Exploration of Branch Splitting and
Tests of Branch Splitting and
Branch-Splitting Independence in Allais Paradoxes
Michael H. Birnbaum*
California State University, Fullerton
09-25-04
Short Title: Branch Splitting and Allais Paradoxes
Contact Info: Mailing address:
Prof. Michael H. Birnbaum
Department of Psychology, CSUF H-830M,
P.O. Box 6846,
Fullerton, CA 92834-6846
Email address: mbirnbaum@fullerton.edu
Phone: 714-278-2102 Fax: 714-278-7134
JEL-classification: C91, D81.
* Support was received from National Science Foundation Grants, SBR-9410572, SES 99-86436, and BCS-0129453.
Abstract
Two experiments with 1043 participants refine and extend previous dissections of constant consequence (Allais) paradoxes. These can be viewed as violations of either restricted branch independence or coalescing. The first study used a number of manipulations intended to “help” cumulative prospect theory (CPT). The second experiment independently split or coalesced upper or lower branches of “safe” or “risky” gambles; this design allows separate tests of upper and lower coalescing and thus of branch-splitting independence. This provides a test of the (lower) Gains Decomposition Utility (GDU) model of Marley and Luce (2001), which violates lower coalescing but satisfies upper coalescing. Violations of upper coalescing were observed, contrary to CPT and GDU, but consistent with Birnbaum’s TAX model. Violations of branch independence in both studies were significant and opposite in direction from predictions of CPT and opposite the pattern needed to account for Allais paradoxes. There were also small, but significant violations of branch splitting independence. Analyses under a true and error model showed that violations of CPT, RDU, and EU cannot be attributed to random error. GDU provided a better fit than CPT, and Birnbaum’s prior transfer of attention exchange (TAX) model provided the best fit, though there were some significant departures.
1. Introduction
Wu and Gonzalez (1998) divided “constant consequence” paradoxes (as in the Allais paradox) into three types, and showed that cumulative prospect theory (CPT) of Tversky and Kahneman (1992), with its inverse-S shaped weighting function, can account for empirical results with all three. Birnbaum (2001) reported that the results of Wu and Gonzalez can be replicated with modest but real consequences, and he noted that his transfer of attention exchange (TAX) model also describes all three of these phenomena.
Constant consequence paradoxes can be decomposed into three simpler properties: transitivity, coalescing, and restricted branch independence. Birnbaum (1999a) proposed testing these properties separately as a way to compare four classes of models of risky decision making: expected utility (EU), original prospect theory (OPT), cumulative prospect theory (CPT) and Birnbaum’s models (including transfer of attention exchange (TAX) and rank-affected multiplicative weights (RAM) models). The three properties are defined as follows:
Transitivity, assumed in all of the models discussed here, is the assumption that A φ B and B φ C [pic] A φ C, where φ denotes the preference relation.
Coalescing is the assumption that if a gamble has two (probability-consequence) branches yielding identical consequence, those branches can be combined by adding their probabilities. For example, if [pic], a gamble with two branches to win $100 and otherwise win zero, then [pic], where ~ denotes indifference. Violations of coalescing combined with transitivity are termed event-splitting effects (Humphrey, 1995; Starmer & Sugden, 1993; Birnbaum, 1999a; 1999b). For example, if [pic] and [pic], we say there is an event-splitting effect. Assuming transitivity, event-splitting effects (aka branch-splitting effects) are violations of coalescing.
For the analysis of Gains Decomposition Utility (GDU) of Marley and Luce (2001), it is useful to distinguish two types of coalescing in three-branch gambles. Let [pic], where [pic].
Upper coalescing assumes:
[pic]
Lower coalescing assumes:
[pic].
The lower GDU model is similar to Birnbaum’s TAX model in properties that it satisfies and violates (Marley & Luce, 2004). However, it differs from TAX in that it satisfies upper coalescing and violates lower coalescing, whereas TAX violates both. Experiment 2 will test these properties separately. When the term “coalescing” without modification is used in this paper, it refers to the assumption of both upper and lower coalescing.
Restricted Branch independence is the assumption that if two gambles have a common probability-consequence (or event-consequence) branch, one can change the value of the common consequence on that branch without affecting the preference induced by the other components.
For three-branch gambles with nonzero probabilities (p + q + r = 1), restricted branch independence can be written as follows:
[pic] (1)
The term “restricted” refers to the fact that the number of branches and probability distribution is the same in all four gambles of Expression 1. Restricted branch independence is a weaker form of “independence” than Savage’s (1954) “sure thing” axiom.
If empirical choices satisfied transitivity, coalescing, and restricted branch independence then they would not show the constant consequence paradox of Allais. For example, consider the following series of choices:
A: $1M for sure φ B: .10 to win $2M
.89 to win $1M
.01 to win $0
⇔ (coalescing & transitivity)
A’: .10 to win $1M φ B: .10 to win $2M
.89 to win $1M .89 to win $1M
.01 to win $1M .01 to win $0
⇔ (restricted branch independence)
A”: .10 to win $1M φ B’: .10 to win $2M
.89 to win $0 .89 to win $0
.01 to win $1M .01 to win $0
⇔ (coalescing & transitivity)
C: .11 to win $1M φ D: .10 to win $2M
.89 to win $0 .90 to win $0
From the first to second step, A is converted to a split form, [pic]; [pic] should be indifferent to A by coalescing, and by transitivity, [pic] should be preferred to B. From the second to third steps, the consequence on the common branch (.89 to win $1M) has been changed to $0 on both sides, so by restricted branch independence, [pic] should be preferred to [pic]. By coalescing branches with the same consequences on both sides, we see that C should be preferred to D.
However, many people choose A over B and D over C; indeed, this finding is the Allais paradox, so we know that at least one of these three assumptions must be false.
If we hope to describe Allais paradoxes, our model must violate at least one of these assumptions. Among transitive theories, subjectively weighted utility theory and “stripped” original prospect theory satisfy branch independence and violate coalescing whereas rank dependent utility and cumulative prospect theory violate branch independence and satisfy coalescing. Birnbaum’s RAM and TAX models violate both branch independence and coalescing.
To complicate matters, both prospect theories allow editing rules of combination (which implies coalescing) and cancellation (which implies branch independence). These editing principles make prospect theories more difficult to test (Kahneman, 2003), because the editing processes contradict implications of the models. Nevertheless, the editing principles can be tested if they are treated as scientific hypotheses rather than as post hoc excuses.
Birnbaum (2004a) performed an experimental dissection, testing coalescing and restricted branch independence separately. In addition, his study included new variations of all three types of common consequence conditions described by Wu and Gonzalez (1998). Although Birnbaum’s TAX model and CPT agree in predicting the three types of constant consequence paradoxes, they disagree when these paradoxes are dissected into tests of coalescing and branch independence. Violations of branch independence were significant, but opposite in direction from predictions of CPT. They also were opposite the Allais paradoxes. Violations of coalescing were significant and in agreement with the Allais paradoxes. These results contradict both original and cumulative prospect theories, with or without their editing principles of cancellation and combination.
Because these results refute models that have been growing in popularity (Starmer, 2000), it is reasonable that every effort must be made to either salvage the popular consensus or to convincingly refute it. Every way of saving CPT that permits experimental test must be ruled out in order to allow conservative scientists to abandon what might seem to others to be a sinking ship.
One objection raised to evidence against CPT is that Birnbaum’s (1999b; 2001; 2004a; 2004b) studies involved small, real chances to win modest prizes, whereas the studies of Kahneman and Tversky (1979), for example, dealt with hypothetical choices among gambles to win large prizes. Perhaps CPT remains viable for hypothetical decisions involving large consequences.
There is a second set of reasons that large prizes are interesting. Birnbaum’s (1999a) RAM and TAX models of evaluation of risky gambles correctly predicted the majority choices in Birnbaum’s (2004a) experiment without estimating any new parameters from the data. Birnbaum’s “prior TAX model” (which uses parameters estimated from previous data; See Appendix A) uses a linear utility function and his study used positive consequences ranging up to $108. Would this same model and utility function work for gambles on consequences in the millions?
2. Experiment 1
Subjectively, it seems that the difference in happiness at winning $1Million rather than $1 is psychologically greater than the difference between winning $2 Million and $1 Million. Therefore, the linear utility function used in the prior TAX model for small prizes seems unlikely to extrapolate to such large values. And if TAX fails here, perhaps another model, like CPT would work better.
In addition, it seems plausible that very large prizes may induce additional psychological considerations or processes, such as satisfying an aspiration level (as in Lopes & Oden, 1999). For example, $1 Million might allow one to retire from that boring 9 to 5 job, whereas prizes from $1 to $100 are not life-changing.
The first experiment therefore investigates Allais paradoxes with large hypothetical prizes to check whether models that worked for small consequences can be extended to account for hypothetical choices involving large prizes.
Experiment 1 also employed a second variation from previous work. Evidence of violation of coalescing is what most strongly contradicts CPT. Therefore, a new form of event-framing was designed in an attempt to facilitate coalescing.
