lecture 06 single attribute utility theory · 2014. 9. 18. · we can show that if the utility...
TRANSCRIPT
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Lecture 06Single Attribute Utility Theory
Jitesh H. Panchal
ME 597: Decision Making for Engineering Systems Design
Design Engineering Lab @ Purdue (DELP)School of Mechanical Engineering
Purdue University, West Lafayette, INhttp://engineering.purdue.edu/delp
September 17, 2014c©Jitesh H. Panchal Lecture 06 1 / 38
http://engineering.purdue.edu/delp
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Lecture Outline
1 Decision Making under UncertaintyAlternate Approaches to Risky Choice ProblemMotivation for Using Utility Theory
2 Uni-dimensional Utility TheoryFundamentals of Utility TheoryQualitative Characteristics of Utility
3 Attitudes to RiskRisk Averse and Risk ProneMeasuring Risk Aversion
4 A Procedure for Assessing Utility Functions
Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Cambridge, UK, CambridgeUniversity Press. Chapter 4.
c©Jitesh H. Panchal Lecture 06 2 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Alternate Approaches to Risky Choice ProblemMotivation for Using Utility Theory
The Structure of a Design Decision
Decision
A1
A2
An
O11
O12
O1k
O21
O22
O2k
On1
On2
Onk
U(O11)
U(O12)
U(O1k)
U(O21)
U(O22)
U(O2k)
U(On1)
U(On2)
U(Onk)
Select Ai
p11
p1k
p21
p1k
pn1
pnk
Alternatives Outcomes Preferences Choice
Slide courtesy: Chris Paredisc©Jitesh H. Panchal Lecture 06 3 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Alternate Approaches to Risky Choice ProblemMotivation for Using Utility Theory
Focus of this Lecture
Problem Statement
Choose among alternatives A1,A2, . . . ,Am, each of which will eventuallyresult in a consequence described by one attribute X .
Decision maker does not know exactly what consequence will result fromeach alternative.
But he/she can assign probabilities to the various consequences that mightresult from any alternative.
c©Jitesh H. Panchal Lecture 06 4 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Alternate Approaches to Risky Choice ProblemMotivation for Using Utility Theory
Alternate Approaches to Risky Choice Problem:a) Probabilistic dominance
FA
FB
Probability
density
Probability
of x or less
fA
fB
0 0
Figure: 4.2 on page 135 (Keeney and Raiffa)
c©Jitesh H. Panchal Lecture 06 5 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Alternate Approaches to Risky Choice ProblemMotivation for Using Utility Theory
Alternate Approaches to Risky Choice Problem:b) Expected value of uncertain outcome
Consider the following acts
Act A: Earn $100,000 for sure
Act B: Earn $200,000 or $0, each with probability 0.5
Act C: Earn $1,000,000 with probability 0.1 or $0 with probability 0.9
Act D: Earn $200,000 with probability 0.9 or lose $800,000 withprobability 0.1
The expected amount earned is exactly $100,000. But not all acts are equallydesirable.
c©Jitesh H. Panchal Lecture 06 6 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Alternate Approaches to Risky Choice ProblemMotivation for Using Utility Theory
Alternate Approaches to Risky Choice Problem:c) Consideration of mean and variance
One possibility is to consider variance, in addition to the expected value ofthe outcome.
But, Acts C and D have the same mean and variance:
Act C: Earn $1,000,000 with probability 0.1 or $0 with probability 0.9
Act D: Earn $200,000 with probability 0.9 or lose $800,000 withprobability 0.1
Are these acts equally preferred?
Therefore, any measure that considers mean and variance only cannotdistinguish between these two acts.
Considering mean and variance imposes additional problem of findingrelative preference between them.
c©Jitesh H. Panchal Lecture 06 7 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Alternate Approaches to Risky Choice ProblemMotivation for Using Utility Theory
Primary Motivation for using Utility Theory
IF an appropriate utility is assigned to each possible consequence,AND the expected utility of each alternative is calculated,
THEN the best course of action is the alternative with the highest expectedutility.
c©Jitesh H. Panchal Lecture 06 8 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Fundamentals of Utility Theory
Assume n consequences labeled x1, x2, . . . , xn such that xi is less preferredthan xi+1
x1 ≺ x2 ≺ x3 ≺ · · · ≺ xnThe decision maker is asked to state preferences about two acts a′ and a′′,where
1 Act a′ will result in consequence xi with probability p′i for i = 1..n2 Act a′′ will result in consequence xi with probability p′′i for i = 1..n
c©Jitesh H. Panchal Lecture 06 9 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Fundamentals of Utility Theory (contd.)
