1. The Allais Paradox & the Ellsberg Paradox
Psychology 466: Judgment & Decision Making Instructor: John Miyamoto 11/09/2017: Lecture 07-2
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2. Outline Brief review of expected utility theory The Allais Paradox
What is the Allais Paradox?
What is the psychological cause of the Allais Paradox?
The Ellsberg Paradox
What is the Ellsberg Paradox?
What is the psychological cause of the Ellsberg Paradox?
EU model of risk aversion
Psych 466, Miyamoto, Aut ‘17
Outline of History of Expected Utility Theory

3. 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?

4. 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 a 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).
Preference Axioms for EU Theory - What Are They?
Psych 466, Miyamoto, Aut '17

5. 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
Psych 466, Miyamoto, Aut '17

6. 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

7. 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

8. 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
Psych 466, Miyamoto, Aut '17

9. 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
Psych 466, Miyamoto, Aut '17

10. 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

11.   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

12. 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

13. 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
Ambiguous versus Sharp Probabilities - What Are They?

14. "Ambiguous" versus "Sharp" Probabilities
Sharp Probability: The chances of winning or losing are very clear.
E.g., This gamble gives you a 10% chance of winning $50, 90% chance of losing $25
E.g., This medical treatment gives you a 2% chance of operative mortalilty, and a 98% chance of full recovery.
Psych 466, Miyamoto, Aut '17
Same Slide with Addition of Definition of Ambiguous Probability

15. "Ambiguous" versus "Sharp" Probabilities
Sharp Probability: The chances of winning or losing are very clear.
E.g., This gamble gives you a 10% chance of winning $50, 90% chance of losing $25
E.g., This medical treatment gives you a 2% chance of operative mortalilty, and a 98% chance of full recovery.
Ambiguous Probability: The chances of winning or losing are not clear.
E.g., This gamble gives you a small chance of winning $50, and a large chance of losing $25
E.g., This medical treatment gives you a very small chance of operative mortalilty, and very good chance of full recovery.
Same Slide with Addition of Hypothesis that People Avoid Ambiguous Probabilities
Psych 466, Miyamoto, Aut '17

16. "Ambiguous" versus "Sharp" Probabilities
Sharp Probability: The chances of winning or losing are very clear.
E.g., This gamble gives you a 10% chance of winning $50, 90% chance of losing $25
E.g., This medical treatment gives you a 2% chance of operative mortalilty, and a 98% chance of full recovery.
Ambiguous Probability: The chances of winning or losing are not clear.
E.g., This gamble gives you a small chance of winning $50, and a large chance of losing $25
E.g., This medical treatment gives you a very small chance of operative mortalilty, and very good chance of full recovery.
Hypothesis that explains the Ellsberg Paradox:
People tend to avoid ambiguous probabilities in the domain of gains.
Psych 466, Miyamoto, Aut '17

17. 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
Point out that ambiguity switches in the two choices. 
unambiguous
ambiguous
ambiguous
unambiguous 
Psych 466, Miyamoto, Aut '17
Neuropsych Evidence for Sensitivity to Ambiguous Probability

18. fMRI Study of Response to Risk & Ambiguity (Huettel et al., 2006)
Four Trial Types Sharp Probabilities Ambiguous Probabilities .
The study compares brain activity in 3 brain areas:
pIFS: Posterior Inferior Frontal Sulcus
aINS: Anterior Insular Cortex
pPAR: Posterior Parietal Cortex
Key findings pertain to fMRI images acquired during the decision stage of the task.
Psych 466, Miyamoto, Aut '17
Same Slide - Focus on Sharp Probabilities

19. fMRI Study of Response to Risk & Ambiguity (Huettel et al., 2006)
Four Trial Types Sharp Probabilities Ambiguous Probabilities .
The study compares brain activity in 3 brain areas:
pIFS: Posterior Inferior Frontal Sulcus
aINS: Anterior Insular Cortex
pPAR: Posterior Parietal Cortex
Key findings pertain to fMRI images acquired during the decision stage of the task.
Psych 466, Miyamoto, Aut '17
Same Slide - Focus on Ambiguous Probabilities

20. fMRI Study of Response to Risk & Ambiguity (Huettel et al., 2006)
Four Trial Types Sharp Probabilities Ambiguous Probabilities .
The study compares brain activity in 3 brain areas:
pIFS: Posterior Inferior Frontal Sulcus
aINS: Anterior Insular Cortex
pPAR: Posterior Parietal Cortex
Key findings pertain to fMRI images acquired during the decision stage of the task.
Psych 466, Miyamoto, Aut '17
Same Slide - Define the Dependent Variable