Consider the format and framing in the choices below. The decision maker selects an urn from which a marble will be blindly and randomly drawn, and the color of marble determines the prize. Each urn contains exactly 100 marbles. Choices 9 and 6 are as follows:
A: 10 blue marbles to win $1M B: 10 red marbles to win $2M
01 blue marble to win $1M 01 white marbles to win $2
89 white marbles to win $2 89 white marbles to win $2
C: 11 blue marbles to win $1M D: 10 red marbles to win $2M
89 white marbles to win $2 90 white marbles to win $2
In this arrangement, it should be easy to see that in A, 10 blue marbles plus 1 blue marble is the same as 11 blue marbles to win $1M in C, and that 89 plus 1 white marbles in B are the same as 90 white marbles to win $2 in D. So, if people have a tendency to combine branches with the same consequences within gambles, this method of event framing (i.e., labeling the events with identical marble colors for equal consequences) ought to facilitate it. If so, that would result in identical choices in these two forms of presenting the same (objective) choice (split versus coalesced).
In addition, if people tend to cancel common branches, it should be easy to see that A and B both have a common branch of 89 white marbles to win $2. Thus, if people have a tendency to edit by combination and cancellation, as postulated by Kahneman and Tversky (1979), this framing should make such editing easier.
The method and procedures were the same as those of Birnbaum (2004a) with five exceptions, all designed to “help” cumulative prospect theory: first, consequences were large and hypothetical; second, events were framed within gambles as well as within choices, to promote coalescing. Third, the majority of choices used event-framing (17 of 20 decisions), to allow people to learn to use this feature; fourth, the event framing was consistent with respect to mapping of colors to magnitude of consequences (red was used for very large consequences, blue and green for medium consequences, and white for small consequences). Fifth, each participant completed the task twice, allowing participants to learn to use the editing rules, and allowing the experimenter to fit models involving random “errors”.
The design also included tests of stochastic dominance in split and coalesced forms. Complete materials can be viewed at the following URL:
Participants viewed instructions and experimental trials via the WWW, using Web browsers. They clicked the button beside the gamble in each choice that they would prefer to play. Instructions read (in part) as follows:
“Although you can't win or lose money in this study, you should imagine that these decisions are for real, and make the same decisions that you would if very large, real stakes were involved.”
There were 200 participants who completed the survey twice, separated by three other intervening judgment tasks that required about 15 minutes. Insert Tables 1 and 2 about here.
3. Results of Experiment 1
The percentages of participants who chose the “risky” gamble, R, in each choice are shown in Tables 1 and 2 for Series A and B respectively. The percentages who chose the R gamble (with 10 red marbles to win $2M and 90 white marbles to win $2) rather than S “safe” gamble (with 11 blue marbles to win $1M and 89 to win $2) were 76% and 77% in the first and second repetitions of Choice 6. However, when the upper branch of S and the lower branch of R are split, as in Choice 9, only 35% and 39% chose the R gamble. If people “spontaneously transform” Choice 9 by coalescing to Choice 6 (Kahneman, 2003), they should make the same decisions, except for error.
Instead, 93 of 200 participants switched from preferring R to S as a result of the splitting from Choice 6 to 9, compared with only 11 who made the opposite switch in the first replicate. In the second replicate, the figures are 90 versus 14. These splits are statistically significant by the binomial test, z = 8.04 and 7.45. They are also statistically significant by a more conservative test; significantly more than half chose R in Choice 6 (z = 7.50 and 7.78) and significantly more than half chose S in Choice 12 (z = 4.10 and 3.05), even though these are (objectively) the same prospects.
The prior TAX model (even with its linear utility function) correctly predicted this reversal. The prior TAX model correctly predicted the other choices in Table 1 as well, except for Choice 16.
According to the prior CPT model (the model fit by Tversky and Kahneman, 1992; see Appendix A), the majority should have chosen the “risky” gamble in every row. This CPT model therefore makes a correct prediction in only one row of Table 1, even failing to predict the basic Allais paradox in the reversal between Choices 6 and 12. And no version of RDU or CPT can explain the reversal between Choices 6 and 9.
In Table 2, splitting again produces a reversal—here between Choices 10 and 17, where 73% and 67% chose R in the coalesced form, and only 24% and 30% chose R when the lower branch of R and upper branch of S were split. In Table 2, prior CPT was able to predict the basic paradox (Choices 10 and 20), though no version of CPT can account for the reversal between Choices 10 and 17. Insert Table 3 about here.
Table 3 presents tests of stochastic dominance and coalescing. According to any version of RDU/RSDU/CPT, people should satisfy coalescing and stochastic dominance (Birnbaum & Navarrete, 1998). Instead, Table 3 shows that from 63% to 79% violate stochastic dominance (significantly more than 50% in every test) in the coalesced form, but only 8% to 15% violate it when gambles are appropriately split so that equal probabilities appear on corresponding branches. The prior TAX model correctly predicts majority preferences in Table 3.
There were 8 tests of stochastic dominance, counting Choices 5, 15, 7, and 18 over two replications. Of the 200 participants, there were 47 who violated stochastic dominance on all 8 tests, compared to only 5 who satisfied it on all 8 tests. The average percentage of violations was 72.4%. The modal choices violate any RDU model and are correctly predicted by the prior TAX model. The effects of having marble colors that matched or did not match on corresponding branches seems to have had little effect, if any, on tests of stochastic dominance.
The use of replications allows one to estimate reliability and fit true and error models to the data (Birnbaum, 2004b; Appendix B). Overall, the agreement between the first and second replicates was 76.3%. For Choices 5 and 15, the true and error model indicates that 84% of the participants “truly” violated stochastic dominance on these choices, and that participants make “errors” 16% of the time when expressing their preferences. The same model fit to Choices 7 and 18 indicated that 80% of participants “truly” violated stochastic dominance on those trials, except that participants make “errors” on 20% of these choices.
Experiment 1 shows that large violations of coalescing are obtained even with very large consequences and with a marble color framing designed to facilitate coalescing. Even with these changes, results continue to violate CPT and largely replicate previous results with small, real consequences. The prior TAX model did not predict all choices accurately; however, when one additional parameter (the exponent of a nonlinear utility function) is estimated from the data, the TAX model with its previously estimated weighting parameters correctly predicted all of the modal choices (Appendix A). Apparently, the use of large hypothetical consequences appears to require no new principle aside from the need to estimate a utility function with diminishing marginal returns.
4. Experiment 2
According to Birnbaum’s TAX model, splitting the branch leading to the highest consequence within a gamble should increase the utility of a gamble, but splitting the branch leading to the lowest consequence of a gamble should decrease the utility of the gamble. Although Birnbaum’s (2004a) study and Experiment 1separated tests of branch independence from tests of coalescing, previous tests did not separate tests of upper and lower coalescing. Whereas the TAX model and GDU model make similar predictions in Experiment 1, they make different predictions in Experiment 2.
The purpose of the second experiment is therefore to decompose the Allais paradoxes into still finer steps, in order to provide separate and independent tests of these two aspects of branch splitting. Thus, this experiment allows a test of branch-splitting independence, a property defined by Birnbaum and Navarrete (1998), and previously tested in only one study of which I am aware (Martin, 1998), which had inconclusive results. Branch splitting independence can be defined as follows:
[pic]
[pic] (2)
[pic]
where [pic]. Conceptually, if splitting the upper branch of G [i.e., (p, y)] into (p – r, y; r, y) improves G then splitting the same branch in the same way should improve H, even though that branch is now the lower branch of H.
Branch splitting independence (BSI) will not be tested in the transparent form of Expression 2, however, but rather the four gambles will be compared indirectly by how they perform against other gambles. In particular, the property will be assessed from four choices as follows:
[pic] (3)
where [pic] is the probability of choosing [pic] over [pic]; [pic] are as defined in Expression 2.
Branch-splitting independence (BSI) is the assumption that splitting the same event leading to the same consequence should always have the same effect no matter what the other branches are in the gamble (splitting the same branch into the same pieces should either improve or diminish the value of any gamble containing that branch). EU and CPT imply no branch-splitting effects at all, so the issue of branch-splitting independence is moot in those theories. SWU and “stripped” prospect theory (Starmer & Sugden, 1993) imply both branch splitting effects and branch-splitting independence. The (lower) GDU model of Marley and Luce (2001) implies lower branch splitting effects (violation of lower coalescing) and no effects of splitting upper branches. RAM and TAX imply branch splitting effects and violation of branch-splitting independence. In particular, RAM and TAX models with their prior parameters, plus the assumption of Thurstone’s law (See Appendix A), imply the following violation of BSI:
[pic]
and yet
[pic]
The GDU model implies upper coalescing but a violation of lower coalescing. Given parameters to make it similar to TAX it implies that the first two probabilities (involving splitting of the upper branch) should be equal, but the second two should show the displayed relation.
5. Method of Experiment 2
As in Experiment 1, people participated via the WWW, and clicked a button beside the gamble in each choice that they preferred. Unlike Experiment 1, participants were informed that three participants in each study would be chosen at random to receive the prize of one of their chosen gambles. The expected value of participation was approximately equal to that of a California State Lottery ticket. Prizes were awarded as promised.
