### 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 = CE \$ TAX2 = CE \$ CPT1 = CE \$ CPT2 = CE \$ Prediction of Priority Heuristic TAX CPT PH Agreement     Frequency that Each Reason is Decisive T1C1P1 Reason 1: min gain T1C1P2 Reason 2: prob min T1C2P1 Reason 3: max gain T1C2P2 Reason 4: prob max T2C1P1 T2C1P2 T2C2P1 T2C2P2 others (undecided by PH)

### Instructions

This calculator is designed to compute predictions for the priority heuristic as described in Brandstätter, Gigerenzer, and Hertwig (2005). The calculator now includes the rounding to prominent numbers, by geometric spacing. The calculator permits an examination of "how often" this model agrees with the TAX and CPT models as fit to previous data. It also counts how often each of the four reasons is used.

According to the priority heuristic, people compare gambles by considering "reasons" for choosing the gambles. First, they examine the lowest consequences in the two gambles, and if the difference is greater than the WeberFraction (default=1/10) of the (rounded) maximum consequence of either gamble, they choose the gamble with the higher lowest consequence. If the difference is less than one JND (WeberFraction) of the maximum, they consider the probability of the lowest consequence, choosing the gamble with the lower probability of receiving the minimum, if the difference is greater than or equal 1 JND in probability (default = 1/10). If there is still no decision, they compare the maximal consequences, and if no decision, they compare the probabilities of the maxima. Instructions for a previous version of the calculator are given below.

This calculator also calculates predictions according to TAX using parameters of Birnbaum and McIntosh (1996); see below. It also calculates the CPT value of each gamble, using parameters fit to data of Tversky and Kahneman (1992) with the weighting function of Tversky and Wakker (1995). The program counts how often among random gambles these three models agree, or pairs of models agree.

If you press the button labeled Choose Pair of Random Gambles, one pair of gambles is selected randomly by the scheme described below, and the calculations are made.

You can also enter a pair of gambles, and check what the models predict. 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 entered in the bottom row. Press Calculate preferences button and you will see that Gamble 2 (which is dominant) should be chosen according to CPT and the PH heuristic, but 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 with Internet Explorer for Mac. To sample 10,000 pairs of gambles, it takes less than 10 minutes in Navigator 4.73 and over one hour in Explorer on a Mac G4 running System 9.

The sampling scheme is as follows: Each consequence is chosen randomly and uniformly from the interval [0, 100]. The values are rounded off to the nearest dollar and .01 in probability before being displayed. This may result in ties in the displayed gambles that are actually different. The branches are ordered according to their 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 default parameters are as follows: delta = -1, gamma = .7, and beta =1. The CPT parameters are listed below. The WeberFraction is used in the Priority Heuristic and is assumed the same in both money and probability, except that probability always assumes the same scale of 1, with the same JND (default is .1); however, the maximal gain in either gamble is first rounded to the nearest prominent number and then multiplied by the WeberFraction to find the JND. For example, if the largest consequence in either gamble is \$96, the nearest prominent number is \$100, and the JND is \$10, so gambles in which the lowest consequence differ by \$10 or more are decided on that basis. If the largest consequence in either gamble is \$60, however, the nearest prominent number is \$50, so the JND is \$5. You can change these in the following display:

 TAX parameters Delta = Gamma = Beta = CPT parameters c = Gamma_c = Beta_c = Difference between TAX and CPT (epsilon) = \$ WeberFraction=

In a sample run of 100,007 trials, it was found that TAX, CPT, and PH agree in their predictions 81% of the time. In addition, TAX and CPT agree with each other 94.4% of the time; TAX and PH agree 84.1% of the time; and CPT and PH agree with each other 83.8% of the time. When the rules are ordered: Min Gain, Prob of Min Gain, Max Gain, Prob of Max gain, the decision is determined by these reasons 72.0%, 19.3%, 7.5%, and 1.1%, respectively; the rule based on 4 reasons does not reach a decision in about .1% of the choices. When the WeberFraction was set to .2, the degree of agreement stayed about the same, but the reasons were less decisive. In a run of 10,002, it was found that the Min Gain, Prob of Min Gain, Max Gain, and Prob of Max gain were decisive in 49.1%, 22.5%, 21.1, and 7.1% of cases, with no decision reached in 0.2% of the cases.

One should not expect to be able to test theories by randomly throwing together some gambles and testing properties, since the models agree so often with each other. The recipe devised by Birnbaum (1997, p. 94) for testing CPT, which also distinguishes TAX and PH, produced a predicted violation of stochastic dominance (according to TAX and RAM), later shown to produce 73% violations, would have been very unlikely to have been 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.
• Eduard Brandstätter, Gerd Gigerenzer, & Ralph Hertwig (2005). The priority heuristic: Choices without trade-offs. Working Paper dated 2004, available from Eduard Brandstätter Johannes Kepler University of Linz, Department of Psychology, Altenbergerstr. 69, 4040 Linz, Austria.
• Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297-323.
• Tversky, A., & Wakker, P. (1995). Risk attitudes and decision weights. Econometrica, 63, 1255-1280.

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.