lecture 06 single attribute utility theory · 2014. 9. 18. · we can show that if the utility...

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



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



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




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.





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)





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.





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.





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.




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





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






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′′






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?





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).





Qualitative Characteristics of Utility

1 Monotonicity2 Certainty equivalence3 Strategic equivalence





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)]





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)





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





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





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.




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.





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.





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)]





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






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.





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






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





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.





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).





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)





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


For non-monotonic utility functions, the certainty equivalent is not necessarilyunique. The risk premium and measure of risk aversion cannot be usefullydefined.




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.




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.




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.




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





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




5. Checking for Consistency

Goal: to uncover discrepancy in a utility function.




Other Considerations for Using the Utility Function

Simplifying the expected utility calculationsParametric/sensitivity analysis




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.


THANK YOU!


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

lecture 06 single attribute utility theory · 2014. 9. 18. · we can show that if the utility...

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