Each participant made 22 choices between gambles. The first four were the same as in Birnbaum (2004a), serving as a warm-up, but the other 18 trials were constructed to allow tests of branch splitting independence. There were two series with 9 choices each, shown in Tables 4 and 5. The position of the gambles within choices was counterbalanced between Series A and B. Within each series, there are two, 2 X 2 factorial designs in which the lower branch of one gamble and the higher branch of the other gamble are either split or coalesced. All combined, there are 8 tests of splitting/coalescing of upper branches and 8 tests of splitting/coalescing lower branches. Each series provides one test of restricted branch independence (Choices 9 and 16 in Table 4 and Choices 17 and 14 in Table 5). Finally, each series has all three types of Allais paradoxes [(Choices 6, 12, and 19) and (Choices 10, 20, and 8)], where RAM, TAX, GDU, and CPT all agree on their predictions. Insert Tables 4 and 5 about here.
There were three studies in Experiment 2, with different participants in each. Study A2 was run first, with 203 participants. Participants were then randomly assigned to Conditions A3 or A4, where A3 had the same method as A2. Condition A4 used small changes in the values of the consequences when a branch was split, designed to investigate if people might be more or less likely to coalesce if the branches led to exactly equal consequences (A3) as opposed to (slightly) different consequences (A4). There were 315 and 325 participants in A3 and A4, respectively. Complete materials, including warm-up trials and instructions can be viewed via the Web from the following link:
6. Results of Experiment 2
The percentages of participants who chose the risky gamble in each choice are displayed in Tables 4 and 5. The difference between conditions A3 and A4 (which used slightly different consequences when a branch was split) were minimal, as were differences between A2 and A3 (not shown). Results of all three conditions (n = 843) are combined under the column labeled “All.”
Note that the first four choices listed in Table 4 involve the same underlying prospects. According to CPT/RDU/RSDU, coalescing or splitting of branches should have no effect, so all decisions in the first four rows should have been the same in each case, apart from error. In violation of these predictions, percentages choosing the “safe” second gamble (S) changed from significantly less than 50% in Choices 6 and 21, where the higher valued branch of the S gamble was coalesced (.2 to win $40), to significantly greater than 50% in Choices 11 and 9, where this same branch was split into two branches of .1 to win $40.
According to EU, the choice percentages should be the same within every row of Table 4, and they should be the same within every row of Table 5. According to CPT, coalescing should have no effect, so the choice percentages should be the same within the first four rows of Table 4, within the last four rows of Table 4, in first four rows of Table 5, and in the last four rows of Table 5. According to GDU, there should be no difference between any two rows where only the upper consequence is split or coalesced. In Table 4, Choices 6 and 11, 21 and 9, 16 and 7, and 13 and 19 should be equal within each pair. In Table 5, Choices 10 and 15, 22 and 17, 14 and 5, and 18 and 8 should be the same within pairs. Predictions for GDU are shown in the last two columns of Tables 4 and 5. According to the prior TAX model, there should be changes from row to row, as indicated by the changing cash equivalent values for the gambles, which are presented in the last columns of the tables. (Appendix A).
In Choice 6 of Table 4, the significant majority prefers the “risky” gamble with 10 marbles to win $98 over the “safe” gamble with 20 marbles to win $40 (the table shows that fewer than 40% choose the second gamble, S, in each study). However, in Choice 11, the same S gamble has now been described as having 10 red marbles to win $40 and 10 blue marbles to win $40. In this case, more than 65% choose the S gamble, even though the (objective) prospects are the same. Here, the prior TAX model correctly predicted that the S gamble should be improved (from a cash equivalent of $9.0 to $11.1), but prior TAX failed to predict that this would be enough to reverse the majority choice, since the R gamble has a higher computed value of 13.3. According to CPT or GDU, there should have been no difference between Choices 6 and 11, since only difference is that the branch with the larger consequence was split in S in Choice 11 and coalesced in Choice 6.
All choice percentages in Table 4, averaged over all 843 participants (“All” column), are significantly different from 50%, except for those in Choice 12. So, the proportion changed from significantly more than 50% in Choice 6 to significantly less than 50% in Choice 11. By the (more sensitive) within-subjects test of correlated proportions, the shift is also significant when tested separately in all three studies, z = 6.18, 6.86, and 7.55 in A2, A3, and A4, respectively.
The novel feature of Experiment 2 is that it separately tests the effects of splitting the branch with the higher or lower valued consequence. According to prior TAX predictions, splitting the higher consequence should improve a gamble and splitting the branch with the lowest consequence should lower the value of the gamble, violating event-splitting independence. There are 8 tests of each of these predictions.
Splitting the branch with the higher consequence significantly increased the proportion choosing that gamble in all eight tests. In Choice 6 X 11, 21 X 9, 19 X 13, 7 X 16, 10 X 15, 22 X 17, 8 X 18, and 5 X 14, the binomial test of correlated proportions yielded z = 11.8, 7.6, 14.3, 14.9, 14.0, 11.8, 15.8, and 11.8, respectively (the critical value of |z| for α = .05 level of significance is 1.96.). All eight are significant and all are in the predicted direction based on TAX. These results are not consistent with EU, CPT, or GDU, which imply no effects of splitting the upper branch.
Splitting the branch with the lower consequence had effects that were smaller and less consistent. Of the eight tests on the combined data, three were statistically significant. The four tests with the largest values of |z| were Choices 6 X 21, 11 X 9, 10 X 21, and 8 X 5, with z = 1.94, –3.01, 3.18, and 5.49, respectively, with three of these four in the direction predicted by TAX.
The difference between Choices 8 and 5 in Table 5 shows that when the lowest outcome is $50, splitting this outcome makes people more likely to choose the other gamble, consistent with TAX. In Choices 10 versus 15, $50 was the highest consequence in the gamble, and splitting it there significantly increased the percentage choosing this gamble (a shift of about 30%). Therefore, we have a case where splitting the same probability to receive the same consequence can either significantly improve or significantly diminish the proportion choosing the gamble, depending on whether that consequence (in this case, $50) is the highest or lowest consequence in the gamble. The effect of splitting $50 in the lower branch, though statistically significant when tested in each of three conditions separately, produced a modest shift of about 10% in the choice percentages.
The prior TAX model correctly predicted the majority choice in all but five cases: Choices 11 and 13 in Table 4, and 15, 20, and 18 in Table 5. In all five cases of discrepancy, the choices involve unequal numbers of branches in the two gambles, and the discrepancy from TAX can be described as follows: the majority chose the gamble with the greater number of (positive) branches.
The Allais paradoxes are comparisons of Choices 6 versus 19 (z = 13.57) and Choices 10 versus 8 (z = 13.46). In both tables, the percentage choosing R decreases from the first to last row, but increases as we increase the common consequence in the pure tests of branch independence (Choices 9 versus 16 and 17 versus 14). According to CPT, Allais paradoxes are produced by violations of restricted branch independence.
In each table, there is a test of restricted branch independence. Note that in Table 4, Choice 9 has a common branch of 80 marbles to win $2 and Choice 16, has a common branch with 80 marbles to win $98. In Table 5 note that Choices 17 and 14 differ only in the consequence on the common branch of 85 marbles to win $7 (Choice 17) or $100 (Choice 14). Because the marble colors on the common branches are the same in Table 4 and not in Table 5, we say choices 9 and 16 are “framed” (supposedly making it easier to notice and cancel the common branch, should people be inclined to do so). Note that Choices 17 and 14 are not “framed” because there are different colors on the common branch.
In both tests, there are significant violations of branch independence (z = 8.03 and 8.87). However, the observed changes go in the opposite direction from that predicted by the prior CPT model and are opposite what is required for CPT to account for the Allais paradox. Both trends agree, however, with the prior TAX model.
Once again, there appears little evidence that the event framing manipulation (using common colors to create event framing that would supposedly make it easy for participants to spot common branches and cancel them during the editing phase) had much effect on tests of branch independence. The failure to find event-framing effects does not disprove their possible existence, of course, but the continued failure to find any substantial effect in several experiments (see also Birnbaum, 2004a; 2004b) appears to show that this type of framing is of minor importance.
In sum, these results contradict both original prospect theory (Kahneman & Tversky, 1979) and cumulative prospect theory (Tversky & Kahneman, 1992), with or without their editing principles of combination and cancellation. Violations of coalescing refute any version of RDU, RSDU, or CPT, with or without the editing rule of combination. Violations of branch independence refute “stripped” original prospect theory, extrapolated to three branch gambles, with or without the editing principle of cancellation.
The GDU model does a much better job than EU or CPT. However, it does not predict violations of upper coalescing.
The prior TAX model does a better job than either prospect theory of predicting the effects of branch splitting and violation of branch independence, but it did not always predict the majority choice in cases where the number of branches differed between the gambles in the choice. Nor did adjustments of parameters allow TAX to perfectly account for this effect (Appendix A). Although there are significant violations of event-splitting independence, consistent with TAX, the effects of splitting the lower branch of a gamble are not as large as predicted by the prior model.
7. Discussion
There are several conclusions that can be drawn.
• First, the evidence against expected utility theory (provided by the significant changes from row to row within each of Tables 1–5) is overwhelming.
• Second, evidence of event-splitting effects, which refute rank-dependent utility theories, including CPT, is also overwhelming. In each series of Experiments 1 and 2, there are significant effects of splitting in each repetition and series of Experiment 1, and in each study of Experiment 2, tested separately. These effects are not only significant, but are large enough to reverse the majority preference in Choices 6 and 9 (Tables 1 and 4), 10 and 17 (Tables 2 and 5), 5 and 11 (Table 3), 7 and 13 (Table 3), 16 and 19 (Table 4) and 14 and 8 (Table 5).
