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
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
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 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.

6. 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.

7. 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.

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 Kahneman and 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 C or 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 C or 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