Assume that for each i , the decision maker is indifferent between thefollowing options:
Certainty Option: Receive xi
Risky Option: Receive xn with probability πi and x1 with probability (1− πi ).This option is denoted as 〈xn, πi , x1〉
Clearly,
π1 = 0
πn = 1
π1 < π2 < π3 < · · · < πn
c©Jitesh H. Panchal Lecture 06 10 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Fundamentals of Utility Theory (contd.)
πi ’s can be thought of as numerical scaling of x ’s.
π1 < π2 < π3 < · · · < πnand
x1 ≺ x2 ≺ x3 ≺ · · · ≺ xn
Fundamental Result of Utility Theory
The expected value of the π’s can be used to numerically scale probabilitydistributions over the x ’s.
The expected π scores for acts a′ and a′′ are as follows:
π̄′ =∑
ip′i πi and π̄
′′ =∑
ip′′i πi
Act a′ ≡ giving the decision maker a π̄′ chance at xn and 1− π̄′ chance at x1Act a′′ ≡ giving the decision maker a π̄′′ chance at xn and 1− π̄′′ chance at x1Now, we can rank order acts a′, a′′ in terms of π̄′, π̄′′
c©Jitesh H. Panchal Lecture 06 11 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Fundamentals of Utility Theory (contd.)
Transforming π’s into u’s using a positive linear transformation
ui = a + bπi , b > 0, i = 1, . . . , n
Then,u1 < u2 < · · · < un
The expected u values rank order a′ and a′′ in the same way as the expectedπ values
ū′ =∑
i
p′i ui =∑
i
p′i (a + bπi ) = a + bπ̄′
Essence of the problem
How can appropriate π values be assessed in a responsible manner?
c©Jitesh H. Panchal Lecture 06 12 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Direct Assessment of Utilities
Define:xo as a least preferred consequence, andx∗ as a most preferred consequence. Assign
u(x∗) = 1 and u(xo) = 0
For each other consequence x , assign a probability π such that x is indifferentto the lottery 〈x∗, π, xo〉. Note that the expected utility of the lottery is:
u(x) = πu(x∗) + (1− π)u(xo) = π
Continue for all x ’s (or fit a curve).
c©Jitesh H. Panchal Lecture 06 13 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Qualitative Characteristics of Utility
1 Monotonicity2 Certainty equivalence3 Strategic equivalence
c©Jitesh H. Panchal Lecture 06 14 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Qualitative Characteristics of Utility: Monotonicity
Definition (Monotonicity)
For a monotonically increasing utility function
[x1 > x2]⇔ [u(x1) > u(x2)]
For a monotonically decreasing utility function
[x1 > x2]⇔ [u(x1) < u(x2)]
c©Jitesh H. Panchal Lecture 06 15 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Qualitative Characteristics of Utility: Certainty Equivalence
Assume lottery L yields consequences x1, x2, . . . , xn with probabilitiesp1, p2, . . . , pn.
Define:x̃ : Uncertain consequence of lottery (i.e., random variable)x̄ : Expected consequence
x̄ ≡ E(x̃) =n∑
i=1
pixi
Definition (Certainty equivalence)
A certainty equivalent of lottery L is the amount x̂ such that the decisionmaker is indifferent between L and the amount x̂ for certain.
u(x̂) = E [u(x̃)], or x̂ = u−1Eu(x̃)
c©Jitesh H. Panchal Lecture 06 16 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Qualitative Characteristics of Utility: Certainty Equivalence (continuousvariables)
If x is a continuous variable, the associated uncertainty is described using aprobability density function, f (x). Then,
x̄ ≡ E(x̃) =∫
xf (x)dx
The certainty equivalent x̂ is a solution to
u(x̂) = E [u(x̃)] =∫
u(x)f (x)dx
c©Jitesh H. Panchal Lecture 06 17 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Qualitative Characteristics of Utility: Strategic Equivalence
Definition (Strategic equivalence)
Two utility functions, u1 and u2, are strategically equivalent (u1 ∼ u2) if andonly if they imply the same preference ranking for any two lotteries.
If two utility functions are strategically equivalent, the certainty equivalents oftwo lotteries must be the same. Therefore,
u1 ∼ u2 ⇒ u−11 Eu1(x̃) = u−12 Eu2(x̃), ∀x̃
c©Jitesh H. Panchal Lecture 06 18 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Fundamentals of Utility TheoryQualitative Characteristics of Utility
Qualitative Characteristics of Utility: Strategic Equivalence (contd.)