• Third, there is evidence of significant violations of restricted branch independence, but these violations are significantly more frequent in the opposite direction from the effects predicted by CPT. The violations are also opposite the direction of the Allais paradox, so it is hard to maintain the position (required by any RDU or CPT model) that Allais paradoxes are produced by violations of branch independence. They are consistent, however, with prior TAX.
• Fourth, there is strong evidence that splitting the branch with the higher valued consequence improves a gamble. This effect contradicts the GDU model, which satisfies upper coalescing as well as EU and CPT, which satisfy all forms of coalescing.
• Fifth, there is statistically significant, but less dramatic evidence that splitting the branch with the lower-valued consequence can make a gamble worse. Splitting a .2 branch to win $50 significantly improves the probability of choosing the gamble when $50 is the highest consequence in the gamble and splitting the same branch significantly lowers the proportion of choosing the gamble when $50 is the lowest consequence in the gamble. Although significant, these violations of event-splitting independence are not as consistent or impressive as the effects of splitting the higher consequence. A possible confounding factor in these studies is the fact that the design involves choices between gambles with differing numbers of branches. Perhaps a bias to prefer gambles with more branches to win positive prizes competes with the effect of splitting the lower valued branch of a gamble.
• Sixth, the prior TAX model predicted successfully the majority choices among gambles with equal numbers of branches, but did not make correct predictions in all cases of gambles with unequal numbers of branches.
• Seventh, the effects of splitting and coalescing appear similar for gambles involving large and small consequences, and the prior TAX model (even with its linear utility function) in most cases correctly describes the main trends in the data, unlike the CPT model.
• Eighth, choices among gambles with large consequences show the same types of violations of stochastic dominance and violations of coalescing as have been reported for gambles with modest consequences. No qualitative difference was found, apart from requiring a nonlinear utility function to extrapolate the TAX model to gambles involving large consequences.
In summary, the present data add to the growing case against rank dependent utility models and cumulative prospect theory as descriptive theories of risky decision making.
Gains Decomposition Utility (GDU) (Marley & Luce, 2001) has some similarities to the TAX model, but is distinct from it. A version of this model was fit to the data, and was found to be more accurate than CPT, but not quite as accurate as the TAX model, even when it used more free parameters. This model was able to correctly reproduce the majority choices in the first experiment, but it failed to describe the effects of splitting the higher consequences in the second experiment.
[Note to Editor and Reviewers: Appendices A and B may need to be reduced in the interest of saving space in the journal. Nevertheless, they contain important information for reviewers.]
8. Appendix A: Model Fitting
Let G = (x1, p1; x2, p2; ... xj, pj; ...; xi, pi; ...; xn, pn) refer to a gamble with n distinct branches, where the consequences are ordered such that x1 ≥ x2 ≥ ... xi ≥ ...≥ xn ≥ 0, and Σ pi = 1.
The CPT model for such nonnegative consequences is the same as that of RDU:
[pic] (4)
where [pic] is the decumulative probability that a prize is equal to [pic] or greater, and W(P) is a monotonic function that assigns decumulative weight to decumulative probability; W(0) = 0 and W(1) = 1
In practice, the decumulative weighting function (Tversky & Wakker, 1995) has been fit with the following equation:
[pic] (5)
where P is decumulative probaility, δ is a constant that reflects risk aversion (δ < 1) or risk-seeking (δ > 1) for fifty-fifty gambles, and γ is a constant that determines whether people will be risk-seeking (γ < 1) or risk-averse (γ > 1) for gambles with small probabilities. Tversky and Kahneman (1992) approximated the utility (“value”) function as a power function, [pic].
To compute the fit of models to observed choice proportions, it is assumed that choice probability depends on the difference in utilities of the gambles, divided by the standard deviation of the difference:
[pic] (6)
where P(A,B) is the probability of choosing gamble A over B; U(A) and U(B) are the average utilities of A and B, and [pic] is the standard deviation of the difference in utilities between the gambles. This model is similar to Thurstone’s (1927) law of comparative judgment, except it uses the logistic instead of the cumulative normal transformation. For simplicity, it is often assumed that the standard deviations are constant, in which case [pic] is the only new parameter needed.
The CPT model thus has four free parameters, α, β, γ, and δ. The “prior CPT model” refers to Equations 4, 5, and 6, with 3 parameters estimated from previous data (Tversky & Kahneman, 1992): β = 0.88, γ = .61, and δ = .724, and with α free.
A “special case” of the TAX model (Birnbaum & Chavez, 1997) can be written as follows:
[pic] (7)
where TAX(G) is the utility of the gamble; t(p) is a function of probability; u(x) is the utility function of money, and δ is the single configural parameter; δ/(n + 1) is the proportion of weight taken from a branch with a higher consequence and transferred to each branch with a lower consequence. Note that the weight transferred is proportional to the (transformed) probability of the branch giving up weight.
The special TAX model (Equations 6 and 7), with [pic] and [pic], also has four free parameters, α, β, γ, and δ. The “prior” TAX model refers to “special TAX” (Equation 7), plus the assumptions that u(x) = x, 0 < x < $150, t(p) = p.7, δ = 1, with α free. Although this special case is equivalent to that in Birnbaum and Chavez (1997), the ordering of the branches here has been reversed to conform to that in CPT, so δ = 1 in this equation corresponds to δ = –1 in Birnbaum and Chavez (1997). These parameters were used by Birnbaum (1999a; 2004a; 2004b) to successfully describe a variety of phenomena for choices between gambles.
Marley and Luce (2001) axiomatized a GDU model in which binary gambles are represented by RDU:
[pic] (8)
where [pic]. To calculate values of three-branch gambles, the gains decomposition rule (Luce, 2000, p. 200-202) is applied as follows:
[pic] (9)
Expression 9 can be applied iteratively to make predictions for multi-branch gambles, without requiring any new parameters beyond those required for binary gambles.
To fit this model, it was assumed that [pic], and the weighting function is approximated by the expression developed by Prelec (1998) and by Luce (2000):
[pic] (10)
With the assumptions of Equations 6, 8, 9, and 10, this model has a total of four parameters, α, β, γ, and δ. These functions match those used by Luce (2000) in his illustration of a “less restrictive” model than RDU, which can violate coalescing and stochastic dominance.
Equation 8 implies that [pic]; i.e., that F = [pic], so this model implies upper coalescing, [pic] [pic] [pic]; however, it need not satisfy lower coalescing, except in the special case where GDU reduces to RDU. There is a rounding error in the calculations in Luce (2000, Chapter 5), which appear to show that this GDU model can account for Wu’s (1994) violations of upper tail independence. Because upper coalescing follows from binary RDU and lower gains decomposition, and because Wu’s upper tail independence follows from upper coalescing and comonotonic restricted branch independence, this GDU model must satisfy upper tail independence.
The same symbols are used here for analogous parameters in CPT, TAX, and GDU models, to show that these models are equally complicated. But keep in mind that the parameters have different meanings (and different values) in the models. With β = 1, γ = .38 and δ = 1.34, GDU, prior TAX (with β = 1, γ = .7 and δ = 1), and prior CPT (with β = 0.88, γ = .61, and δ = .72) make similar predictions for two branch gambles with prizes in the domain of Experiment 2.
Table 6 shows results of fitting the choice proportions for Experiment 1. Both GDU and TAX do a much better job of predicting modal choices than CPT with the same number of free parameters. This should not be a surprise since no version of CPT can predict violations of stochastic dominance. The prior TAX model with its linear utility function fails to predict two choices (#9 and 16). However, by estimating one new parameter from the data (the exponent of the utility function), TAX(2) perfectly reproduces all of the majority choices. Consistent with intuition, the fitted exponent is less than one, [pic]. GDU(4), with parameters estimated from the data, is also able to correctly reproduce all of the majority choices. In Experiment 1, all tests of stochastic dominance and of coalescing are combined tests of both upper and lower coalescing. So, by violating lower coalescing, GDU is able to account for the data of Experiment 1.
Table 7 shows the fit of the models to Experiment 2, which had no tests of stochastic dominance, but does include separate tests of upper and lower coalescing, restricted branch independence, and branch-splitting independence. CPT again fails to account for the tests of coalescing and it cannot account for both the Allais paradoxes and violations of restricted branch independence with the same parameters.
Both TAX and GDU have problems in Experiment 2. GDU cannot account for violations of upper coalescing. TAX fails to predict that people tend to prefer gambles with a greater number of branches with positive consequences.
Insert Tables 6 and 7 about here.
9. Appendix B: True and Error Model
When presented with the same choice, people do not always make the same decision. The fact that people sometimes judge the lighter of two lifted weights to be “heavier” has been attributed to variable and constant “errors” of sensation. Can such “errors” account for violations of stochastic dominance or violations of coalescing in the tests of the Allais paradox?
Various models (aka “stories”) of error have recently been discussed by behavioral economists (Carbone & Hey, 2000; Loomes & Sugden, 1995; Sopher & Gigliotti, 1993). These models of error have a considerable history in psychology; see Luce’s (1994) historical review, featuring the classic work of Fechner and Thurstone. Such models have been applied to tests of stochastic dominance, lower and upper cumulative independence, and branch independence by Birnbaum (2004b), who showed that under any of several error models, violations of these properties are systematic and cannot be attributed to random “error.”
