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Do preference reversals generalise? Results on ambiguity and loss aversion
Journal of Economic Psychology 33 (2012) 48–57
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Journal of Economic Psychology journal homepage: www.elsevier.com/locate/joep
Do preference reversals generalise? Results on ambiguity and loss aversion Linden J. Ball a, Nicholas Bardsley b,⇑, Tom Ormerod a a b
University of Lancaster, Department of Psychology, United Kingdom University of Reading, Department of Food Economics and Marketing, School of Agriculture, Policy and Development, United Kingdom
a r t i c l e
i n f o
Article history: Received 10 February 2011 Received in revised form 31 August 2011 Accepted 2 September 2011 Available online 14 September 2011 JEL classification: C91 D03 PsychINFO classification: 2260 2340
a b s t r a c t Preference reversals are frequently observed in the lab, but almost all designs use completely transparent prospects, which are rarely features of decision making elsewhere. This raises questions of external validity. We test the robustness of the phenomenon to gambles that incorporate realistic ambiguity in both payoffs and probabilities. In addition, we test a recent explanation of preference reversals by loss aversion, which would also restrict the incidence of reversals outside the lab. According to this account, reversals occur largely because the valuation task endows subject with a gamble, activating loss aversion. This contrasts with the choice task, where the reference point is pre-experiment wealth. We test this explanation by holding the reference point constant. Our evidence suggests that reversals are only slightly diminished with ambiguity. We find no evidence supporting their explanation by loss aversion. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Preference reversals External validity Ambiguity Loss aversion
1. Introduction Psychologists often, perhaps even typically, interpret the experimental literature on decision making as providing copious evidence that people do not exhibit the rational consistency in behaviour that mainstream economic theory attributes to them. But many researchers, especially in economics, have been reluctant to endorse this interpretation. It is common instead to question aspects of the lab settings, to run designs which attempt to make the troublesome results disappear, and to devise modifications of decision theory which might rationalise them. This pattern is particularly evident regarding the ‘‘preference reversal’’ phenomenon, first reported by Lichtenstein and Slovic (1971). A preference reversal occurs when a preference is evidenced both for one option over another and vice versa. The most common pattern of reversal, a ‘‘standard reversal’’, occurs as follows. A subject in an experiment chooses a bet which offers a high probability of a small cash prize, denoted the ‘‘P-bet’’, signifying probability, in preference to a bet which offers a lower probability of a larger sum, denoted the ‘‘$-bet’’, signifying money. When asked to value each bet, however, typically by specifying a price for which they would be willing to sell it, the subject attaches a higher monetary value to the $-bet. For example, a majority of subjects might choose a 75% chance of winning £4 (the P-bet) over a 25% chance of winning £12 (the $-bet), but express a higher selling price for the latter than for the former. In doing so, they attach greater
⇑ Corresponding author. E-mail address: [email protected] (N. Bardsley). 0167-4870/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.joep.2011.09.001
L.J. Ball et al. / Journal of Economic Psychology 33 (2012) 48–57
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value to the option which was not chosen. The opposite pattern of reversal ($-bet chosen but P-bet valued higher) is also observed but is relatively uncommon. Preference reversals are especially puzzling to economists as they appear to contradict basic tenets of rationality that are routinely applied in economics. Grether and Plott (1979), amongst the first economists to investigate the phenomenon, complained that ‘‘Taken at face value the data are simply inconsistent with preference theory and have broad implications about research priorities within economics. The inconsistency . . . suggests that no optimisation principles of any sort lie behind even the simplest of human choices . . .’’ Further, the experiments have been heavily replicated, also in designs using monetary incentives, and almost all attempts to make reversals disappear have failed. (For a review of preference reversal studies see Seidl, 2002.) Notwithstanding this, there has only ever been very localised impact of the results on the practice of economics, causing at least one influential methodologist, over 20 years on from the discovery of the phenomenon, to complain of dogmatism (Hausman, 1992). It is plausible that doubts that the results of the experiments generalise to naturally-occurring settings contribute to such resistance. We aim to shed light on the extent to which the findings generalise and to test an apparently compelling, recent attempt to rationalise them. Psychologists’ interpretations of the phenomenon have tended to be in terms of ‘‘compatibility’’ between gamble attributes and the tasks involved in the design (Fischer & Hawkins, 1993; Tversky, Sattath, & Slovic, 1988). This has been specified in terms of the scale of the valuation task being the same as the gamble’s prize dimension (‘‘scale compatibility’’), and the valuation task triggering quantitative rather than qualitative reasoning (‘‘strategy compatibility’’). Either factor supposedly leads to greater weight being placed on the prize dimension of the gamble in the valuation task than in the choice task. Whilst this is an explanation of the phenomenon that is still standing after testing (for example by Cubitt, Munro, & Starmer, 2004), one might question the extent to which the phenomenon is informative about decisions in general under this account. Mechanisms that have been suggested to be consistent with a compatibility hypothesis, including anchoring and adjustment, and imprecision of preferences, are typically expounded with reference to precise monetary values for the prizes (see, for example, Blavatskyy, 2009; Butler & Loomes, 2007; Cox, 2008; Schkade & Johnson, 1989). In the real world, one might argue, we do not often encounter chances of specific sums of money; objects of choice and valuation are either non-monetary or constitute more complex and less well-defined financial prospects. It is debatable philosophically whether an experiment is always undermined by a lack of external validity, and the contrary view expressed by Schram (2005), that theory testing is a sufficient and distinct design justification, is perhaps prevalent amongst experimental economists. Independently of this issue, though, we find it natural to probe how the results might extend to other contexts than the lab, given the large amount of research attention they have attracted over recent decades.1 Even if designs do not have to have external validity, for example because they are testing theories, we presumably learn more from them if they do, so it is important to ask the question. As the quotation from Grether and Plott (1979) suggests, it is possible that preference reversal portends a high degree of variability and/or malleability in decision making quite generally, but given the specific explanations psychologists have proposed it seems debatable how the results might generalise. It is this uncertainty that we seek to address. Bohm’s (1994) field experiment is perhaps the best known test of whether preference reversals generalise to tasks which are more realistic, in that the probabilities and consequences are not transparent. Bohm used real goods in an auction, namely two used cars, to conduct the valuation task of his experiment, finding no evidence of preference reversals. However, although this design clearly departed from transparency, it is perhaps too uncertain how subjects interpreted the options available. A possible interpretation of the results is that subjects had very firm preferences over the cars, which were reflected in both choice and valuation procedures. Even using transparent prospects, it is easy to prevent reversals by making one of the gambles overwhelmingly attractive, either by making its expected value much greater or making it stochastically dominant. But that does not alter the fact that reversals are common when the two gambles are closer in attractiveness. We adopt a different strategy to Bohm, suggested by Bardsley et al. (2010). This is to probe external validity in the lab. We do so by modifying the attributes of the gambles that can be regarded as most artificial, but retaining a degree of control over the gamble characteristics that is difficult to achieve in a field setting. We retain the essential characteristics of the P- and $bets, described above, but explore the introduction of ambiguity in both payoffs and probabilities. Analytically, it seems plausible that ambiguity reduces to uncertainty over probabilities.2 However, it does not follow that experimenters should only ever use tasks that are transparent in consequences. For decision processes, and outcomes, may be different when subjects have to construct for themselves the set of consequences over which probabilities range. We vary the degree of vagueness of the tasks to explore the robustness of preference reversals to departures from transparency. Our use of prospects which are nontransparent in both probabilities and consequences extends work by Trautmann, Vieider, and Wakker (2011) and Maafi (2011), which found reversals for gambles which were ambiguous in probabilities only. We include as a limiting case a gamble pair where even the type of prize is unknown. A second aim of the experiment was to test the loss aversion explanation of preference reversal, first stated by Sugden (2003), and which has been formally derived from Schmidt et al.’s (2008) ‘‘Third Generation Prospect Theory’’ (PT3). The idea
1 Lichtenstein and Slovic (1973) reproduced the phenomenon in a casino. This is somewhat encouraging for external validity, but only concerning generalisation to a (non-student) subject pool and a naturally-occurring setting. Casino gambling has in common with the lab an unusual degree of prospect transparency compared to the generality of normal decisions. The gambles used in the casino replication were regular lab gambles, in that they were fully transparent in both probabilities and consequences. 2 We owe this point to professor Peter Wakker (personal correspondence).
