1. Additional details for Birnbaum and Navarrete Paper.

2. Data

A data file that specifies all of the gambles, except for the check trials, in preparation for the data analysis can be obtained by clinking the following link: DOWNLOAD file for CPTFIT and TAXFIT.

The raw data for each subject, except for the check trials, corresponding to the file specifying the gambles is available by clicking the following link:

Programs in FORTRAN, CPTFIT and TAXFIT, that can be used to fit the CPT model and configural weight, TAX model can be obtained from the following URL

Instructions for using the programs are also available from the above URL.

3. Detailed Method of Birnbaum & Navarrete Paper

Instructions read (in part) as follows:

" . . . On each trial, you will be offered a comparison between two gambles. Your task is to decide which of the two gambles you would prefer to play and to judge how much you would pay to play your preferred gamble rather than the other gamble. . ."

"For each gamble, you can think of a can containing 100 identical slips of paper with different amounts written on them. Since the slips are equally likely and one will be chosen at random, the probability of each outcome is the number of slips with that outcome, divided by 100. Each trial displays the probabilities and values of all possible outcomes for each gamble."

Choices were displayed as in the following example:

```		.34 .33 .33			.20 .25 .55
302. _____	 \$5 \$25 \$100	versus		\$20 \$40 \$60```

"Would you prefer the gamble on the left (34 chances out of 100 to get \$5, 33 chances to get \$25, and 33 chances to get \$100) or the gamble on the right (20 chances to get \$20, 25 chances to get \$40, and 55 chances to get \$60)? "

Judges circled the gamble they would prefer to play, and then judged the strength of their preference in dollars. For purposes of data analysis, a negative sign was associated with choice of the gamble on the left.

3.1 Designs

There were 4 choices between three-outcome gambles designed to test stochastic dominance. These choices were of the form,

G+ = (x, p – q; x+, q; y, 1 – p) versus G– = (x, p; y–, r; y, 1 – p – r),

where 0 < x < x+ < y– < y. The values of (x, x+, y–, y) in the four pairs were (\$12, \$14, \$90, \$96), (\$3, \$5, \$92, \$97), (\$6, \$8, \$91, \$99), and (\$4, \$7, \$89, \$95); the values of (p, q, r) were (.10, .05, .05), (.12, .06, .04), (.05, .03, .03), and (.02, .01, .02), respectively. Note that G+ stochastically dominates G–.

The four trials testing stochastic dominance were embedded among many other choices. The four trials testing stochastic dominance were counterbalanced in position, so consistent choice of the gamble on the right (or left) would produce two violations and two satisfactions of dominance.

The design testing cumulative independence and branch independence was composed of 27 variations of each of the following four choices, making 108 trials:

S = (z, r; x, p; y, q) versus R = (z, r; x', p; y', q);

S'' = (x', r ; y, p + q) versus R'' = (x', r + p; y', q);

S' = (x, p; y, q; z', r) versus R' = (x', p; y', q; z', r); and

S''' = (x, p + q; y', r) versus R''' = (x', p; y', q + r).

There were 4 subdesigns with different probabilities (r, p, q): (.5, .25, .25), (.8, .1, .1), (.6, .3, .1), and (.6, .1, .3). Within each subdesign, there were 6, 7, or 8 levels of (z, x', x, y, y', z'), which were factorially combined with the four types of comparisons. All subdesigns used the following 6 levels of (z, x', x, y, y', z') = (\$2, \$11, \$52, \$56, \$97, \$108), (\$3, \$10, \$48, \$52, \$98, \$107), (\$2, \$11, \$45, \$49, \$97, \$107), (\$2, \$10, \$40, \$44, \$98, \$110), (\$4, \$11, \$35, \$39, \$97, \$111), and (\$5, \$12, \$30, \$34, \$96, \$110). In addition, in the subdesign with (r, p, q) = (.6, .3, .1), a seventh level of (z, x', x, y, y', z') was added: (\$3, \$10, \$25, \$29, \$98, \$109); in the subdesign with (r, p, q) = (.6, .1, .3), the following two levels were added: (\$4, \$10, \$61, \$65, \$98, \$108) and (\$3, \$12, \$56, \$60, \$96, \$107). Note that the major changes are in (x, y), and the other outcomes are nearly constant within each subdesign.