This appendix shows violations of stochastic dominance and branch splitting effects in Allais tests of Experiment 1, which refute CPT, are not due to random “errors.” Recall that each participant in Experiment 1 completed the task twice. We can use inconsistency between replications to estimate the probability of “errors.” As a rough approximation, the mean number of agreements, averaged over participants and over 20 choices in Experiment 1 was 15.26, or 76.3% agreement, similar to the figure reported by Birnbaum (1999b) in a comparable experiment.
Table 8 shows the number of participants who showed each choice pattern for four tests of stochastic dominance (Choices 5 and 15, which are the same choice, with two repetitions). The first column lists each choice pattern, where S and V indicate satisfaction and violation of stochastic dominance, respectively. For example, the last entry in the second column (VVVV) shows that 94 participants violated stochastic dominance on Choices 5 and 15 in both repetitions. The second to last row (VVVS) shows that 18 people violated stochastic dominance on both choices in the first repetition, but satisfied it only on Choice 15 of the second repetition.
The true and error model assumes that each participant is either governed by stochastic dominance or systematically violates it. In addition, it is assumed that each person has a probability of e of making an “error” that results in a response differing from that person’s “true” preference on a particular choice. Let a = the probability that a person is truly governed by stochastic dominance and e = the probability of an error. The probability that a person would show four violations out of four tests is represented as follows:
[pic].
In other words, a person who is truly governed by stochastic dominance has made four errors, and a person who truly violates stochastic dominance has correctly reported his or her preference four times. The probability that a person would show a particular sequence of m violations in n tests, as in Table 8, is as follows:
[pic] (8)
This two parameter model is fit to the observed frequencies, solving for a and e, to minimize the [pic] between observed and fitted frequencies.
For Choices 5 and 15, the model indicates that 84% of participants “truly” violated stochastic dominance on all four presentations (1 – a = 0.84), except for random errors on 16% of the trials. This model has [pic](13) = 18.4. In the analysis of Choices 7 and 18, estimates are 79% true violations, 20% errors, and [pic](13) = 27.8. Fitted frequencies are also shown in Table 8.
Insert Table 8 about here.
To illustrate the connection between this model and another way to conceptualize “error,” participants were divided into two groups according to their internal consistency of responding. There were 87 participants who agreed with their own decisions fewer than 15 times out of 20 repeated trials, and there were 113 who made the same choices on 15 or more of 20 decisions. The observed rates of violation of stochastic dominance on Choices 5 and 15 were 0.647 and 0.841 in the two groups (significantly more than 0.5 in both cases), with the more consistent group showing the higher rate of observed violation. Insert Table 9 about here.
When the true and error model is applied to the 87 less consistent participants, estimates were 1 – a = 0.85 and e = 0.30. In a separate analysis of the 113 with higher consistency, 1 – a = 0.86 and e = 0.09. Note that the estimated rates of true violation are similar in the two groups (85% and 86%), but their inferred error rates are (quite) different (30% and 9%). These analyses further strengthen the case that these observed violations of CPT (and EU) are not due to random error, but rather to systematic intention. Table 9 shows the observed and fitted frequencies for this analysis.
Table 10 shows the true and error model applied to Choices 11 and 13, which are the same as Choices 5 and 7, except for coalescing. Here the data are quite different: no one violated stochastic dominance four times when the gambles were presented in the appropriately split form, and 136 satisfied it on all four choices. When the model is fit, 1 – a = 0.05 and e = 0.13. However, this model overestimates the number who consistently violate stochastic dominance: the model predicts seven should have had three or more violations, whereas the data show only one.
Insert Table 10 about here.
Table 11 shows the number of participants who showed each preference pattern on Choices 6 and 9, in the first and second replicates, which test coalescing apart from stochastic dominance. In this table, S indicates preference for the “safe” gamble (with 11 marbles to win $1 Million, otherwise $2) rather than the “risky” (R) gamble (with 10 marbles to win $2 Million, otherwise $2). Here SRSS refers to a case where the participant chose S on the first replicate of Choice 6, R on Choice 9, and chose S on both of these in the second replicate.
A three parameter true and error model was fit to these data (See Birnbaum, 2004b). In this model, the three parameters are the probability that a person truly prefers R over S in Choice 6, the conditional probability that this person has the same true preference in Choice 9, and the probability of making an “error” expressing preferences. The estimates are that 89.3% of participants truly prefer R over S in Choice 6 and that only 33.8% of these have the same true preference in Choice 9, where the gambles have been split to apparently make the S gamble better and the R gamble worse. The error probability is estimated to be 0.185, and [pic](12) = 14.3. Insert Table 11 about here.
Put another way, this model indicates that 59% of participants truly switch from R in Choice 6 to S in Choice 9 and that no one switched in the other direction, even though the observed data show 46% reversed between #6 and #9 and that 12% made the opposite switch, averaged over replicates. [Table 11 shows that 56 people (28%) switched in this fashion on both replicates, compared to 4 who made the opposite switch both times, z = 6.71]. A two-parameter model that assumes no one truly switched fits much worse, [pic](13) = 236.0.
The analysis of Choices 10 and 17 (Table 2) gave similar results, also shown in Table 11. According to the 3 parameter model, 61% “truly” switched from R to S as a result of splitting the upper branch of S and the lower branch of R and no one switched in the opposite direction, and errors are made on 16.7% of trials; [pic] = 13.0. There were 62 observed (31%) who made this same switch on both replicates compared to only 1 who was observed making the opposite switch.
An analysis of restricted branch independence is shown in Table 12. In this case, one can fit a series of nested models, starting with one that allows all four patterns, just three, or just two (SS’ and RR’; i.e., branch independence). In the observed data, there were 52 and 50 participants who showed the SR’ pattern in replicates 1 and 2, compared to 21 and 16 who showed the opposite reversals. These splits are statistically significant by the binomial test, z = 3.63 and 4.19. There were 20 who showed the SR’ pattern on both replicates. According to the 3-parameter model, which fit best in relation to its number of estimated parameters, the true probability of choosing the S gamble is 0.82, and the conditional probability of choosing S’ given a choice of S is only 0.75, with the error rate estimated to be 0.19. These effects are opposite in direction from the Allais paradox defined as the change from Choice 10 to Choice 8, and the effects are opposite in direction the predictions of prior CPT with its inverse-S weighting function. Similar results were found for Choices 9 and 16, as shown in the last two columns of Table 12.
Insert Table 12 about here.
Table 13 shows how the true and error model can be used to analyze the effects of upper coalescing in Experiment 2. Because participants in Experiment 2 performed each choice only once, one uses lower coalescing as the error term for estimating the effects of upper coalescing. The assumption that people satisfy both upper and lower coalescing yields, [pic] = 242.7; the model allowing violations of upper coalescing and satisfying lower coalescing yields, [pic](11)= 42.1. In this model, the probability of choosing R in Choice 6 or 21 is estimated to be 0.64, the estimated conditional probability to choose R in Choice 11 given R in Choice 6 is 0.482, the conditional probability to choose R in Choice 11 given a choice of S in Choice 6 is 0, and the “error” probability is estimated to be 0.22.
References
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Table 0.
| |Properties Tested |
| |RBI |Upper Coalescing |Lower |BSI |SD |
| | | |Coalescing | | |
|SWU |Satisfies |Violates |Violates |Satisfies |Violates |
|CPT |Violates |Satisfies |Satisfies |Moot |Satisfies |
| |([pic]) | | | | |
|GDU |Violates |Satisfies |Violates |Violates |Violates |
| |([pic]) | |(split worse) | | |
|TAX |Violates |Violates |Violates |Violates |Violates |
| |([pic]) |(split better) |(split worse) | | |
Notes: GDU here refers to the lower gains decomposition model of Marley and Luce (2001), which assumes that a three branch gamble can be decomposed into a binary gamble to win the lowest consequence and a binary gamble to win one of the two higher consequences. This model assumes binary RDU, but does not satisfy coalescing except in a special case where it reduces to full RDU. Luce (personal communication) and Marley are currently working on other models that satisfy gains decomposition (GD) but do not reduce to binary RDU in the two branch case. The more general GD models need not satisfy upper coalescing. RBI = Restricted branch independence; BSI = branch splitting independence; SD = first order stochastic dominance. SWU = subjectively weighted utility; CPT = cumulative prospect theory; GDU = lower gains decomposition; TAX = transfer of attention exchange model.