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Table 1 Hypotheses and predictions. Hypothesis
Prediction
1. Preference reversals are dependent on transparent prospects 2. Preference reversals are encouraged by transparency 3. Preference reversals are dependent on the shifting reference point in the standard design 4. Preference reversals are encouraged by the shifting reference point in the standard design
No systematic reversals in gamble pairs b–f More reversals the lower the – rating of the gamble pair No systematic reversals in the gift treatment More reversals in the gift treatment than in the selling treatment
is that, in the choice task, the reference point from which gambles are evaluated is pre-experiment wealth, so that both gambles are regarded as improvements on the status quo. Since the valuation task is generally conducted under a selling protocol, subjects are, in contrast, placed in the position of already possessing the gamble in question. The theoretical innovation of PT3 is a reworking of prospect theory to accommodate changes in the reference point, including situations where the reference point is a lottery. A key motivation of the authors is to provide an improved explanation of preference reversals. Using this framework, Schmidt et al. (2008) demonstrate that the standard pattern of reversals might be attributed largely to loss aversion. That is, sellers of gambles may be averse to the prospect of having sold a $-bet in the event that it would have paid out, thus ‘‘losing’’ the high payoff. In the choice task, since subjects choose between potential gains, such an eventuality would not be evaluated there as a loss. Given the asymmetry of the valuation function specified in Prospect Theory, one would expect standard preference reversals to be encouraged if the reference point shifts from pre-experiment wealth (in the choice task) to the gamble endowment (in the valuation task). Testing the loss aversion explanation of preference reversals is of interest in its own right, but our two research questions are not entirely unrelated. For if a reference point shift is responsible for the results, one might expect behaviour outside the lab to have greater consistency than it would under the compatibility account.3 The reason for this is related to Bohm’s (1994) reasoning. Bohm notes that the prospects used in preference reversal designs are unusual in that they are such good bets. If a casino used such gambles in its daily business, it would be showering its pundits with money; thus, such bets cannot exist in a market. Be that as it may, for Bohm’s criticism to hold a reason still needs to be found why reversals are especially likely with unusually good bets, since the choice and valuation tasks use the same bets. We share Bohm’s intuition that the standard design might be unrepresentative, but we think that the more likely culprit is the choice task. The reference point implemented by the choice task seems unusual, because in normal economic life choices between free gifts are, sadly, relatively rare. Presents aside, we tend to be giving something up to gain something else, as in buying or selling. The loss aversion account of reversals is tested by holding the reference point constant between choice and valuation tasks. We run both tasks with a free gift protocol comparing this with what happens in the usual protocol where there is gift selection in choice tasks and selling in the valuation task. We use the terms ‘‘gift treatment’’ and ‘‘selling treatment’’ respectively to denote these conditions. This terminology is not used in the instructions however, where we use exactly the kind of language that is standard in the literature. There should be no systematic reversals in the gift treatment if preference reversal depends on loss aversion. Standard reversals should be less frequent in the gift treatment than in the selling treatment if loss aversion is at least a substantial cause. We summarise the foregoing discussion in Table 1 which sets out the hypotheses to be tested. The tasks referred to in the table are described in the next section (Table 2). 