Tests of Prospect Theories 1
Tests of Prospect Theories in Gambles
Displayed in Matrix and Text Formats with
Positive and Negative Consequences
Michael H. Birnbaum
California State University, Fullerton and Decision Research Center, Fullerton
Running head: Tests of Prospect Theories
Date: September 27, 2004
Filename:Birnbaum_tickets_09_27_04.doc
Mailing address:
Prof. Michael H. Birnbaum
Department of Psychology, CSUF H-830M
P.O. Box 6846
Fullerton, CA 92834-6846
Email address:
Phone: 714-278-2102 or 714-278-7653
Fax: 714-278-7134
Author's note: Support was received from National Science Foundation Grants, SBR-9410572, SES 99-86436, and BCS-0129453. I thank Terry Connolly for suggesting the tickets/tables formats and Daniel Kahneman for useful exchanges of views on issues considered in this paper.
PsychInfo:2340 Cognitive Processes (Judgment and Decision Making)
JEL code:C91, D81.
Abstract
Two experiments tested theories of risky decision making. In the first, 443 participants were randomly assigned to one of three conditions in which different formats for display of gambles were used. These formats are variations of one used by Savage as a device to convince himself to satisfy his own axioms. Within each format, five “new paradoxes” that violate EU and prospect theory were tested. These new paradoxes include majority violations of first order stochastic dominance, violations of coalescing, violations of lower and upper cumulative independence, and violations of branch independence that are opposite those predicted by prospect theory. Despite intuitions that these formats would provide a realm in which prospect theories might be descriptive, all five paradoxes persisted in all three new formats. The second experiment, with 200 participants, tested the same properties in gambles with losses instead of gains. These data also violated CPT, but they were approximately compatible with the reflection hypothesis. Combined with other results, there is now a substantial body of evidence against prospect theories, involving more than eight thousand participants tested with 14 formats for displaying gambles.
Keywords: behavioral economics; choice paradoxes; cumulative prospect theory; decision making; expected utility; rank dependent utility; risky decision making; utility theory.
Birnbaum and Navarrete (1998) reported evidence against rank dependent utility (RDU) and Cumulative Prospect Theories (CPT) of risky decision making. They reported evidence of systematic violations of first order stochastic dominance, violations of upper and lower cumulative independence, and violations of restricted branch independence that are opposite the pattern predicted by the inverse-S weighting function used in CPT. Birnbaum (1999) confirmed these violations and reported significant violations of coalescing as well. Definitions of these properties and example violations are displayed in Table 1. The example violations of CPT in Table 1 are consistent with Birnbaum’s prior TAX model, a model fit to previous data that predicted the new results.
Insert Table 1 about here.
These properties in Table 1 can be considered “new paradoxes” of choice because violations of these properties are contradictions within the theory when CPT is used to analyze them. They refute CPT in the same way that the Allais paradoxes refuted expected utility theory. These findings run counter to a growing consensus in economics in favor of cumulative prospect theory (Starmer, 2000). Indeed, the Nobel Prize in economics in 2002 was awarded in part for behavioral investigations of prospect theories.
Kahneman (2003) recalled his collaboration with Amos Tversky that contributed to his share of the 2002 Nobel Prize in Economics. He reviewed their development of the editing principle of combination (Kahneman & Tversky, 1979), which is automatically satisfied by the representation of cumulative prospect theory (Tversky & Kahneman, 1992):
“"To amuse ourselves, we invented the specter of an ambitious graduate student looking for flaws, and we labored to make that student’s task as thankless as possible. The most novel idea of value theory (later named prospect theory) occurred to us in that defensive context. It came quite late, as we were preparing the final version of the paper. We were concerned that a straightforward application of our model implied that the prospect denoted ($100, .01; $100, .01)—two mutually exclusive .01 chances to gain $100—is more valuable than the prospect ($100, .02). The prediction is wrong, of course, because most decision makers will spontaneously transform the former prospect into the latter and treat them as equivalent in subsequent operations of evaluation and choice. To eliminate the problem, we proposed that decision makers, prior to evaluating the prospects, perform an editing operation that collects similar outcomes and adds their probabilities. We went on to propose several other editing operations that provided an explicit and psychologically plausible defense against a variety of superficial counterexamples to the core of the theory. We had succeeded in making life quite difficult for that pedantic graduate student, but we had also made a truly significant advance, by making it explicit that the objects of choice are mental representations, not objective states of the world. This was a large step toward the development of a concept of framing and eventually toward a new critique of the model of the rational agent.” (Kahneman, 2003, p. 727).
When asked if he still believed in the editing principle of combination, as described in this passage, Kahneman (Dec 17, 2003, personal communication) stated that he does not “believe” in either original or cumulative prospect theory in the sense that he would be willing to bet on the prediction of either model to the majority choice in a new experiment. When asked to propose a situation in which combination would be descriptive of the majority, he opined that results of any test would likely be highly dependent on “uninteresting” details of how prospects are displayed to participants. Indeed, Tversky and Kahneman (1992) gave a “pessimistic assessment” of their model in which they doubted that it would generalize to new situations.