Table 1. Dissection of Allais Paradox with large Consequences (Series A). Each entry is the percentage choosing R.
|No |Notes |Choice | | |Prior TAX |Prior CPT |
| | |Second Gamble, S |First Gamble, R |Rep1 |Rep 2 |S |R |S |R |
|6 | |11 blues to win $1,000,000 |10 reds to win $2,000,000 |76 |77 |125 K |236 K |132 K |248 K |
| | |89 whites to win $2 |90 whites to win $2 | | | | | | |
|9 |Split # 6 |10 blues to win $1,000,000 |10 reds to win $2,000,000 |35 |39 |241 K |172 K |132 K |248 K |
| | |01 blues to win $1,000,000 |01 whites to win $2 | | | | | | |
| | |89 whites to win $2 |89 whites to win $2 | | | | | | |
|12 |RBI #9 |100 blues to win $1,000,000 |10 reds to win $2,000,000 |40 |44 |1000 K |810 K |1000 K |1065 K |
| | | |89 blues to win $1,000,000 | | | | | | |
| | | |01 white marble to win $2 | | | | | | |
|16 |RBI #9, 12 |89 reds to win $2,000,000 |89 reds to win $2,000,000 |43 |45 |1400 K |1449 K |1714 K |1825 K |
| | |10 blues to win $1,000,000 |10 reds to win $2,000,000 | | | | | | |
| | |01 blues to win $1,000,000 |01 whites to win $2 | | | | | | |
|19 |Coal #16 |89 reds to win $2,000,000 |99 reds to win $2,000,000 |35 |38 |1541 K |1282 K |1714 K |1825 K |
| | |11 blues to win $1,000,000 |01 whites to win $2 | | | | | | |
Table 2. Dissection of Allais Paradox into Branch Independence and Coalescing (Series B).
|No |Notes |Choice | | |Prior TAX model |Prior CPT model |
| | |First Gamble, S |Second Gamble, R |Rep 1 |Rep 2 |S |R |S |R |
|10 | |15 blues to win $500,000 |10 reds to win $1,000,000 |73 |67 |76 K |118 K |81 K |124 K |
| | |85 whites to win $11 |90 whites to win $11 | | | | | | |
|17 |Split #10 |10 blues to win $500,000 |10 reds to win $1,000,000 |24 |30 |100 K |82 K |81 K |124 K |
| | |05 blues to win $500,000 |05 whites to win $11 | | | | | | |
| | |85 whites to win $11 |85 whites to win $11 | | | | | | |
|20 |RBI #17 |10 blues to win $500,000 |10 reds to win $1,000,000 |28 |31 |500 K |378 K |500 K |470 K |
| | |85 blues to win $500,000 |85 blues to win $500,000 | | | | | | |
| | |05 blues to win $500,000 |05 whites to win $11 | | | | | | |
|14 |RBI # 17, 20 |85 reds to win $1,000,000 |85 reds to win $1,000,000 |40 |47 |684 K |674 K |834 K |791 K |
| | |10 blues to win $500,000 |10 reds to win $1,000,000 | | | | | | |
| | |05 blues to win $500,000 |05 whites to win $11 | | | | | | |
|8 |Coalesce #14 |85 reds to win $1,000,000 |95 reds to win $1,000,000 |13 |14 |757 K |591 K |834 K |791 K |
| | |15 blues to win $500,000 |05 whites to win $11 | | | | | | |
Table 3. Violations of Stochastic Dominance and Coalescing.
|No | |Choice |Percentages of |
| | | |Violations |
| | |G+ |G– |Rep 1 |Rep 2 |
|5 | |90 reds to win $960,000 |85 reds to win $960,000 |79 |76 |
| | |05 blues to win $140,000 |05 blues to win $900,000 | | |
| | |05 whites to win $120,000 |10 whites to win $120,000 | | |
|11 | |85 reds to win $960,000 |85 reds to win $960,000 |8 |12 |
| | |05 blues to win $960,000 |05 blues to win $900,000 | | |
| | |05 greens to win $140,000 |05 greens to win $120,000 | | |
| | |05 whites to win $120,000 |05 whites to win $120,000 | | |
|15 | |90 reds to win $960,000 |85 blacks to win $960,000 |77 |71 |
| | |05 yellows to win $140,000 |05 blues to win $900,000 | | |
| | |05 pinks to win $120,000 |10 whites to win $120,000 | | |
|7 | |94 blacks to win $1,100,000 |91 reds to win $1,100,000 |74 |69 |
| | |03 yellows to win $80,000 |03 blues to win $1,000,000 | | |
| | |03 purples to win $60,000 |06 whites to win $60,000 | | |
|13 | |91 blacks to win $1,100,000 |91 reds to win $1,100,000 |15 |12 |
| | |03 pinks to win $1,100,000 |03 blues to win $1,000,000 | | |
| | |03 yellows to win $80,000 |03 greens to win $60,000 | | |
| | |03 purples to win $60,000 |03 whites to win $60,000 | | |
|18 | |94 reds to win $1,100,000 |91 reds to win $1,100,000 |63 |70 |
| | |03 blues to win $80,000 |03 blues to win $1,000,000 | | |
| | |03 whites to win $60,000 |06 whites to win $60,000 | | |
In Choices 5, 11, and 15, G+ was first, and in Choices 7, 13, and 18 it was second. Choices 5, 11, and 18 are framed.
Table 4. Dissection of Allais Paradox (Series A): Percentages choosing “Risky” Gamble
| |Choice |Condition |Prior |Prior |
|No. | | |TAX model |GDU Model |
| |Gamble, S |Gamble, R |A3 |A4 |All |S |R |S |R |
|6 |20 black win $40 |10 black win $98 |60 |60 |61 |9.0 |13.3 |8.4 |12.9 |
| |80 purple win $2 |90 purple win $2 | | | | | | | |
|11 |10 red win $40 |10 black win $98 |35 |33 |35 |11.1 |13.3 |8.4 |12.9 |
| |10 blue win $40 (39) |90 purple win $2 | | | | | | | |
| |80 white win $2 | | | | | | | | |
|21 |20 black win $40 |10 red win $98 |55 |63 |57 |9.0 |9.6 |8.4 |7.2 |
| |80 purple win $2 |10 blue win $2 (3) | | | | | | | |
| | |80 white win $2 | | | | | | | |
|9 |10 red win $40 |10 red win $98 |39 |44 |41 |11.1 |9.6 |8.4 |7.2 |
| |10 blue win $40 (39) |10 blue win $2 (3) | | | | | | | |
| |80 white win $2 |80 white win $2 | | | | | | | |
|12 |Sure to win $40 |10 red win $98 |49 |50 |49 |40.0 |30.6 |40.0 |31.9 |
| | |80 blue win $40 | | | | | | | |
| | |10 white win $2 | | | | | | | |
|16 |80 red win $98 |80 red win $98 |56 |59 |58 |59.8 |62.6 |65.0 |65.8 |
| |10 blue win $40 (41) |10 blue win $98 (97) | | | | | | | |
| |10 white win $40 |10 white win $2 | | | | | | | |
|13 |80 red win $98 |80 red win $98 |58 |67 |61 |68.0 |62.6 |71.4 |65.8 |
| |20 white win $40 |10 blue win $98 (97) | | | | | | | |
| | |10 white win $2 | | | | | | | |
|7 |80 red win $98 |90 red win $98 |21 |25 |23 |59.8 |54.7 |65.0 |65.8 |
| |10 blue win $40 (41) |10 white win $2 | | | | | | | |
| |10 white win $40 | | | | | | | | |
|19 |80 red win $98 |90 red win $98 |23 |31 |26 |68.0 |54.7 |71.4 |65.8 |
| |20 white win $40 |10 white win $2 | | | | | | | |
Note: Values in parentheses show consequences used in Condition A4.
Table 5. Dissection of Allais Paradox (Series B): Percentages choosing “Risky” Gamble
| |Choice |Condition |Prior |Prior |
|No. | | |TAX Model |GDU Model |
| |Gamble, S |Gamble, R |A3 |A4 |All |S |R |S |R |
|10 |15 red win $50 |10 blue win $100 |78.0 |75.0 |76.9 |13.6 |18.0 |13.1 |17.6 |
| |85 black win $7 |90 white win $7 | | | | | | | |
|22 |15 red win $50 |10 black win $100 |68.3 |73.0 |70.6 |13.6 |14.6 |13.1 |12.6 |
| |85 black win $7 |05 purple win $7 (8) | | | | | | | |
| | |85 green win $7 | | | | | | | |
|15 |10 red win $50 |10 blue win $100 |44.4 |42.2 |45.3 |15.6 |18.0 |13.1 |17.6 |
| |05 blue win $50 (49) |90 white win $7 | | | | | | | |
| |85 white win $7 | | | | | | | | |
|17 |10 red win $50 |10 black win $100 |41.7 |46.4 |44.2 |15.6 |14.6 |13.1 |12.6 |
| |05 blue win $50 (49) |05 purple win $7 (8) | | | | | | | |
| |85 white win $7 |85 green win $7 | | | | | | | |
|20 |Sure to win $50 |10 black win $100 |48.4 |52.2 |50.2 |50.0 |40.1 |50.0 |44.0 |
| | |85 purple win $50 | | | | | | | |
| | |05 green win $7 | | | | | | | |
|14 |85 red win $100 |85 black win $100 |60.5 |65.1 |64.5 |68.4 |69.7 |74.9 |77.5 |
| |10 white win $50 (51) |10 yellow win $100 (99) | | | | | | | |
| |05 blue win $50 |05 purple win $7 | | | | | | | |
|5 |85 red win $100 |95 red win $100 |32.7 |41.0 |36.9 |68.4 |62.0 |74.9 |77.5 |
| |10 white win $50 (51) |05 white win $7 | | | | | | | |
| |05 blue win $50 | | | | | | | | |
|18 |85 black win $100 |85 black win $100 |64.8 |69.8 |68.0 |75.7 |69.7 |79.8 |77.5 |
| |15 yellow win $50 |10 yellow win $100 (99) | | | | | | | |
| | |05 purple win $7 | | | | | | | |
|8 |85 black win $100 |95 red win $100 |26.4 |30.3 |27.5 |75.7 |62.0 |79.8 |77.5 |
| |15 yellow win $50 |05 white win $7 | | | | | | | |
Note: Values in parentheses show consequences used in Condition A4.
