a tax on prospect theories
DESCRIPTION
A TAX on Prospect Theories. Gain-Loss Separability. Michael H. Birnbaum California State University, Fullerton. Two Theories of Risk Aversion. Risk Aversion: preference for sure thing over gamble with equal or higher expected value. - PowerPoint PPT PresentationTRANSCRIPT
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A TAX on Prospect Theories
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Gain-Loss Separability
Michael H. BirnbaumCalifornia State University,
Fullerton
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Two Theories of Risk Aversion
• Risk Aversion: preference for sure thing over gamble with equal or higher expected value.
• EU accounts for risk aversion with a nonlinear utility function.
• Configural weight models, including RAM, TAX, CPT, RSDU, RDU, account for risk aversion mostly in terms of weights.
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Two Theories of Loss Aversion
• Loss Aversion: preference for a sure gain or status quo over a mixed gamble with equal or higher EV.
• EU, CPT account for it with utility function. CPT: u(-x)=-u(x), x > 0.
• RAM, TAX: the negative consequences get greater weight, as do lower positive consequences.
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Testing TAX vs. CPT• Previous talks: properties of non-
negative consequence gambles.• Ten paradoxes refute CPT: violations of
stochastic dominance, coalescing, upper tail independence, lower and upper cumulative independence, violations of restricted branch independence, lower and upper 3-distribution independence, 4-distribution independence, & dissection of Allais paradoxes.
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Coalescing
€
x > y > 0
G = (x, p;x,q;z,1− p − q)
′ G = (x, p + q;z,1− p − q) ~ G
F = (x, p;y,q;y,1− p − q)
′ F = (x, p;y,1− p) ~ F
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Violations of Coalescing
• Violations of coalescing may underlie 5 of the New Paradoxes that violate CPT: SD, ESE, UCI, LCI, & UTI, as well as Allais paradoxes.
• We can deduce each of those properties from other plausible assumptions and coalescing.
• The GLS test involves two choices between 3-branch gambles and one between 4-branch gambles. Maybe coalescing plays a role here as well.
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Coalescing
• CPT, RDU, RSDU, EU satisfy it.• RAM, TAX, GDU, SEU+f(entropy)
violate it.
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GLS is implied by CPT
• EU satisfies GLS• CPT, RSDU, RDU satisfy GLS.• RAM and TAX violate GLS. • Violations are an internal
contradiction in RDU, RSDU, CPT, EU.
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Notation
€
G+ = (0, pii =1
n
∑ ;ym ,pm;K ;y2 ,q2;y1,q1)
€
G − = (x1,p1;x2 ,p2;K ;xn ,pn;0, qi =1
m
∑i)
€
G = (x1,p1;x2 ,p2;K ;xn ,pn;ym ,qm;K ;y2 ,q2;y1,q1)
€
x1 < x2 <K < xn < 0 ≤ ym <K y2 < y1
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Gain-Loss Separability
€
G+ f F +
G− f F−
⇒
G f F
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GLS implied by any model that satisfies:
• Utility of a Gamble is the sum or constant-weight “linear” average of utilities of its positive and negative sub-gambles.
• It will be violated if negative subgambles get greater weight.
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CPT
€
CPU(G ) = [W −(Pii =1
n
∑ )− W −(Pi+1)]u(xi )+ [j =1
m
∑ W +(Qj )− W +(Qj+1)]u(xj )
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Wu & Markle ExampleG+: .25 chance at $1600
.25 chance at $1200
.50 chance at $0
F+: .25 chance at $2000
.25 chance at $800
.50 chance at $0
G-: .50 chance at $0
.25 chance at $-200
.25 chance at $-1600
F-: .50 chance at $0
.25 chance at $-800
.25 chance at $-1000
G: .25 chance at $1600
.25 chance at $1200
.25 chance at $-200
.25 chance at $-1600
F: .25 chance at $2000
.25 chance at $800
.25 chance at $-800
.25 chance at $-1000
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Wu and Markle ResultG F % G TAX CPT
G+: .25 chance at $1600
.25 chance at $1200
.50 chance at $0
F+: .25 chance at $2000
.25 chance at $800
.50 chance at $0
72 551.8 >
496.6
551.3 <
601.4
G-: .50 chance at $0
.25 chance at $-200
.25 chance at $-1600
F-: .50 chance at $0
.25 chance at $-800
.25 chance at $-1000
60 -275.9>
-358.7
-437 <
-378.6
G: .25 chance at $1600
.25 chance at $1200
.25 chance at $-200
.25 chance at $-1600
F: .25 chance at $2000
.25 chance at $800
.25 chance at $-800
.25 chance at $-1000
38 -300 <
-280
-178.6 <
-107.2
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A Bit of Irony• The Wu-Markle example is based on a
paper by Levy & Levy.• That paper criticized CPT based on
comparison of G and F alone.• Wakker replied that CPT with previous
parameters predicts the choice.• But CPT fails to predict choices among
the sub-gambles of G and F, so it is disproved by Wu & Markle’s test.
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Transfer of Attention Exchange (TAX)
• Each branch (p, x) gets weight that is a function of branch probability
• Utility is a weighted average of the utilities of the consequences on branches.
• Attention (weight) is drawn from one branch to others. In a risk-averse person, weight is transferred to branches with lower consequences.
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“Prior” TAX Model
Assumptions:
€
U(G) =Au(x) + Bu(y) + Cu(z)
A + B + C
€
A = t( p) −δt(p) /4 −δt(p) /4
B = t(q) −δt(q) /4 + δt(p) /4
C = t(1− p − q) + δt(p) /4 + δt(q) /4
€
G = (x, p;y,q;z,1− p − q)
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TAX Model
€
u(x) = x
t( p) = pγ
δ =1Assumptions: Same for nonnegative gambles and for mixed gambles. Calculate strictly negative gambles by reflection.
