1. Expected Value, Expected Utility & the Allais and Ellsberg Paradoxes
Psychology 466: Judgment & Decision Making Instructor: John Miyamoto 11/06/2017: Lecture 07-1
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2. Lecture probably ends here
Outline
Expected Value and Expected Utility: What's the Difference?
Expected utility theory Subjective expected utility theory (Bayesian models)
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
Psych 466, Miyamoto, Aut '17
What Is the Expected Value of a Gamble?

3. Expected Value of a Gamble
Same idea but with a different notation for the gamble.
Psych 466, Miyamoto, Aut '17
Expected Value of a Gamble (Cont.): More General Version

4. Expected Value of a Gamble (cont.)
Psych 466, Miyamoto, Aut '17
Would It Be Rational to be an Expected Value Maximizer?

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 gamblers wanting to know, which is the better bet?
Psych 466, Miyamoto, Aut '17
Are You an Expected Value Maximizer? Concrete Example

6. Are You an Expected Value Maximizer
I offer you a choice:
EV( Option 1 ) > EV( Option 2 )
An expected value maximizer would choose Option 1. Do you prefer Option 1 to 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
Psych 466, Miyamoto, Aut '17

7. Are You an Expected Value Maximizer? (Cont.)
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 these 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.)
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Same Slide Without any Rectangular Screens

10. St. Petersburg Paradox (cont.)
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Expected Value of the St. Petersburg Game

11. Expected Value of St. Petersburg Game is Infinite!
Psych 466, Miyamoto, Aut '17

12. Expected Value of St. Petersburg Game is Infinite!
Does It Feel Right that the St. Petersburg Game is Infinitely Valuable?
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13. 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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14. 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 (but Bernoulli did not know how to measure utility)
Next 200 years: Economic theory attempts to get rid of the concept of subjective value.
Psych 466, Miyamoto, Aut '17
Expected Utility Hypothesis: Simplified Mathematical Statement

15. 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 (simplified version): 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 gamble,
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) with respect to every gamble G, G1 and G2 .
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Example: Calculating the EU of Two Gambles

16. 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
How to calculate the Expected Utility (EU) of each option?
To calculate the EU of each option, we need to assign utilities to the outcomes.
U($10,010) = +500
U(-$10,000) = -700
U(+$2) = +1.5
U(-$10) = -1.7
These are just hypothetical values that represent the relative subjective value of the dollar amounts.
Psych 466, Miyamoto, Aut '17
Calculate the EU of Each Option

17. 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  ( ½  -700 ) = Utils
EU( Option 2 ) = (½  1.5) + ( ½  -1.7 ) = Utils
Assume these are the utilities:
U($10,010) =
U(-$10,000) = -700
U($2) = +1.5
U(-$10) = -1.7
Psych 466, Miyamoto, Aut '17
Same Slide Without Red Boxes and Arrows

18. 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  500) + ( ½  (-700) ) = Utils
EU( Option 2 ) = (½  1.5) + ( ½  -1.7 ) = 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) =
U(-$10,000) = -700
U($2) = +1.5
U(-$10) = -1.7
Psych 466, Miyamoto, Aut '17
Rationality Does NOT Demand that We Be Expected Value Maximizers

19. 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.
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Birth of Expected Utility Theory

20. History of Expected Utility (EU) Theory (outline)
Daniel Bernoulli (1738)
People maximize the expected utility of their choices; not the expected value of their choices.
Expected utility provides an explanation for risk aversion, e.g., explains the St. Petersburg Paradox and the desire to buy insurance. Next 200 years: Economic theory attempts to get rid of the concept of subjective value.
von Neumann & Morgenstern ( )
Publish mathematical foundations for EU theory.
EU theory provides an explanation for risk aversion, e.g., explains the St. Petersburg Paradox and desire to buy insurance.
Expected utility (EU) theory embraced as a foundations for economic theory. EU theory explains how a rational agent ought to decide among risky options. Some claim that it describes how people actually behave in economic decisions.
Psych 466, Miyamoto, Aut '17
Reminder: What is the Expected Utility Hypothesis?

21. ReminderExpected Utility Hypothesis (Simplified Version)
The EU hypothesis asserts that a rational agent always satisfies (i) and (ii) with respect to every gamble G, G1 and G2 .
(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 gamble,
then EU(G) = pU(X1) + (1 - p)U(X2)
If a person satisfies the EU hypothesis, (i) and (ii), then this person is said to be an expected utility maximizer, i.e., someone who always chooses the option with the maximum expected utility.
Psych 466, Miyamoto, Aut '17
Preference Axioms - What Are They?