Table 6. Fit of Models to the Group Choice Proportions (Experiment 1).
| |alpha |beta |gamma |delta |SS deviations |Failures to |
| | | | | | |predict Majority Choices |
|Experiment 1: | | | | | | |
|Prior CPT (1) |3.99 E-05 |0.88 |0.61 |0.72 |0.761 |9 (#9,16,17,5,15,7,18, 12, 19) |
|CPT (4) |0.20 |0.23 |0.47 |0.93 |0.508 |6 (#9, 17, 5, 15, 7, 18) |
|Prior GDU (1) |5.23 E-06 |(1) |(1.382) |(0.542) |0.475 |4 (#9, 16, 14, 19) |
|GDU (4) |2.74 E-03 |0.574 |0.917 |0.315 |0.139 |0 |
|Prior TAX (1) |5.41 E-06 |(1) |(.7) |(1) |0.241 |2 (#9, 16) |
|TAX(2) |7.26 E-05 |0.802 |(.7) |(1) |0.236 |0 |
|TAX(3) |6.91 E-06 |0.961 |0.295 |(1) |0.185 |0 |
|TAX (3) |9.70 E-03 |0.498 |0.347 |(0) |0.098 |0 |
| | | | | | | |
Notes: Values in parentheses were fixed. TAX(4) with all parameters free, provided no additional improvement over TAX(3) with delta fixed to zero.
Table 7. Fit of Models to the Group Choice Proportions (Experiment 2).
| |alpha |beta |gamma |delta |SS deviations |Failures to |
| | | | | | |predict Majority Choices |
|Experiment 2: | | | | | | |
|Prior CPT (1) |0.323 |0.88 |0.61 |0.72 |0.769 |8 (#17,9,16,20,11,13,18,15) |
|CPT (4) |0.920 |0.555 |0.810 |0.998 |0.378 |7 (#17, 9, 16, 11,13,18, 15) |
|Prior GDU(1) |0.077 |(1) |(1.382) |(0.542) |0.434 |9 (#20,11,7,21,13,22,18,15,5) |
|GDU (4) |1.864 |0.462 |0.863 |0.909 |0.396 |8 |
| | | | | | |(#17, 16, 11, 21, 13, 15, 5, 8) |
|Prior TAX (1) |0.049 |(1) |(.7) |(1) |0.335 |5 |
| | | | | | |(#20, 11, 13, 18, 15) |
|TAX (3) |0.864 |0.591 |0.583 |(0) |0.271 |6 |
| | | | | | |(#17, 16, 11, 13, 18, 15) |
TAX (4) provided no improvement over TAX (3).
Table 8. Numbers of Participants showing each combination of Choices in tests of Stochastic Dominance in Coalesced Form (Exp 1). Fitted show predicted Frequencies of True and Error model.
| |Choices #5 and 15 |Choices #7 and 18 |
|Choice Pattern |Observed |Fitted |Observed |Fitted |
|SSSS |9 |15.95 |15 |17.36 |
|SSSV |3 |3.70 |3 |5.38 |
|SSVS |4 |3.70 |2 |5.38 |
|SSVV |4 |3.73 |4 |5.21 |
|SVSS |5 |3.70 |4 |5.38 |
|SVSV |1 |3.73 |3 |5.21 |
|SVVS |3 |3.73 |7 |5.21 |
|SVVV |14 |16.16 |13 |16.48 |
|VSSS |8 |3.70 |12 |5.38 |
|VSSV |3 |3.73 |9 |5.21 |
|VSVS |6 |3.73 |4 |5.21 |
|VSVV |9 |16.16 |24 |16.48 |
|VVSS |5 |3.73 |6 |5.21 |
|VVSV |14 |16.16 |10 |16.48 |
|VVVS |18 |16.16 |10 |16.48 |
|VVVV |94 |82.19 |74 |63.91 |
Notes: S = Satisfaction of stochastic dominance, V = violation of stochastic dominance. The first two letters indicate the choice pattern on the first repetition of Choices 5 and 15, respectively, and the second two indicate the choice pattern on the second repetition, respectively.
Table 9. Numbers of Participants showing each choice combination in tests of Stochastic Dominance on Choices 5 and 15. Participants were separated in groups according to their internal consistency.
|Choice Pattern |Inconsistent Group ( n = 87) |Consistent Group ( n = 113) |
| |Observed, |Fitted |Observed, |Fitted to 113 |
|SSSS |3 |3.82 |6 |11.14 |
|SSSV |3 |2.76 |0 |1.12 |
|SSVS |2 |2.76 |2 |1.12 |
|SSVV |3 |3.83 |1 |0.72 |
|SVSS |3 |2.76 |2 |1.12 |
|SVSV |1 |3.83 |0 |0.72 |
|SVVS |3 |3.83 |0 |0.72 |
|SVVV |9 |7.83 |5 |6.45 |
|VSSS |5 |2.76 |3 |1.12 |
|VSSV |3 |3.83 |0 |0.72 |
|VSVS |5 |3.83 |1 |0.72 |
|VSVV |3 |7.83 |6 |6.45 |
|VVSS |5 |3.83 |0 |0.72 |
|VVSV |10 |7.83 |4 |6.45 |
|VVVS |10 |7.83 |8 |6.45 |
|VVVV |19 |17.84 |75 |67.27 |
S = Satisfied, V = violated stochastic dominance on Choices 5, 15, in replicates 1 and 2, respectively.
Participants were separated according to their internal consistency over 20 choices. Inconsistent participants agreed in fewer than 15 of 20 decisions, consistent participants agreed in 15 or more of 20 decisions. Although the two groups showed 65% and 85% observed violations of stochastic dominance, true and error model implies that both groups had approximately equal “true” violation rates, but very different error rates.
Table 10. Numbers of Participants showing each combination of Choices in tests of Stochastic Dominance in Split Form. Predicted Frequencies of True and Error model shown.
|Choice Pattern |Observed |Fitted Frequency, 2 |
| |Frequency |parameters |
| |# 11, 13 | |
|SSSS |136 |111.00 |
|SSSV |10 |15.97 |
|SSVS |12 |15.97 |
|SSVV |7 |2.42 |
|SVSS |12 |15.97 |
|SVSV |1 |2.42 |
|SVVS |2 |2.42 |
|SVVV |4 |1.17 |
|VSSS |6 |15.97 |
|VSSV |1 |2.42 |
|VSVS |3 |2.42 |
|VSVV |1 |1.17 |
|VVSS |5 |2.42 |
|VVSV |0 |1.17 |
|VVVS |0 |1.17 |
|VVVV |0 |5.93 |
Notes: S = Satisfaction of stochastic dominance, V = violation of stochastic dominance. The first two letters indicate the choice pattern on the first and second repetition of Choice 11, respectively, and the second two indicate the choice pattern on the two repetitions of Choice 13. [pic] = 44.5.
Table 11. Numbers of Participants showing each combination in tests of Event Splitting (Exp 1).
|Choice Pattern |Choices #6 and #9 |Choices # 10 and 17 |
| |Observed |Fitted |Observed |Fitted |
|RRRR |31 |29.24 |19 |18.65 |
|RRRS |17 |17.89 |11 |15.24 |
|RRSR |4 |6.72 |5 |3.90 |
|RRSS |7 |4.51 |6 |3.86 |
|RSRR |25 |17.89 |18 |15.24 |
|RSRS |56 |53.88 |62 |60.48 |
|RSSR |3 |4.51 |7 |3.86 |
|RSSS |9 |14.21 |17 |16.17 |
|SRRR |3 |6.72 |3 |3.90 |
|SRRS |3 |4.51 |2 |3.86 |
|SRSR |4 |1.98 |1 |1.59 |
|SRSS |1 |3.04 |1 |4.84 |
|SSRR |5 |4.51 |3 |3.86 |
|SSRS |14 |14.21 |15 |16.17 |
|SSSR |3 |3.04 |2 |4.84 |
|SSSS |14 |12.14 |28 |23.52 |
Notes: S = Chose “safe” gamble with higher probability to win smaller prize, R = chose “risky” gamble. The first two letters indicate the choice pattern on the first replicate, respectively, and the second two indicate the choice pattern on the second replication.
