allais paradox, ellsberg paradox, and the common consequence principle then: introduction to...

Allais Paradox, Ellsberg Paradox,and the Common Consequence PrincipleThen: Introduction to Prospect Theory

Psychology 466: Judgment & Decision MakingInstructor: John Miyamoto

11/12/2015: Lecture 07-2

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Outline

• Review common consequence principle• Ellsberg Paradox• Introduction to Prospect Theory

♦ Risk attitude♦ Reflection effects♦ Framing effects♦ Mental accounting

Psych 466, Miyamoto, Aut '15 2

Lecture probably ends here

Allais Paradox & Common Consequence Principle


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 $1 $1

Option B $2.5 $1 $0

Choice 2 Option A' $1 $0 $1

Option B' $2.5 $0 $0

Statement of the Common Consequence Principle



Chance of Outcome

10% 89% 1%


Option B $2.5 $1 $0


Option B' $2.5 $0 $0


Common Consequence Principle: If two options have the same consequence given some outcome, then you should ignore this consequence.

♦ EU(A) > EU(B) if and only if EU(A') > EU(B')




Expected Utility Calculation for the Allais Paradox

Chance of Outcome

10% 89% 1%

Choice 1 EU(A) =

EU(B) =

Choice 2 EU(A‘) =

EU(B’) =




People's choices with the Allais Problem violate the common concequence principle:

Typical Choices Imply: EU(A) > EU(B) and EU(A') < EU(B')



Same Slide with One More Bullet Point

Common consequence principle imples that both inequalities should

be > or both should be <.



People's choices with the Allais Problem violate the common concequence principle:

Typical Choices Imply: EU(A) > EU(B) and EU(A') < EU(B')

Typical choices in the Allais Paradox Problem violate EU Theory.EU Theory is not descriptively valid.



Ellsberg Paradox

Next: The Ellsberg Paradox

• Who is Daniel Ellsberg?

Psych 466, Miyamoto, Aut '15 8Presentation of the Choices for the Ellsberg Paradox


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 $100 $0 $0

Option B: Bet on blue $0 $100 $0

Choice 2

Option A': Bet on red or yellow$100 $0 $100

Option B': Bet on blue or yellow $0 $100 $100

Repeat this Slide w-o Rectangles


Ellsberg ParadoxEllsberg, 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



Choice 2



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

Ellsberg Paradox and the Common Consequence Principle

Get

Cla

ss R

espo

nses

Number of Balls (60 X 0)

30 Red X Blue 60 - X Yellow


Option B $0 $100 $0


Option B' $0 $100 $100


Ellsberg Paradox & the Common Consequence Principle(a.k.a. 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.

o Base your choice on the aspects of the options that differ.

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’.




Option B $0 $100 $0


Option B' $0 $100 $100

Why Do the Allais & Ellsberg Paradoxes Occur?

Does this feel right? If not, why not?



Choice 1 EU(A) =

EU(B) =

Choice 2 EU(A’) =

EU(B’) =


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’.

Why Do the Allais & Ellsberg Paradoxes Occur?


Ellsberg Paradox & Ambiguous Probabilities

• Ellsberg Paradox occurs because some probabilities are more ambiguous than other probabilities.

Number of balls 30 balls 60 ballsColor Red Blue Yellow

Choice 1



Choice 2



Summary of Allais & Ellsberg Paradoxes

ambiguous

unambiguous

ambiguous

unambiguous


Summary

• Allais Paradox & Ellsberg Paradox:o Strong evidence that expected utility (EU) theory is not descriptively adequate.o Both paradoxes violate a key axiom of EU theory:

The Substitution Axiom, a.k.a., the common consequence principle.

• Hypotheses that explain the Allais Paradox:o People’s decisions are influenced by anticipated regret.o People’s perception of probability is nonlinear.

• Hypothesis that explain the Ellsberg Paradox:o People tend to avoid ambiguous probabilities in the domain of gains.o People tend to seek ambiguous probabilities in the domain of losses.

Where We Are Headed Next


Where We Are Headed Next

• Prospect Theory

• Reflection effect

• Framing effects that result from reflection effects

• Mental accounting

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

• Next: A closer look at risk aversion.

