expected value, expected utility & the allais and ellsberg paradoxes psychology 466: judgment...

Expected Value, Expected Utility

& the Allais and Ellsberg Paradoxes

Psychology 466: Judgment & Decision Making

Instructor: John Miyamoto 11/10/2015: Lecture 07-1

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Outline

• Expected Value and Expected Utility: What's the Difference?

• Allais Paradox

♦ Common consequence principle (a.k.a. Savage’s independence axiom or the sure-thing principle)

♦ Anticipated regret

♦ Nonlinear probability weighting

• Ellsberg Paradox

Psych 466, Miyamoto, Aut '15 2

Lecture probably ends here

What Is the Expected Value of a Gamble?

Expected Value of a Gamble

Psych 466, Miyamoto, Aut '15 3Expected Value of a Gamble (Cont.): More General Version

Expected Value of a Gamble (cont.)

Psych 466, Miyamoto, Aut '15 4Would It Be Rational to be an Expected Value Maximizer?

Would It Be Rational to be an Expected Value Maximizer?

Expected Value Maximizer: Someone who always prefers the gamble that has the higher expected value.

• Discussions pro and con during the 18th and 19th century.

Rich men wanting to know, which is the better bet?

Psych 466, Miyamoto, Aut '15 5Are You an Expected Value Maximizer? Concrete Example

Are You an Expected Value Maximizer”

• I offer you a choice:

• An expected value maximizer would choose Option 1.

• EV( Option 1 ) > EV( Option 2 )

Psych 466, Miyamoto, Aut '15 6Continuation of this Example

Higher Risk

Lower Risk

Are You an Expected Value Maximizer? (Cont.)


Option 1: 50% chance you win $10,010 50% chance you lose $10,000

Option 2: 50% chance you win $2 50% chance you lose $10

Intuitive argument in favor of Option 2: ♦ The pleasure of winning +$10,010 is smaller in absolute magnitude

than the pain of losing -$10,000.

♦ The worst that can happen with Option 2 is the pain of losing -$10.

♦ What really matters is the subjective value of the outcomes,

+$10,010, +$2, -$10, -$10,000. and not the objective monetary amounts.

Psych 466, Miyamoto, Aut '15 7St. Petersburg Paradox - Introduction

St. Petersburg Paradox

Psych 466, Miyamoto, Aut '15 8Illustration of the St. Petersburg Game

St. Petersburg Paradox (cont.)

Psych 466, Miyamoto, Aut '15 9Expected Value of the St. Petersburg Game

Expected Value of St. Petersburg Game is Infinite!

Psych 466, Miyamoto, Aut '15 10EU of St. Petersburg Game is Infinite


Psych 466, Miyamoto, Aut '15 11Does It Feel Right that the St. Petersburg Game is Infinitely Valuable?



Would you give your total wealth for the opportunity to play the St. Petersburg game?

If you are an expected value maximizer, you should be eager to pay everything you own for the opportunity to play the St. Petersburg Game just once.

Bernoulli's Utility Hypothesis

Expected Value & Expected Utility

• Nobody is an expected value maximizer.

Nobody always prefers the gamble with the higher expected value.

• Daniel Bernoulli (1738):People maximize the expected utility of their choices; not the expected value of their choices.

• Utility of X = subjective value of possessing or experiencing X

• Next 200 years: Economic theory attempts to get rid of the concept of subjective value.

Psych 466, Miyamoto, Aut '15 13Expected Utility Hypothesis: Simplified Mathematical Statement


Expected Utility Hypothesis (Simplified Version)

Let U(X) be the utility of X and let EU(G) be the expected utility of a gamble G.

Expected Utility Hypothesis: There exists a function U such that:

(i) for every pair of gambles G1 and G2 ,

G1 preferred to G2 iff EU(G1) > EU(G2)

(ii) If G = (X1, p; X2, 1-p) is any lottery (for money), then

EU(G) = pU(X1) + (1 - p)U(X2)

• The Expected Utility (EU) Hypothesis is the claim that

a rational agent must satisfy (i) and (ii).

Example: Calculating the EU of Two Gambles

Example: Calculating the EU( Option 1 ) & EU( Option 2 )


Option 1: 50% chance you win $10,010 50% chance you lose $10,000

Option 2: 50% chance you win $2 50% chance you lose $10

• Calculate the Expected Utility of Each Option:

♦ EU( Option 1 ) = (½ 8000) + ( ½ (-10,000) ) = -1,000 Utils

♦ EU( Option 2 ) = (½ 1.8) + ( ½ -2.5 ) = -3.5 Utils

♦ EU( Option 1 ) < EU( Option 2 ).

If you are an EU maximizer, you will choose Option 2.

