The Allais Paradox & the Ellsberg Paradox Psychology 466: Judgment & Decision Making Instructor: John Miyamoto 11/09/2017: Lecture 07-2 Note: This Powerpoint presentation may contain macros that I wrote to help me create the slides. The macros aren’t needed to view the slides. You can disable or delete the macros without any change to the presentation.

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

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 (1944 - 1947) 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?

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

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% 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

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

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

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, 10% chance Receive 1 million, 89% chance Receive 0 , 1% chance Choice 2: Option A': Receive 1 million, 11% chance, Receive $0. 89% chance Option B': Receive 2.5 million, 10% chance, Receive $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. Ellsberg Paradox Psych 466, Miyamoto, Aut '17

Time Permitting: Discuss the Ellsberg Paradox Who is Daniel Ellsberg? * Time magazine cover: July 5, 1971. 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 1973. 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

Ellsberg Paradox Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics, 75, 643-649. 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

  Ellsberg Paradox Get Class Responses Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics, 75, 643-649. 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

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

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?

"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

"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

"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

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

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

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

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

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.

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

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

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

Psych 466, Miyamoto, Aut '17

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

Possible Shapes of the Utility Function for Money Psych 466, Miyamoto, Aut '17

Thursday, 9 November, 2017: The Lecture Ended Here Psych 466, Miyamoto, Aut '17

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

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

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

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

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

Psych 466, Miyamoto, Aut '17

Set Up for Instructor Turn off your cell phone. Close web browsers if they are not needed. Classroom Support Services (CSS), 35 Kane Hall, 206-543-9900 If the display is odd, try setting your resolution to 1024 by 768 Run Powerpoint. For most reliable start up: Start laptop & projector before connecting them together If necessary, reboot the laptop Psych 466, Miyamoto, Aut '17