The "check" design consisted of 12 choices with transparent dominance, in which all probabilities and outcomes were the same except one outcome was better in one gamble, or all outcomes were the same but the probability of a higher outcome was better in one gamble. There were 3 comparisons of the form (x, .25; y, .25; \$50, .5) vs. (x, .25; y; .25; \$90, .5), in which the 3 levels of (x, y) were (\$2, \$4), (\$2, \$108), (\$108, \$111); 3 comparisons of the form: (x, .1; y, .1; \$92, .8) vs. (x, .1; y; .1; \$42, .8), in which the 3 levels of (x, y) were (\$3, \$5), (\$3, \$109), (\$109, \$112); 3 choices as follows: (\$5, p; \$50, q; \$90, r) vs.(\$5, p'; \$50, q'; \$90, r'), where (p, q, r; p', q', r') = (.6, .2, .2; .2, .6, .2), (.6, .2, .2; .2, .2, .6), and (.2, .6, .2; .2, .2, .6); 3 trials as follows: (\$4, p; \$40, q; \$93, r) vs. (\$4, p'; \$40, q'; \$93, r'), where (p, q, r; p', q', r') = (.1, .8, .1; .8, .1, .1), (.1, .1, .8; .1, .8, .1), and (.1, .1, .8; .8, .1, .1). Half of the check trials required selecting the gamble on the right and half on the left.

3.2 Procedure and Judges

Each booklet contained 3 pages of instructions with example trials and 10 practice trials, followed by 134 experimental choices. These 134 trials were composed of 108 choices testing cumulative independence, 12 "check" trials, 4 trials testing stochastic dominance, and 5 unlabeled warm-up trials at both the beginning and end of the booklet. The choices were printed in random order, with restrictions that no two successive trials repeat the same design, subdesign, or values of (x, y). Half of the judges worked through the booklet in reverse order.

The experimenter checked responses to the 10 practice trials. Some of these trials were tests of transparent dominance similar to the "check" trials. When a judge violated transparent dominance in practice trials, the experimenter asked the judge to explain the choice, and to reread instructions as needed. In the few cases where subjects violated dominance during practice, violations represented simple misunderstandings of the instructed task; for example, some judges circled the outcomes that they hoped would occur rather than the gamble they would select; occasionally a judge circled both gambles since both were attractive. When the practice trials satisfied transparent dominance, judges were directed to proceed to the experimental trials.

The judges were 100 undergraduates, who completed the experiment within one hour, working at their own paces. Of these, 68% had zero violations of transparent dominance in the 12 check trials; the overall rate of violation was 4%.

An additional 12 judges were tested whose data were excluded prior to analysis; five of these failed to complete the task within the time alotted; 4 violated transparent dominance more than twice out of 12 check trials (2 had 3 violations and 2 had 4); 3 were mostly complete (but left a few trials blank) who had 0 violations. Because all theories under consideration imply transparent dominance, excluding the four who violated transparent dominance more than twice in the "check" trials is neutral with respect to the theories and was intended to remove judges who are careless or confused.

"Check" trials have been routine in previous research to ensure that judges have at least superficial understanding of the task, and to eliminate judges who are confused, careless, or random. If a judge were choosing randomly, then the probability of making 2 or fewer violations of transparent dominance out of 12 check trials is .02. A reviewer asked if the check trials and procedure of eliminating 4 judges who had more than two violations of transparent dominance in the check trials might have biased our results. We tested an additional 38 undergraduates with a shorter version of the booklet without check trials (and none were excluded). Their data were similar to those of the main study. For tests of stochastic dominance, there were 13 people in this group who had 4 violations out of 4, 9 who had 3, 10 who had 2, 3 who had 1 violation, and 3 with 0 violations out of 4. The overall rate of violation for this group is therefore 67.1%, which is significantly greater than 50% and not significantly different from 70%, the rate in our main experiment. This finding indicates that the results can be replicated without the check trials.

This material is based upon work supported by the National Science Foundation under Grant No. SBR-9410572. 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.
Jan 14, 1999