PREFERENCE REVERSAL: A NEW LOOK AT AN OLD PROBLEM. SHU LI Institute of Psychology, Chinese Academy of Sciences, China Peking University, China

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1 The Psychological Record, 2006, 56, PREFERENCE REVERSAL: A NEW LOOK AT AN OLD PROBLEM SHU LI Institute of Psychology, Chinese Academy of Sciences, China Peking University, China A generalized weak dominance approach is used to test the documented preference reversal (PR) phenomenon. This approach simply models risky choice behavior in PR as a choice between the best possible outcomes or a choice between the worst possible outcomes by equating smaller paired outcome difference between bets. The preference reversals are therefore seen as a consequence of the fact that gamble parameters are designed to encourage individuals to differentiate the difference between the worst possible outcomes of the two bets (i.e., to avoid the worse possible outcome of $ bet) rather than to differentiate the difference between the best possible outcomes of the two bets (i.e., to seek the better outcome of $ bet on which people tend to put a higher price). A "matching" task as well as a "pricing" task was designed to examine whether the knowledge of the value difference of the paired possible outcomes will permit prediction of preferential choice. The overall test results favor the equate-to-differentiate explanation. The present data suggest that the anomaly may be not in individuals' inconsistent preferences but rather in our inadequate knowledge of what it is that is being preferred when a question about preference is posed. One of the most frustrating findings in the classical view of preferences is the preference reversal (PR) phenomenon (Stalmeier, Wakker, & Bezembinder, 1997). For more than three decades psychologists and economists have been intrigued by the PR anomaly. The classic example of preference reversal involves the choice-pricing discrepancy in the evaluation of two lotteries with equal expected value. One lottery typically has a high probability of winning a modest cash amount (a P bet). The other, riskier lottery has a small chance of winning a large monetary amount (a $ bet). Participants in a typical preference reversal task tend to state a higher cash equivalent for the $ bet, but they tend to prefer the P bet, when asked to choose between these two lotteries. For example, This research was initiated while the author was doing research at University of New South Wales and completed as a visiting scholar at Department of Psychology, Peking University. The author thanks two anonymous reviewers of this joumal for their helpful comments on the initial version and Elise M. Hopkins for her assistance with the English expression. Correspondence regarding this article should be addressed to Shu Li, Center for Social & Economic Behavior, Institute of Psychology, Chinese Academy of Sciences, Beijing, China.

2 412 LI Lottery A provides a 9/12 chance of winning $110 and a 3/12 chance of losing $10. (P bet) Lottery B provides a 3/12 chance of winning $920 and a 9/12 chance of losing $200. ($ bet) When individuals are asked to choose between these two lotteries, they tend to choose Lottery A over Lottery B. When asked to select a cash equivalent price for each lottery separately they tend to place a higher selling price on Lottery B. This pattern is usually interpreted as indicating an inconsistency, because both measures, choice and pricing, are assumed to be measures of people's well-articulated preferences. The PR phenomenon was first reported by Lichtenstein and Siovic (1971) and by Lindman (1971). Since then, three waves of studies of preference reversal have been described (for a complete description, see Tversky, Siovic, & Kahneman, 1990). There have been three alternative interpretations of PRo These alternatives arise from the violation of one of three principles important to decision theory, namely (a) violations of transitivity (Fishburn, 1985; Loomes & Sugden, 1983); (b) violations of the independence axiom (Holt, 1986; Karni & Safra, 1987); and (c) violations of procedure invariance (Goldstein & Einhorn, 1987; Tversky et ai., 1990). Psychological models have been developed to account for the phenomenon. The majority of the models have focused primarily on the idea that the different psychological processes take place in one context versus the other. For example, expression theory (Goldstein & Einhorn, 1987) postulates that PR is caused by changes in the mapping from a gamble's components to the response. This transformation is assumed to differ predictably for each gamble and for each response mode. Contingent weighting theory (Tversky, Sattath, & Siovic, 1988) attributes PR to variations in the stimulus weighting. The exponential weight of the probabilities is low under the pricing condition and high under the choice or attractiveness judgment conditions, while monetary outcomes are assumed to be weighted more heavily in pricing than in choice. The work by Siovic, Griffin and Tversky (1990) further argued that the shift in weights results from the compatibility between the attributes of the options and the response scale. Gonzalez-Vallejo and Wallsten (1992) used the contingent weighting model to explain PR between both choices and prices, and also between choices expressed with numerical probabilities and those expressed with verbal probabilities. Change-of-process theory (Mellers, Ordonez, & Birnbaum, 1992) attributes PR to variations in the decision strategies used to combine information. The bids are a multiplicative function of probability and amount, whereas ratings are an additive function of probability and amount. According to Tversky et al. (1990), however, the violations of transitivity and independence axiom can account for only a small fraction of observed preference reversals. Therefore, violations of procedure invariance (usually considered as arising from effects of scale compatibility) are nowadays the most widely accepted explanation: Due to the identical scale, the payoffs of a lottery are weighted more heavily in pricing than in choice.