Risk Aversion - Graphical Explanation

Next: Risk Aversion Is Related to the Shape of the Utility Function

• This is a key idea of EU theory.• General idea of risk aversion:

People prefer a sure-thing to a risk, even if the risk has a somewhat greater expected value.

Psych 466, Miyamoto, Aut '15 17Graph of Utility Function

Next: I will explain this idea.

18Psych 466, Miyamoto, Aut '15

Risk Attitude

Risk averse action: A person chooses a sure-thing X over a gamble G where X is less than the expected value of G.

o A risk averse person prefers a sure win of $500

over a 50-50 gamble for $1,010 or $0.

(Note: Expected value of gamble = $505)

Risk seeking action: A person chooses a gamble G over a sure thing X where the expected value of G is less than X.

o A risk seeking person prefers a 50-50 gamble for $1000 or $0

over a sure win of $505.

(Note: Expected value of gamble = +$500)

Examples of Risk Aversion & Risk Seeking


Examples of Risk Aversion & Risk Seeking

• Whenever you buy insurance, you are acting in a risk averse way.o The cost of car insurance is a sure loss that is a bigger loss than

the expected value of the gamble of driving an uninsured car.

• Whenever you play a gamble with a professional casino or state lottery, you are acting in a risk seeking way.

o The cost of the lottery ticket is greater than the expected value of the lottery ticket.

Risk Attitude is Related to Shape of Utility Function

Graphical Explanation of Why($10,010, 0.50; -$10,000, 0.50) Is Not Desirable


The blue dots indicate the utility of -$10,000 and +$10,010.

Graph Showing Expected Utility of Gamble

Fig. 1

EU(gamble)



The blue 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

Fig. 1 Fig. 2



The intermediate red dot indicates the expected utility of the gamble.

The green dot indicates the sure-thing equivalent of the gamble. (Sure-thing equivalent is a.k.a. the certainty equivalent.)

Notice that the green dot is to the left of $0. The decision maker regards this gamble to be equal in preference to a certain loss of money (about -$3,000).

Certainty Equivalent of a Gamble

Fig. 2

EU(gamble)


Assuming EU Theory, risk aversion is related to the curvature of the utility function

• The certainty equivalent (CE) is the sure-thing that is equally desirable as a gamble.

• Panel A shows the CE of an even-chance gamble for $0 and $20,000 when the utility function is concave.

o Because the utility function is risk averse, the CE is pulled to the left (lower values) of the expected value.

o Therefore this person will tend to avoid risks (risk averse).

Same Analysis with respect to Risk Seeking Utility FunctionU

tility

$ in Thousands

This figure shows why a concave utility function implies risk averse choices.

Expected Value




• Panel D shows the CE of an even-chance gamble for $0 and $20,000 when the utility function is convex.

o Because the utility function is risk seeking, the CE is pulled to the right (higher values) of the EV of the gamble.

o Therefore this person will tend to prefer risks over comparable sure-things (risk seeking). U

tility

$ in ThousandsIntroduction to Prospect Theory

This figure shows why a convex utility function implies risk seeking choices.

Expected Value


Prospect Theory

• Proposed in 1979 by Daniel Kahneman & Amos Tversky.

• Attempts to explain patterns of human preference under riskthat are not explained by expected utility (EU) theory.

• Kahneman received the Nobel Prize in Economics in 2001.Prospect theory was a major part of the work for which the Nobel was awarded.

• This lecture will present ideas that are based on prospect theory.This lecture will not try to explain the details of prospect theory.

Key Ideas of Prospect Theory

Key Ideas of Prospect Theory

• Prospect theory is a competitor to EU theory – it is a theory of preference under risk (preference between gambles)

• Altered concept of risk attitude (risk aversion and risk seeking)• Loss aversion• Nonlinear probability weighting• Relativity of value

Coding outcomes as losses and gainsFraming effects & mental accounting

• Next: We need to understand the concept of a risk attitude (risk aversion, risk seeking).

♦ Key contribution of Prospect Theory: It predicts the conditions under which people will be more risk averse or more risk seeking.

Psych 466, Miyamoto, Aut '15 26Risk Attitude

Kahneman & Tversky’s Insights into Risk Attitude

• Important Idea #1:People tend to risk averse for gains and risk seeking for losses.