2. Tasks The task set consisted of 12 gambles, divided into 6 pairs (a–f) for the choice tasks, but presented in randomised order for the valuation tasks. The information about payoffs and probabilities were chosen, on the basis of a pilot study, to produce a mixture of P-bet and $-bet choosers. This was necessary because if either bet were too attractive it would be chosen in both choice and valuation tasks and the preference reversal phenomenon would be excluded merely by imbalanced bet parameters. The paired gambles were as shown in Table 2 below, with a ⁄ rating to indicate the intended degree of ambiguity. A zerostar rating means nothing is ambiguous, so that the gamble is an ‘industry-standard’ prospect, with transparency in both probabilities and payoffs. A one-star rating implies that there is one attribute of the gamble that is ambiguous, a two-star rating that there are two ambiguous attributes, and so on. For the three-star gamble of pair f, neither the probability, type,
3 It should be noted, however, that there is more than one respect in which preference reversal experiments may have or lack external validity. One is the extent to which reversals are potential features of the world outside the lab, in the sense of decisions being influenced by seemingly irrelevant features of framing or response mode. This matter motivates Bohm and Lind (1993) and Bohm (1994). Another is the extent to which the factors responsible for preference reversals also underlie behaviour elsewhere. If preference reversals do stem from loss aversion, for example, and loss aversion is important in other domains, this would be an important kind of external validity even if there were very few naturally-occurring preference reversals, or even none at all. We are concerned here exclusively with the former matter. We owe this point to professor Chris Starmer (personal correspondence).
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L.J. Ball et al. / Journal of Economic Psychology 33 (2012) 48–57 Table 2 Gambles as presented to subjects. (a) P1 A 90% chance of winning. The prize is £6.67 $1 A 20% chance of winning. The prize is £30.00 (b) P2 A fairly good chance of winning £10 $2 A pretty low chance of winning £60 (c) P3 A very high probability of winning. The prize is a modest sum of money $3 A low probability of winning. The prize is a very generous sum of money (d) P4 A reasonable probability of winning a bag of coins $4 A low probability of winning a roll of notes (e) P5 A large sum of money has been divided equally into a large number of equal shares, which are represented by scrabble tiles in a bag. If you draw any of these tiles you will get one of these shares. There are six tiles that do not win: J, K, Q, X and Z (there are two Ks) $5 A large sum of money has been divided into six equal shares, which are represented by scrabble chips in a bag. If you draw any of these tiles you will get one of these shares. The six tiles that win are: J, K, Q, X and Z (there are two Ks) (f) P6 A high probability of winning. The prize (not cash) is modest, but we are confident that most people would be happy to get this $6 A low probability of winning. The prize (not cash) is something extremely desirable, we are confident that most people would be very happy indeed to get this Note: For gamble pair (d) the actual bag of coins and roll of notes were used as stimuli.
Table 3 Underlying gamble payoffs and probabilities. Pair
Gamble
Prize
Probability
a
P1 $1 P2 $2 P3 $3 P4 $4 P5 $5 P6 $6
£6.67 £30 £10 £60 £7 £100 £14 £35 £7.95 £58.30 £10 store voucher £50 store voucher
0.90 0.20 0.70 0.10 0.95 0.05 0.45 0.18 0.80 0.12 0.70 0.10
b c d e f
Note: The store vouchers used were for HMV stores, which are exchangeable for CDs, DVDs, computer games and so on.
