### Calculator to Choose Random Gambles with Prizes between \$0 and \$100

 Gamble 1 Gamble 2 Choice No. = to win \$ to win \$ to win \$ to win \$ to win \$ to win \$ TAX1 = U TAX2 = U No Pred TAX viols= No. 1 doms= No. 2 doms =

### Instructions

This calculator selects random pairs of gambles, checks for stochastic dominance, and then checks if a violation of stochastic dominance is predicted by the TAX model with the parameters of Birnbaum and McIntosh (1996); see below. If you press the button labeled Choose Random Gambles, a pair of gambles is selected. If Gamble 1 dominates, a "1" appears in the small box, "2" indicates that Gamble 2 dominates, and "0" indicates that no stochastic dominance relation exists. The TAX model utility for each gamble is also calculated and shown below each gamble. Click the Choose 100,000 Pairs, and the program will draw 100,000 pairs of gambles and do the same checks.

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
```
To enter this choice, put the branches into the calculator in the positions shown above, with the branches with the lowest ranked consequences put in the top row and the branches with the highest ranked consequence put in the bottom row. Press Test Dominance button and you will see that Gamble 2 dominates and the TAX model predicts a violation (i.e., that people will choose Gamble 1). Birnbaum and Navarrete (1998) tested this prediction and found that 73% of 100 undergraduates violated stochastic dominance on this choice, as predicted by the RAM and TAX models fit to previous data.

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:

 Delta = Gamma = Beta =

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.

### Related Papers

• Birnbaum, M. H. (1997). Violations of monotonicity in judgment and decision making. In A. A. J. Marley (Eds.), Choice, decision, and measurement: Essays in honor of R. Duncan Luce (pp. 73-100). Mahwah, NJ: Erlbaum.
• Birnbaum, M. H. (1999). Paradoxes of Allais, stochastic dominance, and decision weights. In J. Shanteau, B. A. Mellers, & D. A. Schum (Eds.), Decision science and technology: Reflections on the contributions of Ward Edwards (pp. 27-52). Norwell, MA: Kluwer Academic Publishers.
• Birnbaum, M. H. (1999). Testing critical properties of decision making on the Internet. Psychological Science, 10, 399-407.
• Birnbaum, M. H. (2000). Decision making in the lab and on the Web. In M. H. Birnbaum (Ed.), Psychological Experiments on the Internet. San Diego, CA: Academic Press.
• Birnbaum, M. H. (2001). A Web-based program of research on decision making. In U.-D. Reips & M. Bosnjak (Eds.), Dimensions of Internet Science (pp. 23-55). Lengerich, Germany: Pabst Science Publishers.
• Birnbaum, M. H. (2001). Decision and choice: Paradoxes of choice. In N. J. Smelser & P. B. Baltes (Eds.-in-Chief) & A. A. J. Marley (Section Ed.), International Encyclopedia of the Social and Behavioral Sciences. Oxford: Elsevier.
• Birnbaum, M. H., & McIntosh, W. R. (1996). Violations of branch independence in choices between gambles. Organizational Behavior and Human Decision Processes, 67, 91- 110.
• Birnbaum, M. H., & Martin, T. (2003). Generalization across people, procedures, and predictions: Violations of stochastic dominance and coalescing. In S. L. Schneider & J. Shanteau (Eds.), Emerging perspectives on decision research (pp. 84-107). New York: Cambridge University Press.
• Birnbaum, M. H., & Navarrete, J. (1998). Testing descriptive utility theories: Violations of stochastic dominance and cumulative independence. Journal of Risk and Uncertainty, 17, 49-78.
• Birnbaum, M. H., Patton, J. N., & Lott, M. K. (1999). Evidence against rank-dependent utility theories: Tests of cumulative independence, interval independence, stochastic dominance, and transitivity. Organizational Behavior and Human Decision Processes, 77, 44-83.

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.