21. fMRI Study of Response to Risk & Ambiguity (Huettel et al., 2006)
Sharp Probabilities Ambiguous Probabilities
The study compares brain activity in 3 brain areas:
pIFS: Posterior Inferior Frontal Sulcus
aINS: Anterior Insular Cortex
pPAR: Posterior Parietal Cortex
Key findings pertain to fMRI images acquired during the decision stage of the task.
Does the dependent variable show effects in the 3 target brain areas?
A- (ambiguous), R- (risky), -C (certain), -R (risky)
Dependent Variable
(AC + AR) – (RC + RR)
Psych 466, Miyamoto, Aut '17
Results of Huettel et al.

22. Results for Huettel et al. (2006)
pIFS: Posterior Inferior Frontal Sulcus
aINS: Anterior Insular Cortex
pPAR: Posterior Parietal Cortex
Ambiguity Preference: Measure of how often the subject chose the ambiguous option in a choice between one gamble with ambiguous probabilities and one gamble with sharp probabilities.
Risk Preference: Measure of how often the subject chose the riskier option in a choice between a low risk option and a high risk option.
Double Dissociation
pIFS activity correlates with behavioral measure of ambiguity preference.
pPAR activity correlates with behavioral measure of risk preference.
Summary re Allais & Ellsberg Paradoxes
Psych 466, Miyamoto, Aut '17

23. Summary Allais Paradox & Ellsberg Paradox:
Strong evidence that expected utility (EU) theory is not descriptively adequate.
Both paradoxes show that people sometimes violate the common consequence principle (a.k.a. the sure-thing principle)
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.
What is the brain activity that corresponds to disappointment, regret, and the perceived ambiguity of events?
Where We Are Headed in the Next Set of Lectures
Psych 466, Miyamoto, Aut '17

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

26. Time Permitting: Risk Aversion and the Shape of the Utility Function
According to EU theory:
A person's willingness to choose risky options is strongly influenced by the shape of the person's utility function for money.
Psych 466, Miyamoto, Aut '17

27. Possible Shapes of the Utility Function for Money
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28. Thursday, 9 November, 2017: The Lecture Ended Here
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29. Graphical Explanation of Why ($10,010, 0. 50; -$10,000, 0
Graphical Explanation of Why ($10,010, 0.50; -$10,000, 0.50) Is Not Desirable
Fig. 1
see ‘d:\r\notes\risk averse.docm’ for the R-code.
The red dots indicate the utility of -$10,000 and +$10,000.
Graph Showing Expected Utility of Gamble
Psych 466, Miyamoto, Aut '17

30. Graphical Explanation of Why ($10,010, 0. 50; -$10,000, 0
Graphical Explanation of Why ($10,010, 0.50; -$10,000, 0.50) Is Not Desirable
Fig. 1
Fig. 2
see ‘e:\r\notes\risk averse.docm’ for the R-code.
EU(gamble)
The red dots indicate the utility of -$10,000 and +$10,000.
The intermediate red dot indicates the expected utility of the gamble.
Graph Showing Sure-Thing Equivalent of the Gamble
Psych 466, Miyamoto, Aut '17

31. Graphical Explanation of Why ($10,010, 0. 50; -$10,000, 0
Graphical Explanation of Why ($10,010, 0.50; -$10,000, 0.50) Is Not Desirable
Fig. 2
see ‘e:\r\notes\risk averse.docm’ for the R-code.
EU(gamble)
The intermediate red dot indicates the expected utility of the gamble.
Graph Showing Sure-Thing Equivalent of the Gamble
Psych 466, Miyamoto, Aut '17

32. Graphical Explanation of Why ($10,010, 0. 50; -$10,000, 0
Graphical Explanation of Why ($10,010, 0.50; -$10,000, 0.50) Is Not Desirable
Fig. 2
Fig. 3
see ‘e:\r\notes\risk averse.docm’ for the R-code.
EU(gamble)
The intermediate red dot indicates the expected utility of the gamble.
The vertical dotted line at ≈$2.5 indicates the sure-thing equivalent of the gamble.
Graph Showing Sure-Thing Equivalent of the Gamble
Psych 466, Miyamoto, Aut '17

33. Certainty Equivalents for Risk Averse & Risk Seeking Utility
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34. Psych 466, Miyamoto, Aut '17

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