• Even More Important Idea #2:These concepts, risk aversion and risk seeking, apply to gains and losses, not to states of wealth.

Psych 466, Miyamoto, Aut '15 27Reflection Effect Example


Reflection Effect - Example

Choice 1: Which would you prefer?Option A: .80 chance to win $4,000.Option B: 1.0 chance to win $3,000.

Choice 2: Which would you prefer?Option A’: .80 chance to lose $4,000. Option B’: 1.0 chance to lose $3,000.

Expected value of Choice 1, Option A = +$3,200Expected value of Choice 2, Option A’ = -$3,200

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

• This pattern of responses shows that people are risk averse for gains and risk seeking for losses. (This statement is generally true, but there are exceptions to it, to be discussed later.)

typical

typical

Reflection Effect – Simplified Version

Gamblingfor Gains

Gamblingfor Losses


Reflection Effect – The Simple (Slightly False) Version

• Reflection Effect: People are generally risk averse for gains and risk seeking for losses. (Not quite correct, but will be corrected shortly)

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

• Risk Averse for Gains: If all outcomes are zero or positive, people prefer sure things over gambles that have a slightly higher expected value.

o Example: People prefer $3,000 for sure to an 80% chance of $4,000, otherwise $0.

• Risk Seeking for Losses: If all outcomes are zero or negative, people prefer a gamble over a sure loss that is somewhat higher than the expected value of the gamble.

o Example: People prefer an 80% chance of -$4,000, otherwise $0, to -$3,000 for sure.

Examples of Reflection Effect


Examples of Reflection Effect• Bettors at horse track bet on long shots at the end of the day

(many are in a state of trying to recoup losses). • Palestinian political behavior?

Reflection Effect – Fourfold Pattern


Reflection Effect (More Accurate Version) – the Fourfold Pattern

Small Probabilities Medium to Large Probabilities

Gains Risk-Seeking Risk Averse

Losses Risk-Averse Risk-Seeking

• Definition: The reflection effect is the finding that preferences switch from risk averse to risk seeking if we change the outcomes from gains or losses.

o The direction of the change, from risk averse to risk seeking or from risk seeking to risk averse, depends on the size of the probabilities.

Risk Aversion & Utility Curvature


Reflection Effect (More Accurate Version) – the Fourfold Pattern


GainsRisk-Seeking

(buy lottery tickets)Risk Averse

(playing gambles for gains)

LossesRisk-Averse

(buy insurance)Risk-Seeking

(playing gambles for losses)

Examples:o People are risk seeking when they buy lottery tickets

(small probability of large gain). o People are risk averse when they buy home insurance

(small probability of large loss).o People are risk averse w.r.t. gambles for large gains.

(Prefer $5,000 for sure over 50-50 chance of $10,010 or $0)o People are risk seeking w.r.t. gambles for large losses.

(Prefer 75% chance of -$1,000, otherwise $0 over lose -$700

Risk Aversion & Utility Curvature




• Panel A shows the CE of an even-chance gamble for $0 and $20,000 when the utility function is concave.

o Because the utility function is risk averse, the CE is pulled to the left (lower values) of the expected value.

o Therefore this person will tend to avoid risks (risk averse).

Same Analysis with respect to Risk Seeking Utility FunctionU

tility

$ in Thousands

This figure shows why a concave utility function implies risk averse choices.

Expected Value




• Panel D shows the CE of an even-chance gamble for $0 and $20,000 when the utility function is convex.

o Because the utility function is risk seeking, the CE is pulled to the right (higher values) of the EV of the gamble.

o Therefore this person will tend to prefer risks over comparable sure-things (risk seeking). U

tility

$ in ThousandsValue & Prob Weighting Fns of PT

This figure shows why a convex utility function implies risk seeking choices.

Expected Value


Reflection Effect – The Fourfold Pattern of Risk Attitude


Gains Risk-Seeking Risk Averse

Losses Risk-Averse Risk-Seeking

Value Function in Prospect Theory

GainsLosses

Val

ue

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Stated Probability

Pro

babi

lity

Wei

ght

Probability Weighting Functionin 1979 Prospect Theory

Asian Disease Problem

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