3 PREFERENCE REVERSAL 413 In contrast, the view of the author is that the well-documented preference reversal phenomenon is not a matter of "preference reversal" as such. It simply demonstrates another example of a situation where judgment and preferences fail to conform. The present state of the study of PR could be described as one where axiomatic theories dictate that decision makers should prefer the bet for which they demand the most money, but actually they do not. The key issue is whether or not giving a price is the same as giving a preference. Each of the two possible answers to this question leads to a different way of viewing PR data. If the answer is positive, then pricing maximization indicates a preference, and the data does in fact indicate inconsistent preferences. This line of thinking underlies the numerous studies conducted in the three waves of enthusiastic research on this topic. Research attention has been directed to the notion of "reversal" from the very beginning. Little attempt has been made to question the underlying assumption behind the notion of PRo If the answer is negative then pricing maximization can be treated as another artificial or false index of preference. Because the aim of proposing a sole index for preference has long been the motivation behind judgmentbased models, the addition of "pricing maximization" to the long list of maximizations, such as EV (expected value) maximization, EU (expected utility) maximization, SEU (subjective expected utility) maximization, and many others, does not really provide any new insight into the problem. In fact all the known failures of maximization propositions to prescribe human preferences can also be seen as illustrating preference reversals. The situation can be illustrated thus. Imagine that there is a pair of bets with differing expected values. Imagine further that human decisionmakers are presented with each bet separately and asked to compute its expected value (EV), and that they are then presented with the pair of bets and asked to select the preferred one. If they were to choose one member of the pair but set a higher expected value on the other, the standard PR effect would be said to be detected in the result. This analogy is fair in that it seems to suggest that human decision-makers should derive their preference from pricing maximization in the PR task just as they would, say, from EV maximization in this hypothetical task. An alternative way of looking at the documented preference reversal (PR) phenomenon is to view it from the equate-to-differentiate perspective (Li, 1998, 2001, 2003, 2004a, 2004b; Li, Fang, & Zhang, 2000). The equate-to-differentiate decision model posits that the mechanism governing human risky decision making has never been one of maximizing some kind of mathematical expectation, but rather some generalization of dominance detection. Weak dominance states that if alternative A is at least as good as alternative B on all attributes, and alternative A is definitely better than alternative B on at least one attribute, then alternative A dominates alternative B (ct. Lee, 1971; von Winterfeldt & Edwards, 1986). The equate-to-differentiate rule postulates that, in order to utilize the very intuitive or compelling rule of weak dominance to reach

4 414 LI a binary choice between A and B in more general cases, the final decision is based on detecting A dominating B if there exists at least one j such that U Aj.(x) - UBi ~x) > 0 having subjectively treated all U Aj (xi) - UBj (Xj) < 0 as UAj (xi) - UB (x ) = 0, or, detecting B dominating A if there exists at least one J such ~hat UBj (Xj) - U Aj (x) > 0 having subjectively treated all UBj (xi) - U Aj (Xj) < 0 as UBj (x) - U Aj (x) = 0, where Xj (j = 1,..., M) is the objective value of each alternative on dimension j (for an axiomatic analysis, see Li, 2001). In the light of a representation system (with the best possible and the worst possible outcome dimensions) to describe the classic example of preference reversal, P bet is seen as better than $ bet on the worst possible outcome dimension while $ bet is seen as better than P bet on the best possible outcome dimension. In order to utilize weak dominance to reach a decision, people have to "equate" less significant difference between bets on either the best possible or the worst possible outcome dimension, thus leaving the greater one-dimensional difference to be differentiated as the determinant of the final choice. If the classic example of preference reversal is represented by using two dimensions (the best and the worst possible outcome dimensions), the equate-to-differentiate account can most easily be understood with the aid of a graphical representation, as shown in Figure 1. It can be seen that the difference between the worst possible outcome of P bet and $ bet is III $1,000 ~ + $100 GI B (J).2 E os: :t::... co Cl.3.5 ~ 8 $10 "S 0 ~..Q.~ Worst Possible Outcoume in Logarithmic Scale (-) $ j In +P Bet ($110, 9/12; -$10, 3112) 11$ Bet ($920, 3/12; -$200, 9/12) Figure 1. The classic example of preference reversal with P bet and $ bet represented by applying a logarithmic utility function over the best possible and the worst possible outcome dimensions.