22. 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.
Transitivity is an example of a preference axiom.
If you prefer A to B and you prefer B to C, then you must prefer A to C.
E.g., I prefer coffee to tea. I prefer tea to a Coca Cola. Therefore I should prefer coffee to a Coca Cola.
Sure-thing principle (common consequence assumption) is another example of a preference axiom. (To be explained later.)
Preference Axioms Can Be Interpreted as a Normative Claim or a Descriptive Claim
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23. 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.
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.
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Allais Paradox

24. 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,
Maurice Allais discovered that people systematically violate an assumption of EU theory.
Allais Paradox = paradoxical pattern of typical choices in Allais-type choice problems.
Allais discovered violations of the common consequence assumption of EU theory.
Present the Choices for the Allais Paradox
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25. Write Down Your Preferences for these Choices
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.
Continuation of this Slide: Record Student Preferences
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26. Class Results for the Allais Paradox Problem
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 preference
typical preference
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EU Analysis of Choice 1

27. Allais Paradox Violates 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 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 )
Psych 466, Miyamoto, Aut '17
EU Analysis of Choice 2

28. Allais Paradox Violates EU Theory
Choice 2: Option A': Receive 1 million, 11% chance, otherwise $ Option B': Receive 2.5 million, 10% chance, otherwise $0. ()
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 '17
Point Out that these Choices have a Common Consequence

29. Allais Paradox Violates 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.
To be consistent with EU Theory:
Choose A in Choice 1 & A' in Choice 2 OR
Choose B in Choice 1 & B' in Choice 2
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 '17
Same Slide with Both Rectangles Displayed

30. Allais Paradox Violates 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 )
Psych 466, Miyamoto, Aut '17
Tabular Representation of the Choices in the Allais Paradox

31. 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 '17

32. 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. ()
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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33. Common Consequences Principle (Other Names: Sure-Thing Principle, the Independence Axiom)
Common Consequence Principle: If two options have the same probability for a given consequence, then you should ignore this consequence when choosing between the options.
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
Psychological Explanations for the Allais Paradox
Psych 466, Miyamoto, Aut '17

34. Why Do People's Preferences Display 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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35. 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 $
In Choice 1, if you choose option B and get $0, you will feel intense regret. Choosing option A avoids the possibility of regret.
In Choice 2, if you choose option B' and get $0, you will not feel regret regardless of how the gamble turns out because you could have gotten $0 with option A' as well.
Emotions Triggered by Decision Outcomes
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36. Emotions Triggered by Decision Outcomes
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Same Slide with Upper Left Quadrant Displayed

37. Emotions Triggered by Decision Outcomes
Disappointment is experienced when you hope for a good outcome but you get a mediocre or bad outcome.
Psych 466,, Miyamoto, Aut '17
Same Slide with Upper Right Quadrant Displayed

38. Emotions Triggered by Decision Outcomes
Relief is experienced when you fear a bad outcome but you get a neutral or good outcome.
Psych 466,, Miyamoto, Aut '17
Same Slide with Lower Left Quadrant Displayed

39. Emotions Triggered by Decision Outcomes
Regret is experienced when you choose Action A instead of Action B, but once you experience the outcome, you realize that you would have been much better off if you had chosen B.
Psych 466,, Miyamoto, Aut '17
Same Slide with Lower Right Quadrant Displayed

40. Emotions Triggered by Decision Outcomes
Self-congratulation is experienced when you choose Action A instead of Action B, and once you experience the outcome, you realize that you would have been much worse off if you had chosen B.
Psych 466,, Miyamoto, Aut '17
Same Slide with All Quadrants Displayed

41. Emotions Triggered by Decision Outcomes
Allais Paradox & Nonlinear Perception of Probability
Psych 466,, Miyamoto, Aut '17 #

42. Why Do People Have Allais-Type Preferences?
Chance of Outcome
10%
89% 1%
Choice 1
Option A $1
Option B
$2.5 $0
Choice 2
Option A'
Option B'
potential regret
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 '17

