A TAX on Prospect Theories

1

A TAX on Prospect Theories

2

Gain-Loss Separability

Michael H. BirnbaumCalifornia State University,

Fullerton

3

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.

4

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.

5

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.

6

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

7

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.

8

Coalescing

• CPT, RDU, RSDU, EU satisfy it.• RAM, TAX, GDU, SEU+f(entropy)

violate it.

9

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.

10

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

11

Gain-Loss Separability

€

G+ f F +

G− f F−

⇒

G f F

12

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.

13

CPT

€

CPU(G ) = [W −(Pii =1

n

∑ )− W −(Pi+1)]u(xi )+ [j =1

m

∑ W +(Qj )− W +(Qj+1)]u(xj )

14

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


G: .25 chance at $1600

.25 chance at $1200

.25 chance at $-200


F: .25 chance at $2000

.25 chance at $800

.25 chance at $-800


15

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


F-: .50 chance at $0

.25 chance at $-800


60 -275.9>

-358.7

-437 <

-378.6

G: .25 chance at $1600

.25 chance at $1200

.25 chance at $-200


F: .25 chance at $2000

.25 chance at $800

.25 chance at $-800


38 -300 <

-280

-178.6 <

-107.2

16

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.

17

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.

18

“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)

19

TAX Model

€

u(x) = x

t( p) = pγ

δ =1Assumptions: Same for nonnegative gambles and for mixed gambles. Calculate strictly negative gambles by reflection.

20

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.

21

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.

22

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.

23

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

24

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 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 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

25

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.

26

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.

27

28

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.

29

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.

[email protected]








mailto:[email protected]





30

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


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


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


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

31

Additional Results-2R S R S R S


50 white to win $0

50 blue to win $50

50 green to win $500.67

33.3 50 37.4 50


50 white to win $0

100 blue to win $50

(win $50 for sure)0.69

33.3 50 37.4 50


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 red to lose $100

50 white to win $0

50 white to lose $00.53

-33.3 0 -22.3 0

Documents

A TAX on Prospect Theories