�t+12[�t+1;�t+1]

f minbt+12Dt+1

f(1� �t+1)U(lt) + �t+1TX

�=t+1

bt+1� U(l� )gg

= minbt2D0t

TX�=t

bt�U(l� ):

Hence, D0t with U represents �t. The conclusion follows from the uniqueness

of Dt, as implied by ND. �

Finally, (ii) of Proposition 1 follows by recursively applying Lemmas A.3

to A.4 from time 0 to T � 1. �

Appendix B: In�nite Horizon Model

Let Tt � ft; t+ 1; :::g be a countably in�nite number of periods, and T � T0.

Denote by �t the �-algebra that consists of all subsets of Tt. Since �t and

�t0 are identical except for the name of each element, for simplicity, let B

be a collection of all bounded, �t-measurable real valued functions on Tt for

any t 2 T . B is endowed with the sup norm. We use the weak�-topology on

B�, i.e., the dual space of B. Denote by 1 2 B a constant function of one.

De�ne by �A an indicator function on A 2 �t. For each l 2 L, de�ne the

continuation from time t, lt 2 L, by lt = (lt; lt+1; :::). Let0Y�=1

x� � 1 for any

sequence x : T1 ! R++.

We impose regularity and stationarity axioms:5

5For an in�nite horizon extension, Gilboa (1989) and Shalev (1997) require an additional

24

A9 (Best�Worst-BW): For each t 2 T , there exist l� and l� 2 L such

that l� �t l �t l� for all l 2 L.

A10 (Stationarity-ST): For each t 2 T , for all l; l0;el;el0 2 L such that(i) ls = els and l0s0 = el0s0 for all s < t and s0 < t + 1 and (ii) lt+s = l0t+1+s andelt+s = el0t+1+s for all s 2 T , l �t el if and only if l0 �t+1 el0.Proposition B.1: The following statements are equivalent:

(i) f�tg satisfy A1 to A10, where T � 1 in A8.

(ii) There exist (a) a set [�; �] � R satisfying 0 < � � � < 1, (b) a non-empty,

closed, and convex set of strictly positive and countably additive discount fac-

tors on �0, � � B�, each of whose elements, b 2 �, satis�es

b(a) �1Xt=0

btat for all a 2 B,

where bt � t� t+1 and t �tY

�=1

�� based on a sequence of atemporal discount

factors f�tg with �t 2 [�; �], and (c) an a¢ ne function U : M ! R such that:

For each t 2 T , �t is represented by Vt(:); where

Vt(l) = V (lt) � min

b2�f1X�=t

b��tU(l� )g = min�t+12[�;�]

[(1� �t+1)U(lt) + �t+1V (lt+1)]:

Moreover, � and � are unique, U is unique up to a positive a¢ ne transforma-

tion, and maxp2MU(p) and minp2MU(p) exist.

Proof. Given Proposition 4.1 of GS and � de�ned above, (ii) ) (i) is

trivial. We prove (i)) (ii), which consists of several steps.

For a function U :M ! R, de�ne a function U� : L ! B by U�(l)t = U(lt).

axiom to de�ne the continuity in discount factors. Here, dynamic consistency plays a similar

role.

25

Step 1: If f�tg satisfy A1-A9, then for each t 2 T , there exist (a) a set

[�t+1; �t+1] � R satisfying 1 < �t+1 � �t+1 < 1 and (b) a non-empty, closed,

and convex set of �nitely additive discount-factor measures on �t, Dt � B�,

each of whose elements, bt 2 Dt, satis�es bt(1) = 1 and bt(�f�g) > 0 for all

� 2 Tt such that the conditional ordering �t is represented by Vt(:), where

Vt(l) � minbt2Dt

ZTtU � (l)dbt = min

�t+12[�t+1;�t+1][(1� �t+1)U(lt) + �t+1Vt+1(l)];

and U : M ! R satis�es the conditions stated in Proposition B.1. Moreover,

Dt is compact and uniquely determined, and [�t+1; �t+1] is unique.