Psych 466, Miyamoto, Aut '15 15Rationality Does NOT Demand that We Be Expected Value Maximizers

Assume these are the utilities:U($10,010) = +8,000

U($2) = +1.8

U(-$10) = -2.5

U(-$10,000) = -10,000

Rationality Does Not Demand that We Be Expected Value Maximizers

• An insurance company is (approximately) an EV maximizer.An individual person is not an EV maximizer. Why?

• Suppose an insurance company sells 10,000 auto insurance policies for $500/year each.

♦ Insurance company knows that the expected value of each policy is -$420.

♦ An individual auto accident might cost $2,000 to $1,000,000, but they happen rarely.

• Is it rational to buy auto insurance?

• There are many examples where reasonable peopleare NOT EXPECTED VALUE MAXIMIZERS.

Psych 466, Miyamoto, Aut '15 16Transition: Risk Aversion Is Related to Shape of Utility Function

1944 - 1947: The Birth of Expected Utility Theory

• Daniel Bernoulli (1738):People maximize the expected utility of their choices; not the expected value of their choices.

• Utility of X = subjective value of possessing or experiencing X

• Next 200 years: Economic theory attempts to get rid of the concept of subjective value.

• 1944 - 1947: Mathematical work of von Neumann & Morgenstern leads to the discovery of expected utility theory.

Psych 466, Miyamoto, Aut '15 17Preference Axioms - What Are They?

Preference Axioms – What Are They?

• Preference Axioms for EU Theory: A set of assumptions about preference behavior which, if satisfied, imply that a decision maker is an EU maximizer (conforms to EU theory).

• Transitivity is an example of a preference axiom.

• Sure-thing principle (common consequence assumption)is another example of a preference axiom. (To be explained next.)

• Preference axioms can be construed as a normative claim:This is how a rational agent ought to behave.

• Preference axioms can be construed as a descriptive claim:This is how people actually behave.

Psych 466, Miyamoto, Aut '15 18Allais Paradox


Allais Paradox

Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'école Americaine. Econometrica, 21, 503-546.

Choice 1: Option A: Receive 1 million for sure.

Option B: Receive 2.5 million, 10% chance Receive 1 million, 89% chance Receive 0, 1% chance

Choice 2: Option A': Receive 1 million, 11% chance, otherwise $0.Option B': Receive 2.5 million, 10% chance, otherwise $0.

• (Write student responses on the board.)

• Typical choices: Choose A from Choice 1 and choose B' from Choice 2.

Ellsberg Paradox

typical

typical


Tabular Representation of the Allais Choices

Choice 1: Option A: Receive 1 million for sure. ();

Option B: Receive 2.5 million, 10% chance,

Receive 1 million, 89% chance, Receive 0 , 1% chance

Choice 2: Option A': Receive 1 million, 11% chance, otherwise $0.

Option B': Receive 2.5 million, 10% chance, otherwise $0. ()

Chance of Outcome

10% 89% 1%

Choice 1Option A $1 $1 $1

Option B $2.5 $1 $0

Choice 2Option A' $1 $0 $1

Option B' $2.5 $0 $0

Same Slide without the Opaque Grey Rectangles


Allais Paradox Is Based on Common Consequences

Choice 1: Option A: Receive 1 million for sure. ();

Option B: Receive 2.5 million, 10% chance,

Receive 1 million, 89% chance, Receive 0 , 1% chance

Choice 2: Option A': Receive 1 million, 11% chance, otherwise $0.

Option B': Receive 2.5 million, 10% chance, otherwise $0. ()

Chance of Outcome

10% 89% 1%


Option B $2.5 $1 $0


Option B' $2.5 $0 $0

Statement of the Common Consequence Principle

Choice 1: Option A: Receive 1 million for sure. (); Option B: Receive 2.5 million, 10% chance, Receive 1 million, 89% chance, Receive 0 , 1% chance

Choice 2: Option A': Receive 1 million, 11% chance, otherwise $0. Option B': Receive 2.5 million, 10% chance, otherwise $0. ()


Common Consequences Principle(Other Names: Sure-Thing Principle, Savage’s Independence Axiom)

Common Consequence Principle: If two options have the same consequence

given some outcome, then you should ignore this consequence. o Base your choice on the aspects of the options that differ.

Chance of Outcome

10% 89% 1%


Option B $2.5 $1 $0


Option B' $2.5 $0 $0

Allais Paradox Violates the Common Consequence Principle

TypicalChoice

TypicalChoice


Psychological Explanations for the Allais Paradox

Choice 1:

Option A: Receive 1 million for sure.

Option B: Receive 2.5 million, 10% chance

Receive 1 million, 89% chance

Receive 0 , 1% chance

Choice 2:

Option A': Receive 1 million, 11% chance, otherwise $0.

Option B': Receive 2.5 million, 10% chance, otherwise $0.

