You can also enter a pair of gambles, check for dominance, and check if a violation is
predicted by the TAX model with the same parameters. For example, try the pair of
gambles suggested by Birnbaum (1997, p. 94) as a test between the class of rank-dependent
utility models, including Cumulative Prospect Theory and Rank and Sign Dependent
utility theory and the configural weighted TAX and RAM models.
Birnbaum's(1997) choice is as follows:
Gamble 1: .10 to win $12 Gamble 2: .05 to win $12
.05 to win $90 .05 to win $14
.85 to win $96 .90 to win $96
The program seems to run much faster and update the changing display much better with
Netscape Navigator for Mac than Internet Explorer for Mac. To sample 100,000 pairs of
gambles, it takes about 10 minutes in Navigator and over one hour in Explorer on a
Mac running System 9 on a G4.
The sampling scheme is as follows: Each consequence is chosen randomly and uniformly from
the interval [0, 100]. The values are not rounded off for calculations but they are rounded
before being displayed. This may result in ties in the displayed gambles that are actually
different. The branches are ordered according to the consequences. To select probabilities,
random numbers are selected randomly and uniformly from the interval [0, 1]. The sum of these random
numbers might exceed or fall short of 1, so the numbers thus sampled are divided by the
sum of the random numbers to ensure that they behave as proper probabilities.
The TAX model parameters are as follows: delta = -1, gamma = .7, and beta =1. You can change these
in the following display:
With those parameters, based on 500,000 random gambles, the proportion of pairs with a stochastic
dominance relation was .332. The proportion of random pairs where a stochastic dominance relation
holds and the violation is predicted by the TAX model with these parameters is .000176. In other
words, if a person ran an experiment with 1,000 randomly picked gamble pairs, the odds would be more than 5:1
against finding even a single pair in which TAX predicts a violation of stochastic dominance.
In other words, one should not expect to be able to test theories by randomly throwing together some
gambles and testing properties. The recipe devised by Birnbaum (1997, p. 94) produced a predicted violation of
stochastic dominance (according to TAX and RAM, that was later shown to produce 73% violations), would
have been very unlikely to be discovered by trial and error.
Tech Reports, supplementary data, and preprint versions of many of these papers are available
for download from this link.
This material is based upon work supported by the National Science Foundation under Grants No. SBR-9410572, SES-9986436, and BCS-0129453. Any opinions,
findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily
reflect the views of the National Science Foundation.