Proof. This follows from the suitable adaptation of Proposition 1 to the

in�nite horizon setting. In particular, we invoke Proposition 4.1 of GS at

Lemma A.2. Also, we replace (A.4) with

D0t � fbt 2 B�jbt = (1� �t+1; �t+1bt+1); where �t+1 2 �t+1 and bt+1 2 Dt+1g;

and where �t+1 � [�t+1; �t+1]. Each bt 2 D0t de�nes bt : B! R by

bt(at) =

ZTtatdbt = (1� �t+1)att + �t+1

ZTt+1

at+1dbt+1. (B.1)

It is easy to see that D0t is non-empty, closed, convex, and compact in B�. The

existence of maxp2MU(p) and minp2MU(p) follows from BW. �

Step 2: There exist V : L ! R, D � B�, and [�; �] � [0; 1] such that

V (lt) = Vt(l), D = Dt, and [�; �] = [�t+1; �t+1] for all t 2 T .

Proof. The existence of V follows from ST. The uniqueness of Dt proves

other results. �

For the remaining steps, we use V , D, and [�; �]. Let f�tg be a sequence of

atemporal discount factors, where �t 2 [�; �]. From f�tg, de�ne a sequence of

26

discount factors f tg by t �tY

�=1

�� . Construct � � B� by

� � fb 2 B�j For each a 2 B, b(a) �1Xt=0

btat, where bt � t � t+1g: (B.2)

� is non-empty. For any b 2 �,P1

t=0 bt =P1

t=0( t � t+1) = 0 = 1. Hence,

� is a collection of discrete, countably additive, and strictly positive discount

factor sequences. Under these conditions, � is closed and compact in B�.

Step 3: � is convex.

Proof. For any � 2 (0; 1) and any b; b0 2 �, let eb � �b+ (1� �)b0. Then,eb is based on fe�tg such that e�1 = ��1 + (1� �)�01 2 [�; �] and for t > 1,e�t = �t + (1� �) 0t�1(�0t � �t)

� t�1 + (1� �) 0t�12 [�; �];

where f�tg and f tg de�ne b and f�0tg and f 0tg de�ne b0. Hence, eb 2 �. �

Step 4: D =� so that for all l 2 L, Vt(l) = minb2�fP1

�=t b��tU(l� )g.

Proof. For D ��, let bb 2 D. For each k 2 T , let �k � t+kY�=t

��� , where ��� is

an implied discount factor of bb de�ned by (B.1). Then, for each n 2 T , thereexists b(n) 2 � such that b(n)��t = ���t � ��+1�t for all � 2 ft; :::; t + ng. By

recursively applying (B.1), for each l 2 L,

bb(U � (lt)) = t+nX�=t

( ���t � ��+1�t)U(lt� ) + �n+1ZTt+n+1

U � (lt+n+1)dbbt+n+1, andb(n)(U � (lt)) =

t+nX�=t

( ���t � ��+1�t)U(lt� ) + �n+11X

�=t+n+1

b(n)��t �n+1

U(lt� ):

27

Therefore,

jbb(U � (lt))� b(n)(U � (lt))j= �n+1j

ZTt+n+1

U � (lt+n+1)dbbt+n+1 � 1X�=t+n+1

b(n)��t �n+1

U(lt� )j

< �n+1(2jV (l�)j+ 2jV (l�)j):

Since the bound is independent of n, t, and l and �n+1 converges to zero,

bb(U � (lt)) = lim b(n)(U � (lt)): (B.3)

And since (B.3) holds for any l 2 L, bb is either a limit point of � or a point

in �. Hence, bb 2 � because � is closed in B�.

For D ��, let b 2 �. Let f �t g be a sequence of implied discount factors

of b de�ned by (B.2). By (B.1), for each n 2 T , there exists bb(n) 2 D such thatfor each l 2 L,

bb(n)(U � (lt)) = t+nX�=t

( ���t � ��+1�t)U(lt� ) + �n+1ZTt+n+1

U � (lt+n+1)dbb(n);t+n+1.Then b 2 D follows from a similar argument leading to (B.3) because D is

closed in B�. �

Appendix C: Proof of Proposition 2

By a comparison between u = (a; :::; a; a + bt; a; :::; a) and v = (a; :::; a; a +

vt�1; a; :::; a), u '0 v if and only if

(1� �t)vt�1 = �t(1� �t+1)bt; (C.1)

where �T+1 = �T+1 � 0. Similarly, by a comparison between u = (a; :::; a; a�

bt; a; :::; a) and v = (a; :::; a; a� v0t�1; a; :::; a), u '0 v if and only if

(1� �t)v0t�1 = �t(1� �t+1)bt: (C.2)

28

Hence, by rearranging (C.1) and (C.2), yield asymmetry exists between time

t�1 and t if and only if (7) is satis�ed. Given dynamic consistency, this proves

that (7) is a necessary condition.