-----------------------------------------------------------------------

• Class:

Propose psychological explanations for the Allais Paradox.

Anticipated Regret – Explanation for Allais Paradox


Explaining the Allais Paradox in terms of Anticipated Regret

Choice 1: Option A: Receive 1 million for sure. Option B: Receive 2.5 million, 10% chance Receive 1 million, 89% chance Receive 0 , 1% chance

Choice 2: Option A': Receive 1 million, 11% chance, otherwise $0. Option B': Receive 2.5 million, 10% chance, otherwise $0. ------------------------------------------------------------------------

• If you choose option B in choice 1 and get $0, you will feel intense regret.

Choosing option A avoids the possibility of regret.

• If you choose option B' in choice 2 and get $0, you will not feel regret for

your decision because you could have gotten $0 with option A' as well.

What Circumstances Cause Feelings of Regret?

Comment: Decision-Related Emotion

Negative Emotion Positive Emotion

Disappointment(receive bad outcome when you

hoped for a good outcome)

Relief(receive good outcome when you

feared a bad outcome)

Regret(receive a bad outcome when a

different choice would have produced a much better outcome)

Self-Congratulation (?)(receive a good outcome when a

different choice would have produced a much worse outcome)

Psych 466, Miyamoto, Aut '15 25Allais Paradox & Nonlinear Perception of Probability

Why Do People Have Allais-Type Preferences?


• Hypothesis: Choices 1 and 2 differ in terms of anticipated regret.o Regret – comparison between what you have experienced and what you would have

experienced if you made a different choice.o Anticipated Regret – anticipating that a choice will create the possibility of regret.

Clean Version of This Slide

pote

ntia

l reg

ret

no p

oten

tial

reg

ret

Chance of Outcome

10% 89% 1%

Choice 1 Option A $1 $1 $1

Option B $2.5 $1 $0

Choice 2 Option A' $1 $0 $1

Option B' $2.5 $0 $0


Why Do People Have Allais-Type Preferences?

• Hypothesis: Choices 1 and 2 differ in terms of anticipated regret.o Regret – comparison between what you have experienced and what you would have

experienced if you made a different choice.o Anticipated Regret – anticipating that a choice will create the possibility of regret.

Clean Version of This Slide

Chance of Outcome

10% 89% 1%

Choice 1 Option A $1 $1 $1

Option B $2.5 $1 $0

Choice 2 Option A' $1 $0 $1

Option B' $2.5 $0 $0

pote

ntia

l reg

ret

no p

oten

tial

reg

ret

Choice 1: Option A: Receive 1 million for sure, 0% chance of receiving 0 dollars.Option B: Receive 2.5 million, 10% chance

Receive 1 million, 89% chanceReceive 0 , 1% chance

Choice 2: Option A': Receive 1 million, 11% chance, otherwise $0. 89% chance of $0 Option B': Receive 2.5 million, 10% chance, otherwise $0. 90% chance of $0-------------------------------------------------------------------------

• In choice 1-A, the chance of $0 is 0%; in choice 1-B, it is 1%. In choice 2-A', the chance of $0 is 89%; in choice 2-B', it is 90%.

• Psychologically, the difference between a 0% and 1% chance of $0 is greater than the difference between an 89% and 90% chance of $0.


Explaining the Allais Paradox in terms of Nonlinear Perception of Probability

Summary: Explanations of the Allais Paradox

Typical Preferences in the Allais Paradox Violate EU Theory

Choice 1:

EU( Option A ) = (0.10)U( $1 mil ) + (0.89)U( $1 mil ) + (0.01)U( $1 mil )

EU( Option B ) = (0.10)U( $2.5 mil ) + (0.89)U( $1 mil ) + (0.01)U( $0 mil )

Choice 2:

EU( Option A ) = (0.10)U( $1 mil ) + (0.89)U( $0 mil ) + (0.01)U( $1 mil )

EU( Option B ) = (0.10)U( $2.5 mil ) + (0.89)U( $0 mil ) + (0.01)U( $0

mil )Psych 466, Miyamoto, Aut '15 29

Choice 1: Option A: Receive 1 million for sure. (); Option B: Receive 2.5 million, 10% chance, Receive 1 million, 89% chance, Receive 0 , 1% chance

Choice 2: Option A': Receive 1 million, 11% chance, otherwise $0. Option B': Receive 2.5 million, 10% chance, otherwise $0. ()

Time Permitting: Present the Ellsberg Paradox

Time Permitting: Discuss the Ellsberg Paradox

• Who is Daniel Ellsberg?

Psych 466, Miyamoto, Aut '15 30Presentation of the Choices for the Ellsberg Paradox


Tuesday, November 10, 2015: The Lecture Ended Here

expected value, expected utility & the allais and ellsberg paradoxes psychology 466: judgment...

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