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TAX: Violates GLS• Special TAX model violates GLS if
branches with negative consequences receive more weight than those with positive consequences.
• Predictions are calculated with parameters approximated to fit the data of TK 1992 and used since then to predict results of other studies.
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Summary of Predictions
• EU, CPT, RSDU, RDU satisfy Coalescing and GLS
• TAX & RAM violate coalescing and GLS
• Here CPT defends the null hypotheses against specific predictions made by TAX.
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Experiment with Jeff Bahra
• 178 Undergraduates completed a set of 21 choices twice, separated by about 100 other choices. GLS tested in both split and coalesced form. Other tests as well.
• Tested in Lab and via the WWW.
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New Study (n = 178)No. G F TAX
4G+: .25 to win $100
.25 to win $0
.50 to win $0
F+: .25 to win $50
.25 to win $50
.50 to win $0
13.8 20.6
5G- : .50 to lose $0
.25 to lose $50
.25 to lose $50
F- : .50 to lose $0
.25 to lose $0
.25 to win $100
-20.6 -13.8
6G: .25 to win $100
.25 to win $0
.25 to lose $50
.25 to lose $50
F: .25 to win $50
.25 to win $50
.25 to lose $0
.25 to lose $100
-25.0 -25.0
7G’: .25 to win $100
.25 to win $0
.50 to lose $50
F’ : .50 to win $50
.25 to lose $0
.25 to lose $100
-15.5 -34.5
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ResultsChoice % G Prior TAX Prior CPT
G F G F G F
25 black to win $100
25 white to win $0
50 white to win $0
25 blue to win $50
25 blue to win $50
50 white to win $0
0.71 14 21 25 19
50 white to lose $0
25 pink to lose $50
25 pink to lose $50
50 white to lose $0
25 white to lose $0
25 red to lose $100
0.65 -21 -14 -20 -25
25 black to win $100
25 white to win $0
25 pink to lose $50
25 pink to lose $50
25 blue to win $50
25 blue to win $50
25 white to lose $0
25 red to lose $100
0.52 -25 -25 -9 -15
25 black to win $100
25 white to win $0
50 pink to lose $50
50 blue to win $50
25 white to lose $0
25 red to lose $100
0.24 -15 -34 -9 -15
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Violations predicted by TAX (and RAM), not CPT
• EU, CPT, RSDU, RDU are refuted by systematic violations of GLS.
• TAX & RAM, as fit to previous data correctly predicted the modal choices. Predictions calculated in advance of the studies, estimating nothing new.
• Violations of GLS are to CPT as the Allais paradoxes are to EU.
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To Rescue CPT:
• CPT cannot handle the results unless it becomes a configural model. Wu & Markle suggested using CPT with different parameters for different configurations. But this modification does not account for the other 10 “new” paradoxes, nor violations of coalescing.
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Add to the case against CPT/RDU/RSDU
• The case can be made that violations of GLS are due to heavier weighting of branches with negative consequences.
• This pattern is consistent with TAX, using its previously estimated parameters, even with simplifying assumptions that the same configural parameter applies to positive, negative, and mixed gambles.
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For More Information:
http://psych.fullerton.edu/mbirnbaum/
Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers.
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Additional ResultsNo. Choice % G Prior TAX Prior CPT
First Gamble, F Second Gamble, G F G F G
15 25 black to win $100
25 white to win $0
50 white to win $0
25 blue to win $50
25 blue to win $50
50 white to win $0
0.71
13.8 20.6 24.6 18.7
9 25 black to win $100
75 white to win $0
50 blue to win $50
50 white to win $0
0.37 21.1 16.7 24.6 18.7
13 50 white to lose $0
25 pink to lose $50
25 pink to lose $50
50 white to lose $0
25 white to lose $0
25 red to lose $100
0.65
-20.6 -13.8 -20.4 -24.8
5 50 white to lose $0
50 pink to lose $50
75 white to lose $0
25 red to lose $1000.31
-16.7 -21.1 -20.4 -24.8
19 25 black to win $100
25 white to win $0
25 pink to lose $50
25 pink to lose $50
25 blue to win $50
25 blue to win $50
25 white to lose $0
25 red to lose $100
0.52
-25 -25 -8.8 -15.3
11 25 black to win $100
25 white to win $0
50 pink to lose $50
50 blue to win $50
25 white to lose $0
25 red to lose $100
0.24
-15.5 -34.5 -8.8 -15.3
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Additional Results-2R S R S R S
50 black to win $100
50 white to win $0
50 blue to win $50
50 green to win $500.67
33.3 50 37.4 50
50 black to win $100
50 white to win $0
100 blue to win $50
(win $50 for sure)0.69
33.3 50 37.4 50
50 black to win $100
50 white to win $0
100 green to win $45
(win $45 for sure)0.60
33.3 45 37.4 45
50 white to lose $0
50 red to lose $100
50 pink to lose $50
50 orange to lose $500.37
-33.3 -50 -40.8 -50
50 white to lose $0
50 red to lose $100
100 pink to lose $50
(lose $50 for sure)0.31
-33.3 -50 -40.8 -50
50 white to lose $0
50 red to lose $100
100 orange to lose $55
(lose $55 for sure)0.32
-33.3 -55 -40.8 -55
50 black to win $100
50 red to lose $100
50 white to win $0
50 white to lose $00.53
-33.3 0 -22.3 0