nor monetary value of the prize is specified. Subjects were only informed about the desirability of the prize here and that it was not a sum of money. Thus, only for P1 and $1 were the gambles transparent. The underlying probabilities and payoffs for the other gambles were not revealed to subjects until after the experiment. Payoffs and probabilities for each gamble were in fact as shown in Table 3 below. 3. Procedures 3.1. General Experiments were run in May and October 2009 in the Psychology department at the University of Lancaster, UK. 104 subjects took part, 52 in the selling treatment and 52 in the gift treatment. Subjects were undergraduates and postgraduates drawn from across the University. Subjects received a show-up fee of £4 in addition to their earnings from the preference reversal tasks, one of which was paid out for real in accordance with standard experimental economics practice. Expected earnings were approximately £10 in total, consisting of the show-up fee and around £6 from the preference reversal tasks. The experiment lasted around 40 min in total, resulting in a wage rate comparable to that obtainable for ad hoc postgraduate research assistance. Instructions were administered both in written form and verbally, and the procedure determining payouts was demonstrated, both for choice and valuation tasks. Tasks were completed by subjects working through a booklet of answers, recording the choices and valuations in each task. Separate booklets were completed for choices and valuations, and tasks were counterbalanced across sessions. The information for the ambiguous gambles was necessarily incomplete, but the instructions emphasised that the expected payouts were similar in terms of their monetary value. This was explained in terms of the experiment costing roughly the same amount regardless of which task was to be paid out. This served two purposes: to make the comparison between the transparent and ambiguous gambles meaningful, and to ensure that all tasks were taken with roughly equal seriousness. The instructions are given in Appendix.
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3.2. Gamble realisation and payments The sequence proceeded as follows. Before the valuation (choice) tasks were completed, a valuation (choice) task was demonstrated, then the subjects completed the valuation (choice) tasks. At that point the answers to the valuation (choice) tasks were collected. A choice (valuation) task was then demonstrated, and the subjects went on to complete the choice (valuation) tasks. After these answers had been collected, the paid task was selected and played out: a coin flip decided whether this would be a choice or valuation task, then a separate die roll determined which gamble would be paid out. The same gamble (or gamble pair, if a choice task paid out) was selected for everyone. In the event that a gamble paid out, a separate die roll for each subject determined the outcome of the selected lottery, excepting P5 and $5, which were resolved by independent draws of a chip from a bag. Up to 2 rolls of a 10 sided-die, were used, giving a range of numbers from 0 to 99, with the first roll determining the first digit. Thus, for example, for P4, if a subject rolled higher than a 4 on the first roll she lost; if she rolled lower than 4 she won. If she rolled a 4, she rolled again, with numbers 0–4 paying out. In this way, numbers 0–44 paid out and 45–99 lost. For valuation tasks, 12 sealed envelopes were picked from a box by a subject at the beginning of the experiment and assigned as they chose to numbers 1–12, which represented valuation tasks. These envelopes were stuck to a board at the front of the room. Each envelope specified a sum of money, which was to serve as an offer that could be exchanged for a lottery depending on the subject’s reserve price. In the event that a valuation task paid out, a subject drew once from tickets numbered 1–12 to determine which task, for all subjects, was chosen. The number was taken from the envelope to determine whether each subject played the gamble. If their reserve price was lower than the offer, they received the offer in pounds. If their reserve price was higher than the offer they played the gamble. 3.3. Selling versus gift manipulation In the selling treatment, instructions for the valuation tasks were couched in the standard language of prices and offers. That is, subjects were told they had been given a gamble and were asked to specify the price for which they would be willing to sell it. It was explained that an offer had been made for each gamble, contained in the sealed envelopes at the front of the room. If a particular gamble was selected to be played out, then if the offer exceeded their price they would exchange the gamble opportunity for the cash, otherwise they would ‘‘keep’’ and play the gamble. The logic underpinning this procedure is exactly the logic of the more usual Becker Degroot Marshak mechanism, but without the complicating factor of a randomising device to determine the offers. In the gift treatment, both the gamble and the amount of money in the envelope were described as gains, and the instructions avoided the language of buying, selling, offers, keeping or exchanging. Thus, subjects were asked to consider a gamble, the offer was described simply as a number and they were asked to say how much they thought the gamble was worth. In other words, the task was to specify an equivalent gain. Both outcomes were described as prizes – either a subject would receive a gamble to play or they would receive some cash, depending on whether the number in the envelope was less or greater than their valuation respectively. In the gift treatment, therefore, the reference point against which possible outcomes are judged is the same in the choice and valuation tasks, whilst in the selling treatment it differs. There, subjects in the valuation task are asked to put a selling price on something they have already been given, whilst in the choice tasks they choose to receive one of two gambles. 4. Results Results are set out in Table 4 below. We report the result of a two-sided binomial test, where the null hypothesis is that standard and nonstandard reversals are equiprobable. This allows us to distinguish between random and systematic reversals. An exposition of why this is necessary is given in Appendix.4 The proportion of responses that constitute standard preference reversals, that is, the percentage of subjects in each task choosing the P-bet but valuing the $-bet more highly, is shown in the ‘‘reversals’’ entry. This is calculated excluding equal valuations. Summary statistics on valuations are shown in Table 5 below. 5. Analysis 5.1. By task For all gamble pairs, the binomial test is highly significant except for pair e, for which the test statistic is non-significant as reversals are symmetric. In every case where the statistic is significant, standard reversals outnumber nonstandard reversals. Thus, in all cases except pair e there is a statistically significant asymmetry between behaviour in the choice and valuation tasks, which is definitive of the preference reversal phenomenon.