For some constants h and k > 0, if
u1(x) = h + ku2(x), ∀x
then u1 ∼ u2
Theorem
If u1 ∼ u2, there exists two constants h and k > 0 such that
u1(x) = h + ku2(x), ∀x
Example: u(x) = a + bx ∼ x , b > 0
We can show that if the utility function is linear, the certainty equivalent forany lottery is equal to the expected consequence of that lottery.
c©Jitesh H. Panchal Lecture 06 19 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Definition of Risk Aversion
Consider a lottery 〈x ′, 0.5, x ′′〉 whose expected consequence is x̄ = x′+x′′
2 .Assume that the decision maker is asked to choose between x̄ for certainand the lottery 〈x ′, 0.5, x ′′〉.Note: Both options have the same expected consequence
If the decision maker prefers the certain outcome x̄ , then the decision makerprefers to avoid risks⇒ Risk Averse.
Definition (Risk Aversion)
A decision maker is risk averse if he prefers the expected consequence ofany non-degenerate lottery to that lottery.
c©Jitesh H. Panchal Lecture 06 20 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Risk Aversion - An Illustration
Let the possible consequences of any lottery are represented by x̃ , adecision maker is risk averse if, for all nondegenerate lotteries, utility ofexpected consequence is greater than expected utility of lottery, i.e.,
u[E(x̃)] > E [u(x̃)]
Theorem
A decision maker is risk averse if and only if his utility function is concave.
Corollary
A decision maker who prefers the expected consequence of any 50-50 lottery〈x ′, 0.5, x ′′〉 to the lottery itself is risk averse.
c©Jitesh H. Panchal Lecture 06 21 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Risk Prone
Definition (Risk Prone)
A decision maker is risk prone if he prefers any non-degenerate lottery to theexpected consequence of that lottery.
u[E(x̃)] < E [u(x̃)]
c©Jitesh H. Panchal Lecture 06 22 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Risk Premium
Definition (Risk Premium of a lottery)
The risk premium RP of a lottery x̃ is its expected value (x̄) minus itscertainty equivalent (x̂).
RP(x̃) = x̄ − x̂ = E(x̃)− u−1Eu(x̃)
x
u
u(x2)
u(x1)
u(x)
u(x)̂
x1 x2x^x
Risk
Premium
for
Figure: 4.5 on page 152 (Keeney and Raiffa)
c©Jitesh H. Panchal Lecture 06 23 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Risk Premium and Risk Aversion
Theorem
For increasing utility functions, a decision maker is risk averse if and only ifhis risk premium is positive for all nondegenerate lotteries.
The risk premium is the amount of the attribute that a (risk averse) decisionmaker is willing to “give up” from the average to avoid the risks associatedwith the particular lottery.
c©Jitesh H. Panchal Lecture 06 24 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Measuring Risk Aversion
x
u
x-h x+hx
Risk
Premium
x
u
x-h x+hx
Risk
Premium
Certainty equivalent
u = -3e-x
u’ = 3e-x
u’’ = -3e-x
u = 1-e-x
u’ = e-x
u’’ = -e-x
Certainty equivalent
Figure: 4.9 on page 159 (Keeney and Raiffa)
c©Jitesh H. Panchal Lecture 06 25 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
A Measure of Risk Aversion
Definition (Risk aversion)
The local risk aversion at x , written r(x), is defined by
r(x) = −u′′(x)
u′(x)
r(x) > 0⇒ Risk Averser(x) < 0⇒ Risk Prone
Characteristics of this measure:1 it indicates whether the utility function is risk averse or risk prone2 shows equivalence between two strategically equivalent utility functions
c©Jitesh H. Panchal Lecture 06 26 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Local Risk Aversion - Some Results
Theorem
Two utility functions are strategically equivalent if and only if they have thesame risk-aversion function.
Theorem
If r is positive for all x, then u is concave and the decision maker isrisk-averse.
Theorem
If r1(x) > r2(x) for all x, then π1(x , x̃) > π2(x , x̃) for all x and x̃.
c©Jitesh H. Panchal Lecture 06 27 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Constant, Decreasing and Increasing Risk Aversion
Let x denote a decision maker’s endowment of a given attribute X . Now, addto x a lottery x̃ involving only a small range of X with an expected value ofzero. Let the risk premium of this lottery be π(x , x̃).
What happens to π(x , x̃) as x increases?
Example of decreasingly risk averse: As a person’s assets increase, they areonly willing to pay a smaller risk premium for a given task (as people becomericher, they can better afford to take a specific risk).
c©Jitesh H. Panchal Lecture 06 28 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Constant, Decreasing and Increasing Risk Aversion
Implication
Many of the traditional candidates for a utility function (e.g., exponential andquadratic) are not appropriate for a decreasingly risk averse decision maker.
Theorem
The risk aversion r is constant if and only if π(x , x̃) is a constant function of xfor all x̃ .
Theorem
u(x) ∼ −e−cx ⇔ r(x) ≡ c > 0 (constant risk aversion)u(x) ∼ x ⇔ r(x) ≡ 0 (risk neutrality)
u(x) ∼ e−cx ⇔ r(x) ≡ c < 0 (constant risk proneness)
c©Jitesh H. Panchal Lecture 06 29 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Risk Averse and Risk ProneMeasuring Risk Aversion
Non-monotonic Utility Functions
Theorem
For non monotonic preferences, a decision maker is risk averse [risk prone] ifand only if his utility function is concave [convex].
x
u
x
u
Risk Averse Risk Prone
Figure: 4.18 on page 188 (Keeney and Raiffa)
For non-monotonic utility functions, the certainty equivalent is not necessarilyunique. The risk premium and measure of risk aversion cannot be usefullydefined.
c©Jitesh H. Panchal Lecture 06 30 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
A Procedure for Assessing Utility Functions
1 Preparing for assessment.2 Identifying the relevant quality characteristics.3 Specifying quantitative restrictions.4 Choosing a utility function.5 Checking for consistency.
c©Jitesh H. Panchal Lecture 06 31 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
1. Preparing for Assessment
It is important to acquaint the decision maker with the framework that we usein assessing the utility function.
Educate the decision maker (not bias him/her) and hopefully force him/her tothink about his/her preferences.
c©Jitesh H. Panchal Lecture 06 32 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
2. Identifying the Relevant Quality Characteristics
1 Determine monotonicity2 Determine whether the decision maker is risk averse, risk neutral, or risk
prone3 Determine whether the decision maker is increasingly, decreasingly, or
constantly risk averse.
c©Jitesh H. Panchal Lecture 06 33 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
3. Specifying Quantitative Restrictions
Fixing utilities for a few points on the utility function.Involves determining the certainty equivalents of a few 50-50 lotteries.A five point assessment procedure for utility functions.
u(x0.5) =12
u(x1) +12
u(x0)
u(x0.75) =12
u(x1) +12
u(x0.5)
u(x0.25) =12
u(x0) +12
u(x0.5)
x
u
x0 x0.25 x0.5 x0.75 x1
0
0.25
0.5
0.75
1
Figure: 4.22 on page 195 (Keeney and Raiffa)
c©Jitesh H. Panchal Lecture 06 34 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
4. Choosing a Utility Function
Find a parametric family of utility functions that possesses the relevantcharacteristics (such as risk aversion) previously specified for the decisionmaker.
Using the quantitative assessments, try to find a specific member of thatfamily that is appropriate for the decision maker.
An example of monotonically increasing, decreasingly risk averse utilityfunction:
u(x) = h + k(−e−ax − be−cx ), a, b, c, and k ≥ 0
c©Jitesh H. Panchal Lecture 06 35 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
5. Checking for Consistency
Goal: to uncover discrepancy in a utility function.
c©Jitesh H. Panchal Lecture 06 36 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
Other Considerations for Using the Utility Function
Simplifying the expected utility calculationsParametric/sensitivity analysis
c©Jitesh H. Panchal Lecture 06 37 / 38
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Decision Making under UncertaintyUni-dimensional Utility Theory
Attitudes to RiskA Procedure for Assessing Utility Functions
References
1 Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives:Preferences and Value Tradeoffs. Cambridge, UK, Cambridge UniversityPress. Chapter 4.
2 Clemen, R. T. (1996). Making Hard Decisions: An Introduction toDecision Analysis. Belmont, CA, Wadsworth Publishing Company.
c©Jitesh H. Panchal Lecture 06 38 / 38
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THANK YOU!
c©Jitesh H. Panchal Lecture 06 1 / 1
Decision Making under UncertaintyAlternate Approaches to Risky Choice ProblemMotivation for Using Utility Theory
Uni-dimensional Utility TheoryFundamentals of Utility TheoryQualitative Characteristics of Utility
Attitudes to RiskRisk Averse and Risk ProneMeasuring Risk Aversion
A Procedure for Assessing Utility FunctionsAppendix