1. 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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2. Lecture probably ends here
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
Lecture probably ends here
What Is the Expected Value of a Gamble?
Psych 466, Miyamoto, Aut '15

3. Expected Value of a Gamble
Expected Value of a Gamble (Cont.): More General Version
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4. Expected Value of a Gamble (cont.)
Would It Be Rational to be an Expected Value Maximizer?
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5. 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?
Are You an Expected Value Maximizer? Concrete Example
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6. 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 )
Higher Risk
Lower Risk
See ‘e:\p466\nts\stpetersburg.paradox.docm’ for the equations that are used as graphics on this slide.
Continuation of this Example
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7. Are You an Expected Value Maximizer? (Cont.)
I offer you a choice:
Option 1: 50% chance you win $10, % chance you lose $10,000
Option 2: 50% chance you win $ % 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, and not the objective monetary amounts.
St. Petersburg Paradox - Introduction
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8. St. Petersburg Paradox Illustration of the St. Petersburg Game
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9. St. Petersburg Paradox (cont.)
Expected Value of the St. Petersburg Game
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10. Expected Value of St. Petersburg Game is Infinite!
EU of St. Petersburg Game is Infinite
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11. Expected Value of St. Petersburg Game is Infinite!
Does It Feel Right that the St. Petersburg Game is Infinitely Valuable?
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12. Expected Value of St. Petersburg Game is Infinite!
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
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13. 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.
Expected Utility Hypothesis: Simplified Mathematical Statement
Psych 466, Miyamoto, Aut '15

14. 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) = pU(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
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15. Example: Calculating the EU( Option 1 ) & EU( Option 2 )
I offer you a choice:
Option 1: 50% chance you win $10, % chance you lose $10,000
Option 2: 50% chance you win $ % 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 ) = Utils
EU( Option 1 ) < EU( Option 2 ). If you are an EU maximizer, you will choose Option 2.
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
Psych 466, Miyamoto, Aut '15

16. 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 people are NOT EXPECTED VALUE MAXIMIZERS.
Transition: Risk Aversion Is Related to Shape of Utility Function
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17. 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.
: Mathematical work of von Neumann & Morgenstern leads to the discovery of expected utility theory.
Preference Axioms - What Are They?
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18. 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.
Allais Paradox
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19. Allais Paradox Choice 1: Option A: Receive 1 million for sure.
Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'école Americaine. Econometrica, 21,
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.
typical
typical
Ellsberg Paradox
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20. 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 $ Option B': Receive 2.5 million, 10% chance, otherwise $0. ()
Chance of Outcome
10%
89% 1%
Choice 1
Option A $1
Option B
$2.5 $0
Choice 2
Option A'
Option B'
Same Slide without the Opaque Grey Rectangles
Psych 466, Miyamoto, Aut '15

21. 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 $ Option B': Receive 2.5 million, 10% chance, otherwise $0. ()
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 $ Option B': Receive 2.5 million, 10% chance, otherwise $0. ()
Chance of Outcome
10%
89% 1%
Choice 1
Option A $1
Option B
$2.5 $0
Choice 2
Option A'
Option B'
Statement of the Common Consequence Principle
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22. 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.
Base your choice on the aspects of the options that differ.
Chance of Outcome
10%
89% 1%
Choice 1
Option A $1
Option B
$2.5 $0
Choice 2
Option A'
Option B'
Typical Choice
Typical Choice
Allais Paradox Violates the Common Consequence Principle
Psych 466, Miyamoto, Aut '15

23. 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 $
Class: Propose psychological explanations for the Allais Paradox.  
Anticipated Regret – Explanation for Allais Paradox
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24. 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 $
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?
Psych 466, Miyamoto, Aut '15

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

26. Why Do People Have Allais-Type Preferences?
potential regret
Chance of Outcome
10%
89% 1%
Choice 1
Option A $1
Option B
$2.5 $0
Choice 2
Option A'
Option B'
no potential regret
Hypothesis: Choices 1 and 2 differ in terms of anticipated regret.
Regret – comparison between what you have experienced and what you would have experienced if you made a different choice.
Anticipated Regret – anticipating that a choice will create the possibility of regret.
Clean Version of This Slide
Psych 466, Miyamoto, Aut '15

27. Why Do People Have Allais-Type Preferences?
potential regret
Chance of Outcome
10%
89% 1%
Choice 1
Option A $1
Option B
$2.5 $0
Choice 2
Option A'
Option B'
no potential regret
Hypothesis: Choices 1 and 2 differ in terms of anticipated regret.
Regret – comparison between what you have experienced and what you would have experienced if you made a different choice.
Anticipated Regret – anticipating that a choice will create the possibility of regret.
Clean Version of This Slide
Psych 466, Miyamoto, Aut '15

28. Explaining the Allais Paradox in terms of Nonlinear Perception of Probability
Choice 1: Option A: Receive 1 million for sure, 0% chance of receiving 0 dollars. Option B: Receive 2.5 million, % chance Receive 1 million, % chance Receive 0 , % chance
Choice 2: Option A': Receive 1 million, 11% chance, otherwise $ % chance of $0 Option B': Receive 2.5 million, 10% chance, otherwise $0. 90% chance of $
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.
Summary: Explanations of the Allais Paradox
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29. Typical Preferences in the Allais Paradox Violate EU Theory
Choice 1: Option A: Receive 1 million for sure. (); Option B: Receive 2.5 million, 10% chance, Receive 1 million, 89% chance, Receive 0 , % chance
Choice 2: Option A': Receive 1 million, 11% chance, otherwise $ Option B': Receive 2.5 million, 10% chance, otherwise $0. ()
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 )
Time Permitting: Present the Ellsberg Paradox
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30. Time Permitting: Discuss the Ellsberg Paradox
Who is Daniel Ellsberg?
* Time magazine cover: July 5, Article: "The [Vietnam] War Exposés: Battle Over The Right to Know."
* Book cover: "Most Dangerous: Daniel Ellsberg and the Secret History of the Vietnam War," by Steve Sheinkin, 2015.
* Daniel Ellsberg, outside a federal courthouse in 1971, faced 12 felony counts as a result of his leak of the Pentagon Papers; the charges were dismissed in Credit Donal F. Holway/The New York Times. From June 7, 2011 article in the New York Times.
* Ellsberg being arrested at 3/20/2011 protest over treatment of U.S. military prisoner, Bradley (now, Chelsea) Manning.
Presentation of the Choices for the Ellsberg Paradox
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31. Tuesday, November 10, 2015: The Lecture Ended Here
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