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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
3
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
9
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
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.
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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
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}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 E or F
E: X = { 0.25, $240; 0.75, -$760}
F: X = { 0.25, $250; 0.75, -$750}
20
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%)
21
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
22
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}
23
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)]
24
How Could The Expected Utility Model Be So Wrong?
Questions/Comments
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