GainLoss Separability and Reflection In memory of Ward

Gain-Loss Separability and Reflection In memory of Ward Edwards Michael H. Birnbaum California State University, Fullerton 1

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. 2

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)=-lu(x), x > 0. • RAM, TAX: the negative consequences get greater weight, as do lower positive consequences. 3

Reflection 4

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. 5

Coalescing 6

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. 7

Coalescing • CPT, RDU, RSDU, EU satisfy it. • RAM, TAX, GDU, SEU+f(entropy) violate it. 8

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. 9

Notation 10

Gain-Loss Separability 11

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 subgambles. • It will be violated if negative subgambles get greater weight. 12

CPT 13

Wu & Markle Example 14

Wu and Markle Result 15

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 that choice. • But CPT fails to predict choices among the sub-gambles of G and F, so CPT is disproved by Wu & Markle’s test. 16

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. 17

“Prior” TAX Model Assumptions: 18

TAX Model Assumptions: Same d for nonnegative gambles and for mixed gambles. Calculate strictly negative gambles by reflection; i. , e. , d = -1. 19

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. 20

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. 21

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 intermixed. • Tested in Lab and via the WWW. 22

New Study (n = 178) 23

Recall: TAX predictions • Predictions based on u(x) = x for -$100 < x < $100; NO “loss aversion” in utility. • Probability transformation, t(p) = pg with g =. 7. • Configural transfer parameter, d, assumed to be 1 for positive and mixed gambles, -1 for non-positive gambles. 24

Results--% F 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

More General TAX Model • Perhaps the configural weight transfer parameter, d, depends on the configuration of consequences, as in Edwards’ (62) “book of weights: ” 27

Iso-Utility Contours in TAX 28

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. 29

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. 30

For More Information: mbirnbaum@fullerton. edu http: //psych. fullerton. edu/mbirnbaum/ Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers. 31

Additional Results 32

Additional Results-2 33