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 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

The Expected Utility Model. 4 – Maximize Expected (Cardinal) Utility 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

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)]

Risk Aversion; X = { 1-P, $20; P, $0} Utility of dollars U($) U(20) e e’ 18 16 $4 Dollars 20 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.

Risk-preferring Utility of dollars U($) b U(38) d .10(18)+.90(38) 36 .10(18)+.90(38) 18 U(18) Dollars 38 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.

Willingness to sell insurance Utility of dollars U($) U(38) k k’ b b’ 38 18+1.5=19.5 U(18) Dollars 38+1.5=39.5 18+π 38+π 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.

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)

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

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

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

How Could Expected Utility Be wrong? 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.

The Kahneman-Tversky Value Function

The Benefit of Segregating Gains

The Benefit of Combining Losses

The Silver-Lining Effect and Cash Rebates

How Could Expected Utility Be wrong? 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}

How Could Expected Utility Be wrong? Violations of probability rules The reflection effect. Do people value gains different than losses? Prospect Theory. Chose between E or F E: X = { 0.25, $240; 0.75, -$760} F: X = { 0.25, $250; 0.75, -$750}

How Could Expected Utility Be wrong? Violations of probability rules The reflection effect. Do people value gains different than losses? Prospect Theory. A preferred to B (84%) D preferred to C (87%) F preferred to E (100%)

How Could Expected Utility Be wrong? Violations of probability rules The reflection effect. Do people value gains different than losses? Prospect Theory. A(84%) + D (87%) = E B + C = F

How Could Expected Utility Be wrong? 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}

How Could Expected Utility Be wrong? 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)]

How Could The Expected Utility Model Be So Wrong? Questions/Comments