5 PREFERENCE REVERSAL 415 particularly significant, assuming a negatively accelerated utility function (see Figure 1). The construction of the gambles will render the equating of difference on the "best possible outcome" dimension easier than that on the "worst possible outcome" dimension. In olther words, the gamble parameters are designed to encourage individuals to differentiate the difference between the possible outcome (-$10) oj P bet and the possible outcome (-$200) of $ bet on the worst possible outcome dimension rather than to differentiate the difference between the possible outcome (+$110) of P bet and the possible outcome (+$920) of $ bet on the best possible outcome dimension. An inference from the above analysis is that if each dimensional difference of the bets offered could be changed, another way around the preference could be generated. Guided by such thinking, three pairs of the P bet and $ bet (see Table 1) were employed in Li's (1994) study to test the PR with parameter modifications on both the best possible and the worst possible outcome dimensions. The gambles used in Problem 1 are taken from Stevenson, Busemeyer, and Naylor (1990, p. 345). They are in fact modified versions of the gambles from Lichtenstein and Siovic's (1971) study. In the original version, (1.10,.75; -.10,.25) for the P bet and (9.20,.25; -2.00,.75) for the $ bet, this set of gambles is the set which Lichtenstein and Siovic (1971) suggested would result in the greatest number of reversals in their experiment. Problem 2 retains all the gamble probabilities used in Problem 1 but adjusts the payoffs, so that the worst outcomes are closer to each other. It should be noted that Problem 3 is rather unusual in that Lottery B ($ bet) dominates Lottery A (P bet), in the sense that the possible outcome of B is better than that of A on every possible dimension. The inference was confirmed (see Table 2). In the first pair of the P bet and $ bet, (110,.75; -10,.25) for the P bet and (920,.25; -200,.75) for the $ bet, when no effort is made to change the relative difference between the worst possible outcomes, the P bet preferred behavior would remain unchanged, and choice and pricing was inconsistent - only 29% of participants put a higher price on "$ bet" and, at the same time, actually choose the "$ bet,," which is a bet having more to be won as well as more to be lost compared with Lottery A. In the second pair of the $ bet and P bet, (120,.75; -10,.25) for the P bet and (395,.25; -15,.75) for the $ bet, however, when an effort is made to reduce the relative difference between the worst possible outcomes, the preferred choice turns out to be a $ bet preferred one, and choice and pricing, in this case, turn out to be quite consistenlt - 68% of participants put a higher price on "$ bet" and, at the same time, actually choose the "$ bet." Moreover, in the third pair of the P bet and $ bet, (270,.64; -56,.36) for the P bet and (930,.17; -23,.83) for the $ bet, when an effort is made to reverse the relationship of the relative difference between the worst possible outcomes, that is, to make the worst possible outcome of the $ bet better than that of the P bet, the preferre!d choices also turn out to be a $ bet preferred one, and, correspondingly, 66% of the participants chose the bet on which they placed a higher price.

6 416 LI Table 1 Gamble Parameters and Corresponding Expected Monetary Values (EVs) Choice Problem Gambles EVs * 1 A (P bet) = (110,.75; -10,.25) 80 B ($ bet) = (920,.25; -200,.75) 80 * 2 A (P bet) = (120,.75; -10,.25) 87.5 B ($ bet) = (395,.25; -15,.75) A (P bet) = (270,.64; -56,.36) B ($ bet) = (930,.17; -23,.83) Probabilities were expressed as multiples of 1/12. Choice Problem Table 2 Percentages of Participants Giving Consistent Responses Across Each Pair of Bidding Methods 1 Choice vs. Pricing 1 29% (N = 60) 2 68% (N = 52) 3 66%(N = 61) Choice vs. Subsidize 71% (N = 65) 37% (N = 55) 64% (N = 54) Pricing vs. Subsidize 15% (N = 40) 28% (N= 40) 67% (N= 40) The observed discrepancy within biddings demonstrated in Table 2 poses additional difficulty for psychological models proposed for explaining PR and makes the existing explanations irrelevant. As for expression theory, it proposes that choice follows an expected utility rule using a concave utility, but when expressing prices, participants use a scale that is linear to dollar amounts. Although this formulation predicts a choice-pricing discrepancy, it does not predict a pricing-subsidizing2 discrepancy. To do so would require, for instance, different scales (linear or nonlinear to dollar amounts) for the same bidding task. As for contingent weighting theory, the theory assumes that the process of combining probability and outcomes is invariant, but the weights of these factors change in different tasks. Because the price of a bet is expressed in dollars, compatibility entails that the payoffs which are expressed in the same units will be weighted more heavily in pricing than in choice. However, scale compatibility does not imply that monetary outcomes are to be weighted differently across pricing and subsidizing. According to change-of-process theory, preference reversals result from changing decision strategies, but utilities remain constant across tasks. In particular, buying, selling, and avoidance 3 prices are assumed to be consistent with a multiplicative combination of the same scales, P(x, p; 0) = J[s(P) u(x)). If I Part of the results (Choice vs. Subsidize on present Choices 1 and 2) were replicated with student subjects from the School of Information Systems at the University of New South Wales. 2For the subsidizing task, subjects were asked to imagine that they can be furnished with a subsidy to play the gamble and then to determine the minimum subsidy with which they would be willing to play each gamble (Li, 1994). 3For avoidance prices, subjects were told to assume that they owned the gamble and were asked to state the maximum amount they would be willing to pay to avoid playing it (Mellers, Chang, Birnbaum, & Ordonez, 1992). A task akin to subsidizing was used in Li (1994).

7 PREFERENCE REVERSAL 417 this is in fact the reason for PR, reversal of the rank order of two bidding tasks should not arise when the same scale is applied to the pricing and the subsidizing. On the whole, the data illustrated in Table 2 suggest that the normative way to represent preference by judgment will not work, not only because of inadequate knowledge of preference but also because of inadequate knowledge of judgment. In searching for further evidence of whether the judgment-based model was capable of prescribing preferential choice betlavior, the following two experiments were designed in the present research. One was designed to test PR in its standard or risky version, that is, the P bet and $ bet were to be tested in their original format (with the precise probabilities), and the other was designed to test PR in its uncertain v!rsion, that is, the P bet and $ bet were modified by removing all of the precise probabilities from the original problems. It was hoped that the testing of the compliance of the equate-to-differentiate account with the resulting data would provide additional insights into the underlying cognitive mechanisms that govern choice behavior in the PR task. Experiment 1 In this experiment, the P bet and $ bet were modified by removing all of the precise probabilities from the original problems, replacing them with vague odds of winning and losing. This 'fuzzy-trace' way of treating gamble probabilities was triggered by Reyna and Brainerd's (1995) study, which proposed that detailed nuances of problem information were presumably not central to reasoning and that reasoners tended to operate on representations that were at the lowest level of precision that permitted a task-relevant response. According to Reyna and Brainerd (1995), removing all of the numbers from the Asian disl3ase problem (Tversky & Kahneman, 1981), and replacing them with vague phrases such as "some people will be saved or no one will be saved" did not eliminate framing effect. In fact, framing effects were not only detected, but they were larger in magnitude when the numbers were absent than when they were present. Be that as it may, it appears that the explanation for the "true" or "larger" effect of PR should be the one which is able to apply to the lotteries with absent numbers. The reader will note that the P bet and $ bet used are uncertain versions of the three pairs of bets presented in Table 1 respectively. A so-called "matching" task was designed to examine whether the knowledge of the value difference of the paired outcomes along each dimension will permit prediction of preferential choice. Operationally, the outcomes of lotteries on both the best and the worst possible outcome dimensions are paired in the present task. Participants were then asked to choose the pair with outcomes which are, for them, the most different. If the equate-to-differentiate theory's one-dimensional difference account is correct, then the knowledge of the matching results will permit explanation or prediction of option prefl3rence. That is, if $ bet is chosen, the participant should choose the pair of two "best possible

8 418 LI outcomes" as most different, thus leading to the maximization of the best possible outcomes (possible +$930 and possible +$270). However, if P bet is chosen then the participant should choose the pair of two "worst possible outcomes" as most different, thus leading to the minimization of the worst possible outcomes (possible -$23 and possible -$56). The justification for proposing the present experiment lies in the fact that removing all of the precise probabilities will render the identification of which best (worst) possible outcome is his or her subjectively better (worse) one with greater ease. Participants Chinese and Australian students were employed as experimental participants. The Australian participants were 32 students from the School of Psychology at the University of New South Wales; the Chinese participants were 50 students majoring in Planning and Statistics at the College of Finance and Economics, Fuzhou University. Materials and Procedure The proposed problems involved a test of the equate-to-differentiate account, in that participants were asked to make a choice first and then to carry out a matching task with respect to the lotteries involved. The stimuli were presented in booklets. The three problems designed in this experiment (see Table 3), with the first two each coupled with a matching task, were presented in questionnaire form to both Australian and Chinese participants, with appropriate changes in currency. For each problem, the choice task was a choice between two lotteries (Lottery A and Lottery B) while the matching task was a choice between two alternatives (labeled C and D), in which either the best outcomes of lotteries or the worst outcomes of lotteries were paired. Table 3 Stimuli Used in Experiment 1 for Choice and Mmatching Tasks Task Problem 1 Problem 2 Problem 3 Choice Lottery A an unknown chance an unknown chance an unknown chance of winning $110 and of winning $110 and of winning $270 and an unknown chance an unknown chance an unknown chance of losing $10 of losing $10 of losing $56 Lottery B an unknown chance an unknown chance an unknown chance of winning $920 and of winning $395 and of winning $930 and,an unknown chance an unknown chance an unknown chance of losing $200 of losing $15 of losing $23 Alternative C "an unknown chance "an unknown chance of winning $110" vs. of winning $110" vs. "an unknown chance "an unknown chance of winning $920" of winning $395" Matching Alternative D "an unknown chance "an unknown chance of losing $10" vs. "an of losing $10" vs. "an unknown chance of unknown chance of losing $200" losing $15"

9 PREFERENCE REVERSAL 419 Participants were instructed that there were no right or wrong answers, and that the experimenters were interested in their thoughtful answers. Participants were asked to "circle the lottery you would prefer to have" for the choice task and to "circle the one whose alternatives are most different" in the matching task respectively. When the completed questionnaires were collected, the participants wme then debriefed. Results and Discussion The observed modal responses from Australian and Chinese participants were perfectly consistent across the three problems. Just as removing all of the numbers from the classical Asian Disease Problem and replacing them with vague phrases such as "some people" did not eliminate the framing effect (Reyna & Brainerd, 1995), so removing all of the precise probabilities in Problem 1 did not prevent people from preferring the so-called lop bet." That is, a majority (82%) of the participants (85% for Australian; 79% for Chinese; p <.01) preferre~d Lottery A in Problem 1,4 in contrast with only 21 % of the participants (~~O% for Australian; 22% for Chinese; p <.01) who preferred the same lottery in Problem 2. This suggested that the higher probability was not necessary for choosing the lop bet" but the smaller difference between worst possible outcomes was. Problem 3 involving uncertainty was deemed by the equate-todifferentiate model as an instance of one alternative dominating the other, in the sense that, Lottery B is better than Lottery A on both the best possible outcome dimension (possible $930 > possible $270) and the worst possible dimension (possible -$23 > possible -$56). It is unambiguously the best alternative available and therefore no further analysis was required. In fact, 100% of the 82 participants chose the "dominating" alternative (Lottery B) as their preferred alternative. For the results obtained from Problems 1 and 2, an analysis revealed that matching significantly accounted for 5.7%, X 2 (1) = 4.92, phi squared =.057, P <.03, and 18.5%, X2(1) = 15.26, phi squared =.185, P <.01, of the choice variance in Problems 1 and 2 respectively. These data, coupled with the straightforward result obtained from Problem 3 (100% chose B), were generally consistent with the predictions of such a rule of dominance detecting, but not necessarily with those of EV, EU, or SEU based rules. Experiment 2 The findings in Experiment 1 fitted nicely with the equate-todifferentiate approach in the sense that matching information could permit prediction of the lottery chosen in its uncertain version. In Experiment 2, the P bet and $ bet were to be tested in their original format (with the precise probabilities). A "partial" pricing task rather than an overall pricing task was designed to help test the equate-to-differentiate account in 41n Problem 1, it was later found by using 175 Singaporean student subjects that people also tend to put a higher price on the uncertain version of the "$ bet" (M = $89.62) than on the uncertain version of the "P bet" (M = $17.88), t(174) = 7.379, P <.001.

10 420 LI further detail. This was because the ordinal relations along each possible outcome dimension were no longer clear for the risky version of the P bet and $ bet. For the uncertain version, the "possible outcome" dimensions were assumed to have, at least, an ordinal scale representation that was a strictly increasing function of objective possible outcome. However, for the risky version, the introduction of known probabilities did not necessarily increase the metric level. Even if you could determine, by means of a matching task, the dimension on which the utility difference was the greatest, you still could not determine which outcome along this dimension was subjectively the better one. That is, even if you could know whether the participant was a "best possible outcome" maximizer or a "worst possible outcome" minimizer, you still had no idea which best possible outcome was his or her subjectively better one, or which worst possible outcome was his or her subjectively worse one. Strictly speaking, you still had no idea, for a specific participant, which outcome along the "best possible outcome" dimension, a 3/12 chance to win $920 or a 9/12 chance to win $110, was the subjectively better one, or which outcome along the "worst possible outcome" dimension, a 9/12 chance to lose $200 or a 3/12 chance to lose $10, was the subjectively worse one.. There was a plethora of studies which explore an individual's impression of probability and outcome evaluations. Some experimental evidence suggested the existence of a kind of dependency between probability and outcome evaluations (Edwards, 1962). Some risky theories hypothesized a weighting function which translates probabilities contains discontinuity at two endpoints (0, 1) (Kahneman & Tversky, 1979). Some theorists suggested that the pessimistic weighting function gave greater weight to lower outcome (i.e., to outcomes with lower ranks) while the optimistic weighting function gave greater weight to larger outcomes (i.e., to outcomes with higher ranks) (Quiggin, 1982). Some research presented evidence that people had different comfort levels along the hope for good outcomes and fear of bad outcomes continuum (Lopes, 1987). Some descriptive models postulated that the attractiveness of the lottery offering the higher probability (to gain or lose) was weighted by its probability advantage, while the attractiveness of the lottery offering the larger payoff (either a loss or a gain) was weighted by its payoff advantage (Shafir, Osherson, & Smith, 1993). Some approaches concerned primarily with how poor people were at interpreting low probabilities (Stone, Yates, & Parker, 1994). Behavior analysts considered the standard probability learning paradigm as matching to sample without a sample (Fantino, 1998). All these suggested that the judgment was so subjective that the valence of outcomes and evaluation of probabilities were unlikely to be mapped into their mental counterparts by way of simple reflection. In practice, one available method to determine the ordinal relation between possible outcomes was to ask participants to set the buying or selling price for those outcomes, the assumption being that the higher the winning prize and the higher the winning probability, the higher will be the price. A challenging question was that, if the ordinal relation

11 PREFERENCE REVERSAL 421 of those outcomes could be determined by using the pnclng method, could the equate-to-differentiate approach account for the individual's preferred behavior both on P bet and on $ bet? The present experiment represented an attempt to carry out such a test to see whether the equate-to-differentiate approach could provide a possible explanation and prediction for the observed preference reversals. Participants Participants were 36 undergraduate students enrolled in Introductory Psychology courses at the University of New South Wales, who participated for course credit and 47 undergraduate students from Fuzhou University, who participated as volunteers. Materials and Procedure To further test how the judged information was incorporated into the choice process, participants were presented with Problems 4 and 5 which were Problems 1 and 2 used in U's (1994) study. An inconsistency involving choice-pricing discrepancy as well as an inconsistency involving choice-subsidy discrepancy was found by utilizing the gamble parameters in Problems 1 and 2 respectively (U, 1994). In addition, two simple gambles (Problems 6 and 7) were used as decision stimuli. The gamble parameters in Problem 6 were from Tversky and Thaler (1990). They reported that 71 % of their participants chose the alternative with the higher chance of winning a relatively small prize, while 67% put a higher selling price on the alternative with the lower chance of winning a larger prize. The Problem 7 gambles were used by Tversky et al. (1990) as Table 4 Stimuli Used in Experiment 2 for Choice, Matching, and Pricing Task Task Problem 4 Problem 5 Problem 6 Problem 7 Lottery A a 9/12 chance of a 9/12 chance of an 8/9 chance of a 28/36 chance of winning $1 10 and winning $120 and winning $4, but a winning $10, but an Choice a 3112 chance of a 3/12 chance of 1/9 chance of 8/36 chance of losing $10 losing $10 winning nothing winning nothing Lottery B a 3112 chance of a 3112 chance of a 1/9 chance of a 3/36 chance of winning $920 and winning $395 and winning $40, but an winning $100, but a a 9112 chance of a 9112 chance of 8/9 chance of 33/36 chance of losing $200 losing $15 winning nothing winning nothing Alternative C "9/12 chance to win "9112 chance to win "8/9 chance to win '28/36 chance to win $110" vs. "3/12 chance $120" vs. "3/12 chance $4" vs. "1/9 chance to $10" vs. "3/36 chance Matching to win $920" to win $395" win $40" to win $100" Alternative 0 "3/12 chance to lose "3112 chance to lose ' 1/9 chance to win "8/36 chance to win $10" vs. "9112 chance $10" vs. "9/12 chance nothing" vs. "8/9 chance nothing" vs. "33/36 chance to lose $200" to lose $15" to win nothin!l" to win nothing" Pricing a 9112 chance of winning $110; a 3/12 chance of winning $10; a 3/12 chance of winning $920, a 9112 chance of winning $200; a 9/12 chance of winning $120; a 3112 chance of winning $395; a 9112 chance of winning $15; an 8/9 chance of winning $4; a 1/9 chance of winning $40; a 28/36 chance of winning $10; a 3/36 chance of winning $100

12 422 LI examples of bets which have typical characteristics for illustrating the preference reversal phenomenon. Each of the four problems employed in Experiment 2 was summarized in Table 4. Participants were asked to make three decisions concerning the above pai rs of lotteries: (a) to state a direct preference by making a straight pairwise choice between the two; (b) to match one of the pairs of possible outcomes; and (c) to place a buying price on every possible outcome. The choice and matching instructions were the same as those given in Experiment 1. The instruction for pricing task read as follows: In this task you are asked to think of each of the following gambles as representing a lottery ticket with the odds of winning and the amount to be won being as given in those gambles. Your task is to determine the maximum price at which you would be willing to buy each ticket. Write this price in the blank space to the left of each alternative. When the completed questionnaires were collected, the participants were also debriefed. Results and Discussion The results obtained for Problems 4 and 5 were summarized in Tables 5 and 6. For columns C and D of the matching task, the cells in the upper right and lower left corners (data underlined) represented choices consistent with the equate-to-differentiate approach, while the cells in the upper left and lower right corners represented choices inconsistent with it. The testing results were reasonably supportive of the equate-to-differentiate approach. A 2 (matching) x 2 (rating) x 2 (choice) analysis revealed a significant relation between matching, rating, and choice for both Problem 4, X 2 (3) = 15.08, P <.01, and Problem 5, X2 (3) = 11.00, P <.025. As for Problems 6 and 7 (see Tables 7 and 8), the pricing task was not posed to the worst possible outcome pairs. This was because when a null-outcome was used for both bets, in Problem 6, it was clear that an Table 5 Summary of Results from the Three Tasks of Problem 4 Matching C 0 Rating Rating Rating Rating (920,.25) > (110,.75) > (10,.25) > (200,.75) > (110,.75) (920,.25) (200,.75) (10,.25) A 1 (4) Lill 0(0) ~ Choice B QJ1.m 0(1 ).o...q). 1 (10L Note. The Chinese data are presented in parentheses. The data underlined are those consistent with the position of the equate-to-differentiate approach. Those who assigned the same buying price either to (920,.25) and (110,.75), or to (200,.75) and (10,.25) were not included in the present table. A 2 (matching) x 2 (rating) x 2 (choice) analysis revealed a significant relation between matching, rating, and choice, X2 (3) = 15.08, P <.01.

13 PREFERENCE REVERSAL 423 Choice Table 6 Summary of Results from the Three Tasks of Problem 5 Matching C D Rating Rating Rating Rating (395,.25) > (120,.75) > (10,.25) > (15,.75) > (120,.75) (395,.25) (15,.75) (10,.25) A 0(4) @ () (1).ill B Q...@ 5 (3) ;U1). 3(4L Note. The Chinese data are presented in parentheses. The data underlined are those consistent with the position of equate-to-differentiate approach. Those who assigned the same buying price either to (390,.25) and (120,.75) or to (15,.75) and (10,.25) were not included in the present table. A 2 (matching) x 2 (rating) x ~~ (choice) analysis revealed a significant relation between matching, rating, and choice, X 2 ((l) = 11.00, P <.025. Choice Table 7 Summary of Results from the Three Tasks of Problem 6 Matching C Rating Rating (40,.11»(4,.89) (4,.89»(40,.11) A 4 (6) 1.ill 1QJZl. B ll(.8). 1 (2) 2 (4) Note. The Chinese data are presented in parentheses. The data underlined are those consistent with the position of equate-to-differentiate approach. Those participants who assigned the same buying price to (40,.11) and (4,.89) were not included in the present table. A X2 analysis revealed a significant relation between matching, rating, and choice, X2(2) = 7.13, P <.05. Choice A B Tabje 8 Summary of Results from the Three Tasks of Problem 7 Rating (100,.08»(10,.78) 5 (4) 9...(W Matching C Rating (10,.78»(100,.08) am 4 (1) D D I...(ID Note. The Chinese data are presented in parentheses. The data underlined are those consistent with the position of equate-to-differentiate approach. Those participants who assigned the same buying price to (100,.08) and (10,.78) were not included in ttle present table. A X2 analysis revealed a significant relation between matching, rating, and choice, X2(2) = 6.35, P <.05. "8/9 chance of winning nothing" was the worse one of the worst possible outcome pairs (a "1/9 chance to win nothing" vs. an "8/9 chance to win nothing") and that, in Choice 7, a "33/36 chance to win nothing" was the worse one of the worst possible outcome pairs (an "8/36 chance to 2 (9)

14 424 LI win nothing" vs. a "33/36 chance to win nothing") for a "worst possible outcome" minimizer. In this circumstance, a X 2 analysis revealed a significant relation between matching, rating, and choice for both Problem 6, X2(2) = 7.13, P <.05, and Problem 7, X2(2) = 6.35, P <.05. General Discussion The risky choice behavior was seen by the equate-to-differentiate model not as a compensatory process of maximizing some kind of expectation (EV, EU, or WU), but as a noncompensatory process of dominance detection, that is, a choice between the best possible outcomes or a choice between the worst possible outcomes. It was shown in the present experiments that the empirical evidence in relation to individuals' risk preference could be satisfactorily accounted for by the generalized weak dominance strategy revealed by the "matching" results. Given that the resulting data is supportive of the hypothesis being tested, the question arises as to whether the judgment-based model can ensure that the overall value judged is not a biased measurement when the requirement for an overall judgment is forced upon the participant. At present, the evidence that the resulting judgment is an unbiased measurement over all dimensions is not convincing. From the position of the equate-to-differentiate approach, when a simple gamble is involved (with pure gain or loss prospects), there seems to be no evidence that the null-outcome is integrated into the overall utility. When a complex gamble (where gain and loss are mixed) is involved, the bidding involved in the gamble might be conceived of as not taking into account subjective values with respect to all of the dimensions. That is, the overall pricing provides a measurement biased towards the possible gain dimension, while the overall subsidizing is biased towards the possible loss dimension. The results therefore raise doubts about whether people behaved as if they were trying to maximize "something." The present study makes a further contribution to the understanding of how the perceived difference between the possible outcomes exerts influence on individuals' risk preference. In proposing a nonlinear additive difference (NLAD) model to explain the violation of the transitivity axiom, Tversky (1969) suggests that there are several general considerations which favor the NLAD rule. For example, intradimensional comparison may simplify the evaluation and may be more natural than interdimensional ones and so on. The NLAD account and the "equate-to-differentiate" explanation have one feature in common, that is, the comparison is intradimensional. However, whether the evaluation is compensatory or noncompensatory is a clear distinction that should be made between the nonlinear additive difference rule and the "equate-to-differentiate" rule. The NLAD account is proposed on the grounds that the individual has the cognitive apparatus to construct an overall ordinal utility function (8) and is able to choose according to an ordering implied by an aggregate of multidimensional net difference so that intransitivity is originated. The NLAD rule is compensatory

15 PREFERENCE REVERSAL 425 in that relative judgments of all dimensions are taken into account, these relative judgments being nonlinear functions of intradimensional differences [i.e., o(a + b) *- o(a) + o(b)]. The "'equate-to-differentiate" model, on the contrary, is developed on the grounds that the individual does not possess such a well-defined value (o) and is cognitively unable to perform a multidimensional integration, so that a noncompensatory rule which results in intransitivity is adopted as an alternative to deal with multidimensional choices. The present model's noncompensatory property will not allow deficiencies on one dimension to be compensated for by high values on another in a bi-choice process. Thus, the transitivity dictated by axiomatic theories is no longer considered necessary for the model to hold. Such a fundamental difference might serve to provide some guidance as to the route falsifying research might take. The lexicographic semi order (LS) rule has much in common with the "equate-to-differentiate" rule. This is because the~ predictions of the two models are both based on noncompensatory characterization. Empirically, the Tversky's (weak) stochastic intransitivity has been demonstrated only with simple lotteries. The lexicographic semi order (LS) strategy suggested by Tversky (1969) and the simplification "editing" strategy invoked by Tversky and Kahneman (1986) might explain intransitivity in simple lotteries by assuming that the choices between the above first four lotteries are all based on payoff (x) dimension but the choice between lotteries (a) and (e) is based on chance (P) dimension. That is, it might be assumed that the probabilities of adjacent lotteries are considered to be identical, whereas the probabilities of (a) and (e) differ enough to affect evaluation and choice. However, Tversky's account of intransitivity is based on a representing system that treats the gamble outcome (x) and chance of winning it (P) as two risk dimensions on which pairs of simple bets vary (Ranyard, 1982). If the analysis for present data followed exactly the same logic, there are at least three pairs of choice problems, Problems1 to 3, which cannot be explained, given that the two dimensions, payoff dimension and chance dimension, remain used. Moreover, such a construction of two dimensions will cause some difficulty in applying either the additive rule or the additive difference rule to!~eneral risk problems. In reality, however, the state decision-makers are left with is that the nonlinear additive difference rule is merely studied under theoretical conditions, whereas the construction of the observed stochastic intransitivity is empirically guided by a non-nlad model. The present notion does not address the soundness of its axiomatic justification for producing an intransitive ordering, or the question of whether edementwise processing is inconsistent even with prospect theory (Kahneman & Tversky, 1979; Tversky & Kahneman, 1992), although there is evidence that the form of the difference functions applied to the various intradimensional differences cannot be deduced from the form of the value and weighting functions (Budescu & Weiss, 1987). The problem raised is if an intradimensional evaluation is possible for an additive difference model, then the possibility of forming an interdimensional evaluation through a simple additive model that guarantees transitivity should not be logically excluded.

16 426 LI The present results involving matching and "partial" pricing, together with those obtained by Li (1994) involving "overall" pricing, indicate that preference data from bidding responses are insufficient for deducing anything further. As generalized expectation maximizers claimed that the gambler's choice of an option is taken as evidence that the gambler is maximizing the overall worth of a gamble, the PR advocate made a parallel claim that the information derived from bidding alone is indeed the true preference. It appears that both are somewhat like one of the blind philosopher's claims about the true nature of an elephant based on his experience of different parts of not one but two elephants. Overall, various PR explanations locate the cause of the phenomenon at either framing, strategy selection, weighting of information, or expression of preferences stage of decisions (Payne, Bettman, & Johnson, 1992). However, the present data suggest that the anomaly may not be in participants' inconsistent preferences, but rather in researchers' inadequate knowledge of what it is that is being preferred when a question about preference is posed. The experiments reported in this paper have led to gaining some new insight into the general issue of human risky decision-making behavior. It is suggested that there could be an alternative way of seeing the inconsistency between the preferences shown by human participants and the ordering implied by value- or utility-integration calculations. When being confronted with such an inconsistency, the theoretical avenue might not necessarily be to modify the maximizing criterion so as to make the chosen alternative the greatest "something," but to seek to understand what other role the utility will play if it has not been one of maximizing mathematical expectation. Considering the limited cognitive ability of a human decision-maker, it is the present contention that inconsistency is inevitable and predictable because the equate-to-differentiate rule best reflects actual human practice. References BUDESCU, D. V., & WEISS, W. (1987). Reflection of transitive and intransitive preferences: A test of prospect theory. Organizational Behavior and Human Decision Processes, 39, EDWARDS, W. (1962). Utility, subjective probability, their interaction, and variance preferences. Journal of Conflict Resolution, 6, FANTINO, E. (1998). Behavior analysis and decision making. Journal of the Experimental Analysis of Behavior, 69, FISHBURN, P. C. (1985). Nontransitive preference theory and the preference reversal phenomenon. Rivista Internazionale di Science Economiche e Commerciali, 32, GOLDSTEIN, W. M., & EINHORN, H. J. (1987). Expression theory and the preference reversal phenomena. Psychological Review, 94, GONZALEZ-VALLEJO, C., & WALLSTEN, T. S. (1992). Effects of probability mode on preference reversal. Journal of Experimental Psychology: Learning, Memory, & Cognition. 18,

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