But if these “uninteresting” manipulations are the variables that determine human behavior in economic experiments, then they are the important variables that psychologists must understand if they hope to predict, manipulate, and explain behavior.
Birnbaum (2004b) distinguished three variables of such manipulations: form, format, and framing. To understand the concept of form, we need to define the branches of a gamble. The term branch refers to a probability (or event) -consequence pair that is distinct in the presentation to the decider. In the above example from Kahneman (2003), the first prospect is a three branch gamble to win $100 with probability .01, win $100 with probability .01, or receive $0 with probability .98. The second is a two branch gamble with one branch of .02 to win $100, and a second branch with probability .98 to win $0. The first display is a called a “split” form of the gamble and the second is the “coalesced” form of the same prospect. A prospect can be presented in one of many possible split forms or in coalesced form, in which branches leading to the same consequence are combined. According to the editing principle of combination in original prospect theory and the equation of cumulative prospect theory, form should have no effect.
The term format is used to denote how prospects and choices between prospects are displayed to the participants. For example, one might represent probability by means of pie charts representing wheels with spinners, in terms of the relative numbers of marbles of various colors in an urn, as decimal fractions, or in other ways. Similarly, a choice may be arranged with gambles of a choice presented side by side or one placed above the other. Branches may be arranged vertically or horizontally, and they may be juxtaposed or not. Format and form have often been confounded in previous studies. However, within any format, form can be manipulated, so these variables need not be confounded.
Kahneman and Tversky (1979) and Tversky and Kahneman (1992) theorized that form (coalesced or split) should have no effect, whereas Kahneman (2003, personal communication) conjectured that format would have large effects. Perhaps the effect of form depends on format. If so, then there may be a format in which CPT can be retained as a descriptive model.
Tversky and Kahneman (1986) conjectured that event-framing might be an important variable. Event-framing was confounded with branch splitting (and with other variables including the number of branches with positive and negative consequences, the arrangement of branches, etc.) in their test of stochastic dominance. It is important to maintain the distinction between consequence and event framing. Consequence framing refers to how a given consequence is described; for example, one might describe the same (objective) result of a gamble framed as gains or losses. Consequence framing has been shown to have large effects (Tversky & Kahneman, 1979). Event-framing, in contrast, refers to how the “events” that determine consequences are described. Birnbaum (2004) found that an event-framing manipulation(marble colors on corresponding branches) had miniscule effects, and this variable is not pursued here.
Birnbaum (2004) tested nine different formats to see if there was some format for presentation of choices in which prospect theories might provide correct predictions of majority choices. Birnbaum and Martin (2003) reviewed studies that included two other formats. In all 11 formats studied so far, violations of prospect theories as in Table 1 were obtained. Contrary to both versions of prospect theory, data refute the editing principles of combination and cancellation, as well as dominance detection in three branch gambles, if these editing principles are treated as testable scientific hypotheses. None of the 11 formats tested yielded data that could be represented accurately with either version of prospect theory.
Terry Connolly described (personal communication, March 9, 2004) a small study in which no one violated stochastic dominance. He used a matrix format that displayed how prizes were mapped to ticket numbers, where a ticket would be drawn to determine the prize. This format had previously been used by Savage (1972, p. 103) as a device for helping to convince himself to satisfy his own “sure thing” axiom and thus avoid Allais paradoxes. Connolly theorized that he had found a format in which CPT could be rescued as a descriptive model.
However, Connolly’s study did not test coalescing, and his format used strictly split events. In other words, his study created a new situation in which Birnbaum’s (1999; 2004) results and theories would agree that the majority should satisfy stochastic dominance. However, it has not yet been determined whether the effects of form might be different in this type of format. Perhaps in matrix format, evidence against CPT would vanish. Experiment 1 of this study therefore explores three variations of Connolly’s “tickets” formats to see if prospect theories can be saved as descriptive of risky decision making in one or more of these previously unexamined formats.
Harless (1992) compared such a matrix format for presentation of choices with a text (“tickets”) format. In his matrix format, juxtaposition of branches was confounded with event-splitting: when branches were juxtaposed, the larger prize of one choice was also split, and when they were not juxtaposed, branches yielding the same prize were coalesced (Harless, Figure 1). However, in the text (“tickets”) format used by Harless (1992, Figure 3), branches were always coalesced, whether juxtaposed or not. Because the juxtaposition effect was theorized to be a “regret” effect, Harless reached the conclusion that “regret” effects depend on problem representation (i.e., format). This study will use both split and coalesced forms of both text and matrix formats like those of Harless (1992), to disentangle these variables.
