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How Could The Expected Utility Model Be So Wrong?

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.

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How Could The Expected Utility Model Be So Wrong?

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  1. How Could The Expected Utility Model Be So Wrong? Tom Means SJSU Department of Economics Math Colloquium SJSU December 7, 2011

  2. 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

  3. 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

  4. 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)]

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

  6. Risk-preferring Utility of dollars U($) b U(38) .10(18)+.90(38) d U(18) 38 Dollars 0 b’ d’ 36 18 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.

  7. Willingness to sell insurance Utility of dollars U($) U(38) k b U(18) 38+1.5=39.5 Dollars 0 k’ b’ 38 18+1.5=19.5 38+π 18+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.

  8. 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)

  9. 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

  10. 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 Kahnemanand Amos Tversky) • Ambiguity aversion • The Ellsberg Paradox (known vs. ambiguous distribution) • Base rates • Nonlinear probability weights • Framing

  11. Framing Problem One. Pick A or B A:X = { P = 1, $30; 1-P = 0, $0} B: X = {0.80, $45; 0.20, $0}

  12. 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}

  13. 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

  14. 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.

  15. The Kahneman-Tversky Value Function

  16. The Benefit of Segregating Gains

  17. The Benefit of Combining Losses

  18. The Silver-Lining Effect and Cash Rebates

  19. 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 Cor D • C: X = { 1.0, $-750; 0.0, $0} • D: X = {0.75, $-1000; 0.25, $0}

  20. 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}

  21. 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%)

  22. 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

  23. 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 Cor D • C: X = { 0.25, $6000; 0.75, $0} • D: X = { 0.20, $8000; 0.80, $0}

  24. 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)]

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

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