To show that it is su¢ cient, consider a comparison between time s and

time s0, where s > s0. Then

vs0 = bs

sY�=s0+1

��(1� �s+1)(1� �s0+1)

= bs

sY�=s0+1

��(1� �� )

(1� ��+1), and

v0s0 = bs

sY�=s0+1

��(1� �s+1)(1� �s0+1)

= bs

sY�=s0+1

��(1� �� )

(1� ��+1).

By (7), vs0 < v0s0. �

References

1. Anscombe, F., and R. Aumann (1963): �A De�nition of Subjective Prob-

ability,�Annals of Mathematical Statistics, 34(1), 199-205.

2. Eichberger, J., and D. Kelsey (1996): �Uncertainty Aversion and Dy-

namic Consistency,�International Economic Review, 37(3), 625-640.

3. Epstein, L. (1983): �Stationary Cardinal Utility and Optimal Growth

under Uncertainty,�Journal of Economic Theory, 31(1), 133-152.

4. Epstein, L. (1987): �The Global Stability of E¢ cient Intertemporal Al-

locations,�Econometrica, 55(2), 329-355.

5. Epstein, L., and M. Schneider (2003): �Recursive Multiple-Priors,�Jour-

nal of Economic Theory, 113(1), 1-31.

6. Epstein, L., and T. Wang (1994): �Intertemporal Asset Pricing under

Knightian Uncertainty,�Econometrica, 63(2), 283-322.

29

7. Frederic, S., G. Loewenstein, and T. O�Donoghue (2002): �Time Dis-

counting and Time Preference: A Critical Review,�Journal of Economic

Literature, 40(2), 351-401.

8. Geo¤ard, P. (1996): �Discounting and Optimizing: Capital Accumula-

tion Problems as Variational Minmax Problems,�Journal of Economic

Theory, 69(1), 53-70.

9. Gilboa, I. (1989): �Expectation and Variation in Multi-Period Deci-

sions,�Econometrica, 57(5), 1153-1169.

10. Gilboa, I., and D. Schmeidler (1989): �Maxmin Expected Utility with

Non-Unique Prior,�Journal of Mathematical Economics, 18(2), 141-153.

11. Grant S., A. Kajii, and B. Polak (2000): �Temporal Resolution of Un-

certainty and Recursive Non-Expected Utility Models,�Econometrica,

68(2), 425-434.

12. Kahneman, D., and A. Tversky (1979): �Prospect Theory: An Analysis

of Decision under Risk,�Econometrica, 47(2), 263-292.

13. Koopmans, T. (1960): �Stationary Ordinal Utility and Impatience,�

Econometrica, 28(2), 287-309.

14. Loewenstein G., and D. Prelec (1992): �Anomalies in Intertemporal

Choice: Evidence and an Interpretation,� Quarterly Journal of Eco-

nomics, 107(2), 573�597.

15. Samuelson, P. (1937): �A Note on Measurement of Utility,�Review of

Economic Studies, 4(2), 155-161.

30

16. Sarin B., and P. Wakker (1998): �Dynamic Choice and NonExpected

Utility,�Journal of Risk and Uncertainty, 17(2), 87-119.

17. Schmeidler, D. (1989): �Subjective Probability and Expected Utility

without Additivity,�Econometrica, 57(3), 571-587.

18. Shalev, J. (1997): �Loss Aversion in a Multiple-Period Model,�Mathe-

matical Social Sciences, 33(3), 203-226.

19. Shi, S., and L. Epstein (1993): �Habit and Time Preference,� Interna-

tional Economic Review, 34(1), 61-84.

20. Thaler, R. (1981): �Some Empirical Evidence on Dynamic Inconsis-

tency,�Economics Letters, 8, 201-207.

21. Tversky, A., and D. Kahneman (1991): �Loss Aversion in Riskless Choice:

A Reference-Dependent Model,�Quarterly Journal of Economics, 106(4),

1039-1061.

22. Uzawa, H. (1968): �Time Preference, the Consumption Function, and

Optimum Asset Holdings,�in J.N. Wolfe, ed., Capital and Growth: Pa-

pers in Honour of Sir John Hicks (Chicago: Aldine, 1968).

23. Wakai, K. (2002): �Linking Behavioral Economics, Axiomatic Decision

Theory and General Equilibrium Theory,�Ph.D. Dissertation, Yale Uni-

versity.

31