Table 12. Numbers of Participants showing each choice combination in tests of Restricted Branch Independence. (Experiment 1)
| |Choices # 17 and 14 |Choices # 9 and 16 |
|Choice Pattern |Observed |Fitted Frequency, |Observed Frequency, |Fitted Frequency, |
| |Frequency | | | |
|SS'SS' |67 |53.45 |53 |44.20 |
|SS'SR' |18 |16.67 |10 |14.94 |
|SS'RS' |5 |12.80 |10 |11.58 |
|SS'RR' |10 |4.75 |5 |5.15 |
|SR'SS' |17 |16.67 |9 |14.94 |
|SR'SR' |20 |21.14 |21 |18.35 |
|SR'RS' |2 |4.75 |5 |5.15 |
|SR'RR' |13 |8.44 |16 |9.97 |
|RS'SS' |4 |12.80 |13 |11.58 |
|RS'SR' |4 |4.75 |6 |5.15 |
|RS'RS' |4 |3.83 |5 |4.29 |
|RS'RR' |8 |4.57 |10 |6.61 |
|RR'SS' |3 |4.75 |5 |5.15 |
|RR'SR' |8 |8.44 |4 |9.97 |
|RR'RS' |5 |4.57 |9 |6.61 |
|RR'RR' |11 |16.63 |16 |23.39 |
Notes: S = Chose “safe” gamble with higher probability to win smaller prize, R = chose “risky” gamble, S’ and R’ are the same gambles with the consequence on the common branch increased from the smallest consequence to the largest.
Table 13. Numbers of Participants showing each combination of Choices 6, 11, 21, and 9 in Experiment 2.
|Choice Pattern |Observed |Fitted Frequency, 4 |
| |Frequency |parameters |
| |# 6, 11, 21, 9 | |
|RRRR |114 |105.52 |
|RRRS |55 |57.55 |
|RRSR |30 |31.11 |
|RRSS |22 |23.66 |
|RSRR |72 |57.55 |
|RSRS |98 |120.05 |
|RSSR |31 |23.66 |
|RSSS |86 |61.62 |
|SRRR |23 |31.11 |
|SRRS |18 |23.66 |
|SRSR |9 |16.41 |
|SRSS |16 |35.18 |
|SSRR |30 |23.66 |
|SSRS |65 |61.62 |
|SSSR |28 |35.18 |
|SSSS |132 |121.46 |
Notes: Choices are 6, 11, 21, and 9, respectively. S = Choice of “Safe” gamble with 20 marbles to win $40, R = choice of “Risky” gamble with 10 marbles to win $98.
Supplementary Materials: These materials are included here as a convenience for editor and reviewers, rather than for publication in the journal. The materials are available via the Web, and will remain so for reference by journal readers.
Experiment 1:
Decision-Making Experiment: Choices between Gambles
This is a study of decision making involving important decisions with large prizes. Although you can't win or lose money in this study, you should imagine that these decisions are for real, and make the same decisions that you would if very large, real stakes were involved.
Email address:__________________________________________
We will notify you by email if you are a winner.
Country:
Age: You must be over 18 years to participate.
Are you Male or Female?
Female
OR
Male
Education (in years). ___________
If you are a college graduate, put 16.
If you have a Ph.D., put 20.
Education: Years.
Now, look at the first choice, No. 1, below. Would you rather play:
A: fifty red marbles to win $1,000,000 and fifty white marbles to win $2 ($2 means just "two dollars"),
OR
B: fifty blue marbles to win $350,000 and fifty green marbles to win $250,000.
Think of probability as the number of marbles in one color in an urn (container) containing 100 otherwise identical marbles, divided by 100. Gamble A has 50 red marbles and 50 white marbles; if a marble drawn at random from urn A is red, you win $1,000,000 (one million dollars). If a white marble is drawn, you win $2 (two dollars). So, the probability to draw a red marble and win $1,000,000 is .50 and the probability to draw a white marble and get $2 is .50. If someone reaches in urn A, half the time they draw red and win $1,000,000 and half the time they draw white and win $2. But in this study, you only get to play a gamble once, so the prize will be either $2 or $1,000,000. Gamble B's urn has 100 marbles also, but 50 of them are blue, winning $350,000, and 50 of them are green and win $250,000. Urn B thus guarantees at least $250,000, but the most you can win is $350,000. Some will prefer A and others will prefer B. To mark your choice, click the button next to A or B. Notice that the dot next to No. 1 will empty and fill in the button next to your choice. There are always 100 marbles in each urn, but the number of each color and their prizes for each color differ.
For each choice below, click the button beside the gamble you would rather play, if you really faced these decisions.
1. Which do you choose?
A: 50 red marbles to win $1,000,000
50 white marbles to win $2
OR
B: 50 blue marbles to win $350,000
50 green marbles to win $250,000
2. Which do you choose?
C: 50 red marbles to win $1,000,000
50 white marbles to win $2
OR
D: 50 green marbles to win $450,000
50 blue marbles to win $350,000
3. Which do you choose?
E: 20 red marbles to win $1,000,000
30 blue marbles to win $960,000
50 white marbles to win $500,000
OR
F: 20 red marbles to win $1,000,000
30 green marbles to win $620,000
50 white marbles to win $500,000
4. Which do you choose?
G: 10 red marbles to win $1,000,000
50 green marbles to win $120,000
40 white marbles to win $20
OR
H: 10 red marbles to win $1,000,000
50 blue marbles to win $960,000
40 white marbles to win $20
5. Which do you choose?
I: 90 red marbles to win $960,000
05 blue marbles to win $140,000
05 white marbles to win $120,000
OR
J: 85 red marbles to win $960,000
05 blue marbles to win $900,000
10 white marbles to win $120,000
[Trials 5-20 are described in Tables of the paper.]
-----------
21. Have you ever read a scientific paper
(i.e., a journal article or book) on the theory of
decision making or on the psychology of decision making?
No. Never.
OR
Yes, I have.
COMMENTS: _______________ [a text box was provided here].
Please check to make sure that you have answered all of the Questions.
When you are finished, push this button to send your data:
FINISHED
---------------------------------------------------------------------------------------------------------------------
Experiment 2 materials
Decision-Making Experiment: Choices between Gambles
This is a study of decision making in which you can gamble without risk and perhaps even win some money, if you are lucky. At the same time, your participation in this study can help scientists in the Decision Research Center learn more about how people make choices. Decide first if you want to participate. You must be at least 18, and you may participate only once. Scroll down and look over the questionnaire. It usually takes about 5 minutes.
Email address_____________________________________
We will notify you by email if you are a winner.
Country: _____________________________________
Age: You must be over 18 years to participate. _____________________________________
Are you Male or Female?
Female
OR
Male
Education (in years).
If you are a college graduate, put 16.
If you have a Ph.D., put 20.
Education: __________ Years.
Now, look at the first choice, No. 1, below. Would you rather play:
A: fifty red marbles to win $100 and fifty white marbles to win $0 (nothing),
OR
B: fifty blue marbles to win $35 and fifty green marbles to win $25.
Think of probability as the number of marbles in one color in an urn (container) containing 100 otherwise identical marbles, divided by 100. Gamble A has 50 red marbles and 50 white marbles; if a marble drawn at random from urn A is red, you win $100. If a white marble is drawn, you win $0. So, the probability to draw a red marble and win $100 is .50 and the probability to draw a white marble and get $0 is .50. If someone reaches in urn A, half the time they draw red and win $100 and half the time they draw white and win $0. But in this study, you only get to play a gamble once, so the prize will be either $0 or $100. Gamble B's urn has 100 marbles also, but 50 of them are blue, winning $35, and 50 of them are green and win $25. Urn B thus guarantees at least $25, but the most you can win is $35. Some will prefer A and others will prefer B. To mark your choice, click the button next to A or B. Notice that the dot next to No. 1 will empty and fill in the button next to your choice. There are always 100 marbles in each urn, but the number of each color and their prizes for each color differ.
For each choice below, click the button beside the gamble you would rather play. On Mar. 21, 2003, after people have finished their choices, three people will be selected randomly to play one gamble for real money. One trial will be selected randomly from the 20 trials, and if you were the lucky person, you will get to play the gamble you chose on the trial selected. You might win as much as $108. Any one of the 22 choices might be the one you get to play, so choose carefully. Winners will be notified by email. Offer void where prohibited by law.
1. Which do you choose?
A: 50 red marbles to win $100
50 white marbles to win $0
OR
B: 50 blue marbles to win $35
50 green marbles to win $25
2. Which do you choose?
C: 50 red marbles to win $100
50 white marbles to win $0
OR
D: 50 green marbles to win $45
50 blue marbles to win $35
3. Which do you choose?
E: 20 red marbles to win $100
30 blue marbles to win $96
50 white marbles to win $50
OR
F: 20 red marbles to win $100
30 green marbles to win $62
50 white marbles to win $50
4. Which do you choose?
G: 10 red marbles to win $108
50 green marbles to win $12
40 white marbles to win $2
OR
H: 10 red marbles to win $108
50 blue marbles to win $96
40 white marbles to win $2
5. Which do you choose?
I: 85 red marbles to win $100
10 white marbles to win $50
05 blue marbles to win $50
OR
J: 95 red marbles to win $100
05 white marbles to win $7
[Items 5-22 are described in the Tables]
22. Which do you choose?
k: 15 red marbles to win $50
85 black marbles to win $7
OR
l: 10 black marbles to win $100
05 purple marbles to win $7
85 green marbles to win $7
-----------
23. Have you ever read a scientific paper
(i.e., a journal article or book) on the theory of
decision making or on the psychology of decision making?
No. Never.
OR
Yes, I have.
Winners will be notified by email. If you share an email address, include your
name in the comments below, so we can identify the winner.
COMMENTS: ________________________________________________________
Please check to make sure that you have answered all of the Questions.
When you are finished, push this button to send your data:
FINISHED
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