43. Why Do People Have Allais-Type Preferences?
Chance of Outcome
10%
89% 1%
Choice 1
Option A $1
Option B
$2.5 $0
Choice 2
Option A'
Option B'
potential regret
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.
Explaining the Allais Paradox in terms of Nonlinear Probability Weighting
Psych 466, Miyamoto, Aut '17

44. 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, Receive $ % chance
Option B': Receive 2.5 million, 10% chance, Receive $0, % 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.
Ellsberg Paradox
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45. 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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46. Tuesday, 7 November, 2017: The Lecture Ended Here
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47. Ellsberg Paradox
Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics, 75,
We are going to draw a ball from an urn. The urn contains 30 red balls, and 60 balls that are either blue or yellow, but you do not know the relative proportion of blue and yellow balls. Payoffs are based on the following payoff matrix.
Number of balls  30 balls 60 balls Color  Red Blue Yellow
Choice 1
Option A: Bet on red $1000 $0 $0
Option B: Bet on blue $0 $1000 $0
Choice 2
Option A': Bet on red or yellow $1000 $0 $1000
Option B': Bet on blue or yellow $0 $1000 $1000
Repeat this Slide w-o Rectangles
Psych 466, Miyamoto, Aut '17

48.   Ellsberg Paradox Get Class Responses
Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics, 75,
We are going to draw a ball from an urn. The urn contains 30 red balls, and 60 balls that are either blue or yellow, but you do not know the relative proportion of blue and yellow balls. Payoffs are based on the following payoff matrix.
Number of balls  30 balls 60 balls Color  Red Blue Yellow
Choice 1
Option A: Bet on red $1000 $0 $0
Option B: Bet on blue $0 $1000 $0
Choice 2
Option A': Bet on red or yellow $1000 $0 $1000
Option B': Bet on blue or yellow $0 $1000 $1000
Typical choices: Choose A from Choice 1 and choose B' from Choice 2.
Get Class Responses  
Ellsberg Paradox and the Common Consequence Principle
Psych 466, Miyamoto, Aut '17

49. Ellsberg Paradox & the Common Consequence Principle (Other Names: Sure-Thing Principle, Savage’s Independence Axiom)
Common Consequence Principle: If two options have the same consequence given some outcome, you should ignore this common consequence. You should base your choice on the aspects of the options that differ.
Number of Balls (60  X  0)
30 Red
X Blue
60 - X Yellow
Choice 1
Option A
$1000 $0
Option B
Choice 2
Option A'
Option B'
Psych 466, Miyamoto, Aut '17
Allais Paradox Violates the Common Consequence Principle

50. Ellsberg Paradox (cont.)
Common consequence principle says:
you should prefer A to B and A’ to B’ OR you should prefer B to A and B’ to A’.
Does this feel right? If not, why not?
Number of Balls (60  X  0)
30 Red
X Blue
60 - X Yellow
Choice 1
Option A
$1000 $0
Option B
Choice 2
Option A'
Option B'
Psych 466, Miyamoto, Aut '17
Why Do the Allais & Ellsberg Paradoxes Occur?

51. Ellsberg Paradox & Ambiguous Probabilities
Common consequence principle says:
you should prefer A to B and A’ to B’ OR you should prefer B to A and B’ to A’.
Number of balls  30 balls 60 balls Color  Red Blue Yellow
Choice 1
Option A: Bet on red $1000 $0 $0
Option B: Bet on blue $0 $1000 $0
Choice 2
Option A': Bet on red or yellow $1000 $0 $1000
Option B': Bet on blue or yellow $0 $1000 $1000
Define concept of an ambiguous probability.
unambiguous
ambiguous
ambiguous
unambiguous
Psych 466, Miyamoto, Aut '17
Summary re Allais and Ellsberg Paradoxes

52. Summary
Allais Paradox & Ellsberg Paradox: Strong evidence that expected utility (EU) theory is not descriptively adequate.
Hypotheses that explain the Allais Paradox:
People’s decisions are influenced by anticipated regret.
People’s perception of probability is nonlinear.
Hypothesis that explain the Ellsberg Paradox:
People tend to avoid ambiguous probabilities in the domain of gains.
People tend to seek ambiguous probabilities in the domain of losses.
Psych 466, Miyamoto, Aut '17
Where We Are Headed in the Next Set of Lectures - END

53. Where We Are Headed in the Next Set of Lectures
Prospect Theory
Reflection effect
Framing effects that result from reflection effects
Mental accounting
Psych 466, Miyamoto, Aut '17
END