how could the expected utility model be so wrong? tom means sjsu department of economics math...

How Could The Expected Utility

Model Be So Wrong?

Tom Means

SJSU Department of Economics

Math Colloquium SJSU

December 7, 2011

The Expected Utility Model.

1 – Decision Making under conditions of uncertainty

Choose option with highest expected value.

2 – Utility versus Wealth. (St. Petersburg paradox)

Flip a fair coin until it lands heads up. You win

2n dollars. E(M) = (1/2)($2) + (1/4)($4) + …. = ∞

3 – Ordinal versus Cardinal Utility

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The Expected Utility Model.

4 – Maximize Expected (Cardinal) Utility.

E[U(M)] = Σ Pi × U(Mi)

(The Theory of Games and Economic Behavior, John von Neumann and Oskar Morgenstern

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Attitudes Toward Risk.

1 – Comparing E[U(M)] with U[E(M)]

Risk-Aversion; E[U(M)] < U[E(M)]

Risk-Neutrality; E[U(M)] = U[E(M)]

Risk-Seeking; E[U(M)] > U[E(M)]

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Risk Aversion; X = { 1-P, $20; P, $0}

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Tom is indifferent between not buying insurance and having an expected utility equal to height e’e, and buying insurance for a premium of $4 and having a certain utility equal to the value of the utility function at an income of $16.

0 Dollars 20

Utility of

dollars U($)

16

U(20)

$4

e

e’18

Risk-preferring

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Kathy will not sell insurance to Tom at a zero price because she prefers a

certain utility equal to height b’b rather than an expected utility equal to height d’d.

0 Dollars 38

Utility of

dollars U($) U(38)

d

d’3618

U(18)

b’

b

.10(18)+.90(38)

Willingness to sell insurance

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Kathy is indifferent between not selling insurance and having a certain utility equal to height b’b, and selling insurance at a premium of $1.50 and having an expected utility equal to height k’k.

0 Dollars 38+1.5=39.5

Utility of

dollars U($) U(38)

k

k’18+1.5=19.5

U(18)

b

b’38

38+π18+π

Measuring Risk Aversion

- U[E(M) – π) = E[U(M)]

- Arrow/Pratt Measure π ≈ -(σ2/2)U’’(M)/U’(M)

- Absolute Risk Aversion r(M) = U’’(M)/U’(M)

- Relative Risk Aversion r(M) = M × r(M)

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Some Observations

- Individuals buy insurance (car, life, fixed rate mortgages, etc) and exhibit risk aversion

- Risk-averse individuals go to casinos.

- Freidman/Savage article

- Ellsberg, Allais, Kahneman/Taversky

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How Could Expected Utility Be wrong?

E[U(M)] = Σ Pi × U(Mi)

Violations of probability rules– Conjunction law

• The Linda problem. P(A & B) > P(A), P(B)• (Daniel Kahneman and Amos Tversky)

– Ambiguity aversion• The Ellsberg Paradox (known vs. ambiguous

distribution)

– Base rates – Nonlinear probability weights– Framing

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Framing

Problem One. Pick A or B

A: X = { P = 1, $30; 1-P = 0, $0}

B: X = {0.80, $45; 0.20, $0}

Framing

Problem Two.

Stage One.

X = { 0.75, $0 and game ends; 0.25, $0 and move to second stage}

Stage Two. Pick C or D.

C: X = { P = 1, $30; 1-P = 0, $0}

D: X = {0.80, $45; 0.20, $0}

Framing

Problem Three. Pick E or F

E: X = {0.25, $30; 0.75, $0}

F: X = {0.20, $45; 0.80, $0}

X + C = E

X + D = F


Violations of probability rules– Constructing values – Absolute or Relative,

Gains versus Losses

Choosing an option to save 200 people out of 600.

Choosing an option where 400 out of 600 people will die.

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The Kahneman-Tversky Value Function

The Benefit of Segregating Gains

The Benefit of Combining Losses

The Silver-Lining Effect and Cash Rebates


Violations of probability rules– The reflection effect. Do people value gains

different than losses? Prospect Theory.

Chose between A or B

A: X = { 1.0, $240; 0.0, $0}

B: X = {0.25, $1000; 0.75, $0}

Chose between C or D

C: X = { 1.0, $-750; 0.0, $0}

D: X = {0.75, $-1000; 0.25, $0}19




Chose between E or F

E: X = { 0.25, $240; 0.75, -$760}

F: X = { 0.25, $250; 0.75, -$750}

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A preferred to B (84%)

D preferred to C (87%)

F preferred to E (100%)

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A(84%) + D (87%) = E

B + C = F

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Violations of probability rules– Scaling of Probabilities.

Chose between A or B

A: X = { 1.0, $6000; 0.0, $0}

B: X = {0.80, $8000; 0.20, $0}

Chose between C or D

C: X = { 0.25, $6000; 0.75, $0}

D: X = { 0.20, $8000; 0.80, $0}

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Violations of probability rules– Scaling Probabilities.

A preferred to B and D preferred to C

E[U(A)] > E[U(B)] implies E[U(C)] > E[U(D)]

E[U(A)]/4 > E[U(B)]/4 and add 0.75U(0) to both sides to show E[U(C)] > E[U(D)]

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How Could The Expected Utility Model Be So Wrong?

Questions/Comments

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how could the expected utility model be so wrong? tom means sjsu department of economics math...

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