Starmer and Sugden (1993) tested juxtaposition against event-splitting in a matrix format, concluding that event-splitting effects were prominent; however, Luce (2000, p. 183) considered those tests unconvincing because they did not employ within-subjects designs. This study will use within-subjects tests of coalescing within each of three formats, which are varied between subjects.
The second experiment used the same choices as the first, except that all prizes were converted to hypothetical losses. As noted in Starmer’s (2000) review, it is often found that gambles composed of chances to lose or break even have the opposite preference order as gambles composed of chances to win or break even, when the losses are simply converted to wins. This empirical generalization is known as the reflection effect (Kahneman & Tversky, 1979),and it has been reported to be a good approximation for choices that satisfy prospect theories, such as choices between binary gambles (Tversky & Kahneman, 1992).
The reflection hypothesis cannot hold in general; however, because the assumption leads to self-contradiction for symmetric mixed gambles (see Appendix).
The reflection hypothesis has never been tested, however, with the types of choices used in this study. There are three (simple) possibilities for this experiment. It is possible that the reflection hypothesis will be satisfied, in which case choices between gambles on losses will also violate CPT. A second possibility is that gambles on losses will satisfy CPT, in which case they must violate the reflection hypothesis. It is also possible that both CPT and the reflection hypothesis will fail to hold for these choices.
Methods
Experiment 1: Three New Formats
Participants viewed the studies via the WWW by means of Web browsers. They chose between gambles by clicking a button beside the gamble in each pair that they would prefer to play, knowing that 3 people would be selected to play one of their gambles for real cash. In all three new formats, there were always 100 numbered tickets and the number on a randomly drawn ticket would determine the prize.
Participants were randomly assigned to one of three different format conditions by a JavaScript routine. Instructions were the same as in the “tickets” condition of Birnbaum (2004), except modified to accommodate each new format. Complete materials can be viewed from the following URLs:
The new tickets format is shown in Figure 1. This format specifies the ticket numbers corresponding to each prize, like the format of Harless (1992), rather than the number of tickets to win each prize, as in Birnbaum (2004). For example, instead of “5 tickets to win $14”, the new tickets formats states, “Tickets # 91-95 win $14.”
Insert Figure 1 about here.
The unaligned matrix format is illustrated in Figures 2 and 3. Figure 2 shows the coalesced form (Choice 5) and Figure 3 shows the split form (Choice 11) of the same objective choice. Note that in the unaligned format, each branch has equal horizontal spacing within each choice, independent of the number of tickets associated with that branch. The unaligned matrix format is similar to that used by Savage (1972, p. 103) and Connolly (2004, personal communication).
Insert Figures 2 and 3 about here.
The aligned matrix format is illustrated in Figures 4 and 5. In the aligned format, horizontal spacing was constrained to vary monotonically with the number of tickets in each branch. The HTML was, in fact, programmed to make the horizontal spacing proportional to numbers of tickets; however, proportionality cannot be guaranteed for all browsers, systems, monitors, and window settings. Figure 4 illustrates how Choice 5 is displayed in the coalesced form. Note that in Figure 4, Tickets #86-90 in Gamble J appear to the left of tickets #91-95 in Gamble I, unlike the unaligned format shown in Figure 2. Like Birnbaum’s (2004) pie chart formats, the aligned format should in theory reveal stochastic dominance visually.
Insert Figures 4 and 5 about here.
Birnbaum (2004) noted that the cash values chosen for tests of restricted branch independence in that study were not optimally “tuned” for tests of restricted branch independence. In the new tickets format, values of consequences were adjusted to provide a more optimal test; in particular, the values $40 and $44 were replaced with $43 and $47. In addition, these studies use mostly undergraduates in lower division psychology, 61% of whom were female. This resembles the population in which parameters for the prior RAM and TAX models had been estimated.
Participants were 433 people (85% of whom were recruited from the “subject pool” of CSUF) and 15% recruited via the WWW. About half of the undergraduates were tested in labs containing computers connected to the Web, and half participated via the Web at times and places of their own choosing. No systematic differences were observed between “in lab” and Web results, once participant demographics were factored out, so data are combined in the analyses presented here. Participants were randomly assigned to one of three conditions: 141, 141, and 151 completed the new tickets, aligned matrix, and unaligned matrix formats, respectively.
Experiment 2: Gambles to Lose
In Experiment 2, the 20 choices were the same as those in Experiment 1, except that all prizes were changed from gains to losses. Each gamble was displayed in the reversed text format as used by Birnbaum (2004), except a branch would be written as “.50 probability to lose $100” rather than “.50 probability to win $100.” In this format, branches are listed in descending order of absolute value of the consequences. There were 200 undergraduates who performed the task twice, separated by three intervening tasks that required about 15 minutes. Complete materials for the study, including instructions, can be viewed from the following URL: