Are Preference Reversals Errors?

Transcription

1 Are Preference Reversals Errors? An Experimental Investigation Ulrich Schmidt University of Kiel John D. Hey* University of York. Ulrich Schmidt Institut für Volkswirtschaftslehre Abteilung Finanzmarkttheorie Universität Hannover Königsworther Platz Hannover Telephone: (0049) (511) Fax: (0049) (511) us@bwl.uni-kiel.de John D. Hey Department of Economics, University of York YORK YO1 5DD, U.K Telephone: (+44) (0) Fax: (+44) (0) Dipartimento di Scienze Economiche, Universita Degli Studi di Bari, Via Camillo Rosalba 53, BARI, ITALY Telefono: (+39) (0) Fax: (+39) (0) jdh1@york.ac.uk Corresponding author: John D. Hey Corresponding address: Department of Economics, University of York YORK YO1 5DD, U.K Corresponding telephone number: (+44) (0)

2 Are Preference Reversals Errors? An Experimental Investigation Abstract: This paper investigates whether some part of the preference reversal phenomenon can be attributed to errors in the responses of subjects in experiments. Such errors have been well documented in other investigations of behaviour in risky decision problems, but their relevance to the preference reversal phenomenon has not been explored. Building on earlier work, we develop an extended error model and apply it to the results of an experiment in which subjects tackle risky choice problems on five separate occasions. In this experiment subjects had to answer choice questions in three occasions and to state selling and buying prices in the remaining two occasions. Our results indicate that scale compatibility can be ruled out as a significant sole explanation of the preference reversal phenomenon. Moreover, we can show that a considerable fraction of observed preference reversals can be classified as pricing errors, whereas choice errors turn out to play a minor role. Key words: preference reversal, errors, scale compatibility. JEL-classification: C91, D81. 2

3 Introduction The preference reversal phenomenon, first observed by Slovic and Lichtenstein (1968), occurs if, for a given pair of lotteries, the choice ordering of a subject is systematically different from his or her price ordering. The usual case observed in practice is that subjects tend to state a preference for the safer lottery but state a higher selling price for the riskier one. Nowadays, preference reversals are established in the literature as a rather robust phenomenon that clearly contradicts traditional preference theory. As a typical example, consider two lotteries: one a relatively safe lottery (usually referred to as the P-bet) that offers a 90% chance to win $10 and a 10% chance to win nothing; and the other a relatively risky lottery (usually referred to as the $-bet) which offers a 10% chance to win $90 and a 90% chance to win nothing. When offered a choice between the two lotteries many subjects choose the safe lottery over the risky one. However, when asked for the minimal selling price, many subjects state a higher price for the safe lottery than for the risky one. This evidence seems to indicate that the preference between both lotteries depends on the method employed to elicit preferences. An important aspect of the preference reversal phenomenon is the fact that the evidence cannot be regarded as a random deviation from traditional preference theory. This is because subjects who prefer the safe lottery state an inconsistent price ordering (i.e. a higher selling price for the risky lottery) relatively frequently, whereas among subjects who prefer the risky lottery the frequency of inconsistent price orderings (i.e. a higher selling price for the safe lottery) is comparatively low. For instance in the well known study of Grether and Plott (1979), inconsistent price orderings were observed with a frequency of 69% among subjects who chose the safe lottery but only with a frequency of 22% among subjects who chose the risky lottery. A number of attempts to explain preference reversals have been proposed, of which four seem to command particular respect (see Seidl (2002) for a review): (1) violations of transitivity (Loomes and Sugden (1983) and Fishburn (1985)); (2) violations of the independence axiom (Holt (1986) and Karni and Safra (1987)); (3) violations of the reduction axiom (Segal (1988)); 3

4 and (4) violations of procedure invariance (Goldstein and Einhorn (1987) and Tversky et al. (1990)). In their seminal paper, Tversky et al. (1990) presented experimental evidence that apparently shows that the first three explanations can account only for 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. The importance of scale compatibility is also stressed in a recent study of Cubitt, Munro, and Starmer (2004) although a part of their evidence can only explained by scale compatibility in the presence of prominence effects. Suppose an individual has not only to state his or her minimal selling price as in the usual experimental design for which preference reversals have been observed but also his or her maximal buying price for a given lottery. Since both the selling and the buying price have to be stated in monetary units, scale compatibility should have the same effect on both prices. More precisely, if observed preference reversals are only due to scale compatibility, the evidence should be, at least in our theoretical framework, independent of whether selling prices or buying prices are elicited. However, our experimental results clearly show that this is not true: while the preference reversal phenomenon is evident for selling prices, for buying prices no systematic pattern can be observed. At this stage, it is interesting to note that for other observed violations of Expected Utility theory, like, for example, the Allais paradox, the notion of errors (in subjects responses in experiments) has been proposed as a plausible and simple explanation. In the light of the result mentioned above, it seems reasonable to analyse the preference reversal phenomenon from the point of view of errors. Surprisingly, the only analysis of this kind we are aware of is one conducted by Lichtenstein and Slovic (1971). However, since Lichtenstein and Slovic (1971) apply their error model only to between-subjects data several points can be criticised. One important point is the fact that they assign identical error probabilities to all subjects, a fact which 4

5 is clearly rejected by the data from other experimental studies, for example, that of Hey and Orme (1994). In our view, a sensible analysis of preference reversals and errors requires a repeated experiment in order to separate out the stochastic and the deterministic components of behaviour. Accordingly, we designed an experiment in which the subjects had to perform five different tasks on five separate occasions, answering choice questions on three occasions and pricing questions on the other two occasions. The main argument behind our analysis is as follows. Suppose individuals have a deterministic choice and price ordering for all lottery pairs. But suppose further that subjects express these orderings with some error, which means that they sometimes express the wrong ordering. Rather obviously, we assume that the error rate is some positive number but less than 50% - that is, subjects state more often their true preference than the opposite preference. Now consider the set of all choices an individual makes in the three choice occasions. For our analysis we partition this set into two disjoint subsets, first the set of those choices which were, for a given lottery pair, identical in all three choice occasions and secondly the set of those choices for which inconsistencies occurred between the three choice occasions. In the second subset at least one of the three observed choices for a given lottery pair is an error. In the first subset the three choices are either all correct or all errors. However, if the probability of errors is not too high, it is rather improbable that a subject made the same error for a given lottery pair three times in a row. Therefore, we can exclude the bulk of choice errors from our data set by considering only the first subset in which the choice was identical in all three choice occasions. Analogously, we can exclude most of the pricing errors if we restrict attention to the cases in which the order of prices for a given lottery pair was identical in both pricing occasions. Obviously, if observed preference reversals are errors, they can be due to either choice errors, or pricing errors, or both types of errors. This gives rise to three hypotheses: 5

6 Hypothesis 1: Preference reversals occur less frequently in those cases in which the pricing order is identical in both pricing occasions. Hypothesis 2: Preference reversals occur less frequently in those cases in which the choice is identical in all three choice occasions. Hypothesis 3: Preference reversals occur less frequently in those cases in which the choice is identical in all three choice occasions and the pricing identical in both pricing occasions. These three hypotheses are investigated in this paper. The next section presents our experimental design. Section 3 explains our theoretical framework and section 4 contains the results: Our data show that the exclusion of pricing errors does indeed reduce the frequency of preference reversals significantly whereas choice errors seem to play a minor role. Some concluding remarks appear in section 5. 1 Experimental Design The experiment was conducted at the Centre of Experimental Economics at the University of York with 24 participants. Each participant had to attend five separate occasions, A, B, C, D, and E with occasions A and B being identical (except for the way that the questions were portrayed on the computer screens). During five days of one week one of each four different occasions was offered on every single day with varying chronological order. Consequently, 20 occasions were offered altogether and the participants could choose on which days they attended which occasions. Each of the occasions lasted between 25 and 40 minutes. The time varied not only between the single occasions but also across the subjects since they were explicitly encouraged to proceed at their own pace. After a subject had completed all five occasions one question of one occasion was selected randomly and played out for real. The average payment to the subjects was with 80 being the highest and 0 being the lowest payment. 6

7 On each of the five occasions the subjects were presented with the same 30 lottery pairs - 28 risky and 2 uncertain. The probabilities of the 28 risky lottery pairs are given in Table 1. The 2 uncertain pairs were expressed as in the Ellsberg Paradox with 30 as the possible prize. Note that in each pair in Table 1 the left lottery is safer than the right lottery, though in the experiment the left-right juxtaposition was randomised. The lotteries were presented as segmented circles on the computer screen. Figure 1 presents an example in which there is a 50% chance of getting 10, a 20% chance of getting 30, and a 30% chance of getting 40. If a subject received a particular lottery as reward he or she had to spin a wheel on the corresponding circle. The amount won was then determined by the segment of the circle in which the arrow on the wheel stopped. We now describe the single occasions. For expositional reasons we begin with occasions D and E and then proceed to describe occasions A, B and C. On occasion D each of the 60 lotteries appeared separately in randomised order on screen and subjects were asked for each lottery: Submit your bid for this lottery in a second-price sealed-bid auction. That is subjects were asked to assume they did not have the lottery and had to bid to get it. They had to type in their bid and confirm it by pressing the return key. At the beginning of the experimental session subjects received a three-page instruction sheet. Then an audio-tape of these instructions was played which took approximately ten minutes. The instructions explained clearly the rules and the incentive compatibility of second-price sealed-bid auctions. If a question of occasion D was selected for the reward, the subject received a payment of y where y is the highest amount in the corresponding lottery. Moreover, if the subject submitted the highest bid among all subjects in the group with whom she completed occasion D, he or she would additionally play out the lottery and had to pay the second highest bid. Summarising, we elicited in occasion D for each lottery the maximal buying price which will be referred to as bid in the following. Occasion E was identical to occasion D except that for each lottery a different question was asked: Submit your offer for this lottery in a second-price offer auction. That is subjects 7

8 were asked to assume that they owned the lottery and had to make an offer to dispose of it. Again subjects received a handout and had to listen to an audio-tape of the three-page instructions which explained clearly the rules and the incentive compatibility of the second-price offer auction. If a question from occasion E was selected for the reward, the subject could play out the corresponding lottery. However, if he or she submitted the lowest offer among all subjects in the group with whom she completed occasion E, he or she received the second lowest offer instead of the lottery. Occasion E allowed for the elicitation of minimal selling prices - which will be called asks in the following. Note that the second-price auctions used for the elicitation of bids and asks are certain to be incentive compatible only if expected utility holds [see Grimm and Schmidt (2000) for a detailed analysis], i.e. it is optimal for an individual to reveal the true values if this individual is an expected utility maximiser, but not necessarily if the individual is not an expected utility maximiser. Due to this fact we assumed consistency with expected utility in the theoretical framework presented in the next section. Moreover, it is well known that the bid for a given lottery may differ from the ask due to income effects. Although these income effects can only explain a very small difference between bids and asks [see Willig (1976) and Harless (1989)] they are ruled out in our theoretical framework by assuming constant absolute risk aversion. In occasions A, B, and C there were two different screens for every lottery pair, though the second of these screens does not concern the analysis of this paper. On the first screen both lotteries of the pair appeared as circles on screen and subjects had to indicate whether they preferred the left lottery, or the right lottery, or neither. After pressing the corresponding key they had to confirm their choice by pressing the return key. The order of the single lotteries and the left-right positioning of each lottery pair were randomised. (On the second screen (conditional) certainty equivalents were elicited employing a BDM mechanism. This data will be analysed in a separate paper.) As in occasions D and E, in occasions A, B, and C subjects received a handout and had to listen to an audio-tape of the instructions at the beginning of the experimental session. 8

9 If the question on the first screen was selected for their reward they could play out the preferred lottery. In the case of indifference one of the lotteries was selected randomly by the experimenter. 2 The Theoretical Framework Before we analyse the alternative hypotheses mentioned in the introduction we briefly explain our underlying theoretical framework which is based on the theory of errors developed in Hey (1995). If we disregard the possibility of errors for a moment, individuals are assumed to be expected utility maximisers who exhibit constant absolute risk aversion (CARA). This assumption guarantees that under our incentive mechanism preference reversals are ruled out in the absence of errors for both selling and buying prices. However, in the presence of errors preference reversals may occur in our framework. Our assumptions are, therefore, motivated by the desire to separate intrinsic preference reversals from preference reversals that are simply caused by errors. In the following we denote the utility difference between lottery S and lottery R by V(S, R). Consequently, V(S, R) > 0 (< 0) indicates that the safe lottery S is strictly preferred (strictly not preferred) to the risky lottery R. Since we assume expected utility with CARA, V(S, R) is given by (1) V(S, R) = i [1 - exp(-cx i )]s i - i [1 - exp(-cx i )]r i = i exp(-cx i )[r i - s i ], where the values of x i denote the possible consequences of the lotteries while the values of s i (r i ) are their probabilities in the lottery S (R). Initial wealth has been normalised to zero in (1) and the parameter c determines the degree of absolute risk aversion of the considered individual. Let us now consider the asks for lotteries. The ask a S for the safe lottery S is determined by (2) i [1 - exp(-cx i )]s i = 1 - exp(-ca S ), while the ask a R for the risky lottery R is the solution to (3) i [1 - exp(-cx i )]r i = 1 - exp(-ca R ). 9

10 Subtracting (3) from (2) yields (4) V(S, R) = exp(-ca R ) exp(-ca S ) and, thus, a S > (<) a R if and only if V(S, R) > (<) 0. Consequently, in the absence of errors, preference reversals with respect to asks are ruled out in our framework. Let us now consider the bids for lotteries. The bid b S for the safe lottery S is determined by (5) i [1 - exp(-c(x i -b S ))]s i = 0. Analogously, the bid b R for R is the solution to (6) i [1 - exp(-c(x i -b R ))]r i = 0. Combining (5) and (6) we get (7) i exp(-cx i )[r i exp(cb R ) s i exp(cb S )] = 0, which shows that b S > (<) b R if and only if V(S, R) > (<) 0. Therefore, preference reversals are also ruled out for bids in the absence of errors. We assume expected utility with CARA in our theoretical framework in order to guarantee that preference reversals for both, bids and asks, can only be due to errors. Errors are now integrated in our framework as follows. There is some error term ε such that in practice not V(S, R) but the value of V(S, R) + ε determines the choice between S and R. More precisely, the individual will choose S (R) if and only if V(S, R) + ε > 0 (< 0). Furthermore, let D i (S, R) denote for i = a, b denote the difference of asks and bids, i.e. D a (S, R) = a S a R and D b (S, R) = b S - b R. We assume that also the price ordering is stated only with some error ϕ i, i.e. the individual will assign a higher price to S (R) if and only if D i (S, R) + ϕ i > 0 (< 0). We assume that both error terms are random variables with a median of zero. In this case the probability of making an error is always less than 50%, i.e. if V(S, R) > 0 the individual will state with a probability of more than 50% a preference and a higher price for S. Moreover, it is 10

11 easily verified that, in this case, the validity of all the three hypotheses of the introduction is implied by our model. 3 Results For each lottery pair and each subject we will analyse the following five observations of our experiment: the three stated preferences between both lotteries of occasions A, B, and C, the relation of the maximal buying prices (bids) for both lotteries of occasion D, and the relation of the two minimal selling prices (asks) of occasion E. In the following we indicate a preference for the safe (risky) lottery by S (R). A predicted preference reversal (PPR) occurs if the preference is S but the ask (bid) is higher for the risky than for the safe lottery. Conversely, an unpredicted preference reversal (UPR) consists of a preference for R and a higher ask (bid) for the safe lottery. Table 2 shows the relative frequency of preference reversals for asks (respectively bids in parentheses) in the single occasions. The second column in Table 2 reports the absolute frequency of S and R choices. The third column shows the relative frequency of consistent price orderings, i.e. the cases in which subjects assigned a higher ask (bid) to the chosen lottery. The relative frequency of preference reversals, i.e. inconsistent price orderings is stated in the fourth column while the fifth column report the cases in which an identical price was assigned to both lotteries. Over all occasions, the frequency of PPRs, i.e. inconsistent price ordering given that the preference is S, amounts to 33% for asks and 26% for bids. Recall that the explanation of preference reversals by scale compatibility implies an identical pattern for asks and bids in our framework because in both cases the response scale is given by monetary units. However, the fourth column of Table 2 shows that there is a clear difference between the results for asks and bids: while for asks the frequency of PPRs is roughly two times higher than the frequency of UPRs, both frequencies are nearly identical for bids. We can conclude that, in contrast to asks, for bids no systematic deviation from traditional preference theory can be observed. This clearly contradicts the explanation of preference reversal by scale compatibility, at least in our 11

12 framework. More precisely, scale compatibility is not a significant sole explanation of preference reversals, however, it can possibly be a good explanation only if we deviate from expected utility with CARA. Let us now turn to Hypothesis 1. Suppose for simplicity that the individual makes no errors in the choice questions (i.e. ε equals always zero) and that V(S, R) > 0. In this case the probability of a preference reversal is obviously given by p = Prob[D i (S, R) + ϕ i < 0]. In Table 3 only those cases are considered in which the relation of asks for a lottery pair was identical to the relation of bids. For this cases our model predicts a probability for preference reversals of p 2 / [p 2 + (1 p) 2 ] which is strictly less than p for p < 0.5. In other words, the bulk of pricing errors should have been eliminated from consideration in Table 3 and, according to Hypothesis 1, the frequency of preference reversals should decrease. Comparing Tables 2 and 3 reveals that this is indeed the case since the frequency of both PPRs and UPRs is roughly 20% lower in Table 3 than in Table 2 for asks. If the cases with identical prices are removed from Table 2, the number of PPRs for asks amounts to 41%. Thus, the number of PPRs if the pricing order is identical is 15 percentage points lower than in the case if the pricing order is different. A chi-square test confirms that this difference is significant at the 1% level (χ 2 = 28.02). Thus, Hypothesis 1 is confirmed by our data: (predicted) preference reversals occur less frequently if the pricing order is identical in both pricing occasions. From this evidence one can conclude that observed preference reversals are partly not intrinsic preference reversals but due to pricing errors. Let us now turn to Hypothesis 2. In Table 4 attention is restricted to those cases in which the stated preference was identical in occasions A, B, and C. Again it can be easily verified that, according to our model, the probability of choice errors is lower for these cases than for all cases. Compared to Table 2 the number of PPRs in Table 4 is slightly more than 10% lower. If the cases with identical preference are removed from Table 2, the number of PPRs for asks amounts 12

13 to 34%. Thus, the number of PPRs if preference is identical is 5 percentage points lower than in the case if the pricing order is different. However, a chi-square test shows that this difference is insignificant (χ 2 = 2.18). Thus, contrary to Hypothesis 2, considering only cases with identical preference in all three choice occasions does not lead to a significant reduction of PPRs. Let us finally consider Hypothesis 3. In Table 5 only those cases are considered in which the ordering of prices was identical in both pricing occasions and the preference was identical in all three preference occasions. Compared to Table 2 this leads to a reduction of PPRs of 42%. If the cases with identical pricing and preference are removed from Table 2, the number of PPRs for asks amounts to 35%. Thus, the number of PPRs if pricing and preference is identical is 16 percentage points lower than in the case if pricing or preference (or both) are different. This difference is significant at the 1% level (χ 2 = 15.21). Thus, Hypothesis 3 is confirmed by our data: (predicted) preference reversals occur less frequently if the pricing order and the preference is identical in the single occasions. Taken together the evidence seems to indicate that the influence of pricing errors is higher than the influence of choice errors. In order to analyse this question we exclude the cases with identical pricing and preference of Table 5 and compare the remaining cases of identical pricing with the remaining cases of identical preference. The relative number of PPRs is in the former case given by 28% and in the latter case by 42%. Since this difference is significant at the 1% level we can conclude that more PPRs are due to pricing errors than to choice errors. 4 Conclusions The goal of the present paper was to analyse the role of errors in explaining the preference reversal phenomenon. We developed a simple model assuming expected utility preferences with constant absolute risk aversion, and showed that, even in this standard framework, preference reversals will be observed in the presence of errors. In order to test the 13

14 implications of our model we ran a repeated experiment in which subjects had to make decisions concerning the same 30 lottery pairs in five consecutive occasions. In three occasions they just had to choose the preferred lottery from each pair whereas in the remaining two occasions they had to assign prices to each of the 60 lotteries. In contrast to the standard experimental design, subjects had to state minimal selling prices (willingness-toaccept or asks) in one of the pricing occasions, but maximal buying prices (willingness-to-pay or bids) in the other pricing occasion. As a first result, we observe that the evidence concerning asks and bids is rather different: while for asks the frequency of predicted preference reversals is, as in previous studies, much higher than the frequency of unpredicted preference reversal, both frequencies are roughly the same for bids. Thus, no systematic deviation from standard choice theory can be observed for bids. Since the hypothesis of scale compatibility has identical implications for bids and asks it can be ruled out as a sole explanation of preference reversals, at least in our framework. In other words, preference reversals can only be explained by scale compatibility, if one deviates from a framework of expected utility with constant absolute risk aversion. This result is somewhat similar to the conclusion of Cubitt, Munro, and Starmer (2004), that their data cannot be explained by scale compatibility alone, but only in conjunction with some other deviation from standard theory like prominence effects. Concerning the role of errors, we find that the exclusion of pricing errors leads to a significant reduction of the frequency of preference reversals, while the exclusion of choice errors does not. If we exclude both pricing and choice errors, the frequency of predicted preference reversals can be reduced to 19%. In view of this comparatively low number one may conclude that intrinsic preference reversals are perhaps somewhat less important than earlier studies suggest. However, even if both pricing and choice errors are excluded, the frequency of predicted preference reversals is nevertheless much higher than the frequency of unpredicted preference reversals (19% versus 7%). Thus, a systematic deviation from 14

15 standard theory remains. There are, of course, a number of attempts to explain this phenomenon that could be labelled psychological. In addition, an explanation within standard preference theory has recently been offered by Sugden (2003): in this, Sugden develops a model of preferences with state-dependent reference points in which preference reversals result from loss aversion. However, an empirical test of this explanation remains to be done. 15

16 *Acknowledgements The experiment was conducted while the first author was visiting EXEC at the University of York. Financial support for this visit by the Deutsche Forschungsgemeinschaft under contracts Schm1396/1-1 and 1-2 is gratefully acknowledged. Financial support for the running of the experiment and for the payment of the subjects was obtained from the TMR ENDEAR project of the European Union under contract CT We are indebted to an anonymous referee who pointed out some flaws in an earlier version of this paper and to the editor for very helpful comments. We are also indebted to Norman Spivey for writing the software for this experiment, and would also like to thank Stefan Traub for help with the data processing. For critical and helpful comments on the instructions we wish to thank Marie Edith Bissey, Carmela Di Mauro, and Anna Maffioletti. 16

17 References Cubitt, R., Munro, A., and Starmer, C.V. (2004). Testing Explanations of Preference Reversal, Economic Journal, forthcoming. Fishburn, P.C. (1985). Nontransitive Preference Theory and the Preference Reversal Phenomenon, Rivista Internazionale di Science Economiche e Commerciali 32, Goldstein, W., and Einhorn, H.J. (1987). Expression Theory and the Preference Reversal Phenomena, Psychological Review 94, Grether, D., and Plott, C. (1979). Economic Theory of Choice and the Preference Reversal Phenomenon, American Economic Review 69, Grimm, V., and Schmidt, U. (2000). Equilibrium Bidding without the Independence Axiom: A Graphical Analysis, Theory and Decision 49, Harless, D.W. (1989). More Laboratory Evidence on the Disparity between Willingness to Pay and Compensation Demanded, Journal of Economic Behavior and Organization 11, Hey, J.D. (1995). Experimental Investigations of Errors in Decision Making under Risk, European Economic Review 39, Hey, J.D., and Orme, C. (1994). Investigating Generalizations of Expected Utility Theory Using Experimental Data, Econometrica 62, Holt, C.A. (1986). Preference Reversals and the Independence Axiom, American Economic Review 76, Karni, E., and Safra, Z. (1987). Preference Reversals and the Observability of Preferences by Experimental Methods, Econometrica 55, Lichtenstein, S., and Slovic, P. (1971). Reversals of Preferences Between Bids and Choices in Gambling Decisions, Journal of Experimental Psychology 89, Loomes, G., and Sugden, R. (1983). A Rationale for Preference Reversal, American Economic Review 73,

18 Segal, U. (1988). Does the Preference Reversal Phenomenon Necessarily Contradict the Independence Axiom?, American Economic Review 78, Seidl, C. (2002). Preference Reversal, Journal of Economic Surveys 16, Slovic, P., and Lichtenstein, S. (1968). Relative Importance of Probabilities and Payoffs in Risk Taking, Journal of Experimental Psychology 78, Sugden, R. (2003). Reference-dependent subjective expected utility, Journal of Economic Theory 111, Tversky, A., Slovic, P., and Kahneman, D. (1990). The Causes of Preference Reversal, American Economic Review 80, Willig, R.D. (1976). Consumer Surplus without Ap.ology, American Economic Review 66,

19 Table 1: The Lottery Pairs

20 Table 2: Preference Reversals Occasion Number of Choices Consistent Inconsistent Equal A S: (57) 36 (29) 16 (14) R: (51) 18 (28) 18 (21) B S: (61) 31 (24) 18 (15) R: (56) 17 (24) 17 (20) C S: (58) 33 (26) 17 (16) R: (53) 18 (28) 19 (20) Aggregated S: (59) 33 (26) 17 (15) R: (53) 17 (27) 18 (20) 20

21 Table 3: Preference Reversals if Pricing is Identical Occasion Number of Choices Consistent Inconsistent Equal A S: R: B S: R: C S: R: Aggregated S: R:

22 Table 4: Preference Reversals if Preference is Identical Number of Consistent Inconsistent Equal Choices S: (65) 29 (22) 15 (13) R: (56) 13 (22) 16 (22) 22

23 Table 5: Preference Reversal if Preference and Pricing is Identical Number of Consistent Inconsistent Equal Choices S: R:

24 Figure 1: The Presentation of Lotteries 30 20% 10 50% 30% 40 24