4
We owe this point to professor Robert Sugden (personal correspondence).
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L.J. Ball et al. / Journal of Economic Psychology 33 (2012) 48–57 Table 4 Results. Pair
a
Treatment
Selling
Gift
b Selling
Gift
c Selling
Gift
d Selling
Gift
e Selling
Gift
f Selling
Gift
Sample size Choice % Choosing P % Choosing $ Choice and Valuation Consistent, prefer P Consistent, prefer $ Standard Reversal Counter Reversal Binomial p-value Standard reversals
52
52
52
51
51
52
51
52
51
52
52
51
0.85 0.15
0.81 0.19
0.71 0.29
0.73 0.27
0.84 0.16
0.88 0.12
0.53 0.47
0.40 0.60
0.65 0.35
0.60 0.40
0.73 0.27
0.73 0.27
19 5 25 3 0 0.48
15 8 27 2 0 0.52
24 13 13 2 0.01 0.25
19 7 18 7 0.04 0.35
23 4 20 4 0 0.38
30 6 16 0 0 0.31
10 20 17 4 0.01 0.33
9 26 12 5 0.14 0.23
21 9 12 9 0.66 0.23
20 10 11 11 1 0.21
16 9 22 5 0 0.42
20 10 17 4 0.01 0.32
Notes: (i) ‘Consistent, prefer P ($)’ denotes P ($) was chosen and valued at least as highly as $ (P). (ii) Different sample sizes reflect missing data, from one subject in each treatment.
Table 5 Valuations. Selling
Mean Median Sample standard deviation
Gift
P
$
P
$
£7.43 £6.58 £2.89
£9.39 £8.38 £5.39
£7.06 £6.25 £3.20
£8.64 £7.29 £4.87
Note: Median and standard deviation are shown for subjects’ mean valuations over the 12 gambles.
By inspection of Table 4 there is no evidence of a straightforward relationship between the degree of ambiguity of the tasks and the incidence of reversals as defined by their rating in Table 2. There is a significant difference, however, between the percentage of standard reversals for the transparent gambles and across the other tasks (50% versus a mean of 31%; 2tailed Z-test, p < 0.01). This comparison uses subjects’ decisions averaged over the other gambles; on average standard preference reversals made up 31% of observations for pairs b–f. 5.2. By treatment Asymmetric preference reversals are a prominent feature of the results in both treatments, as indicated by the binomial test. The standard pattern of preference reversal results is generally reproduced in both treatments, meaning that there is a greater propensity for subjects to favour P in the choice task than in the valuation task. Differences in the proportions of standard reversals across treatments are not significant for any gamble pair (2-tailed Z-test, ps > 0.10). Differences in valuations across treatments are non-significant according to standard tests (2-tailed T-test; Mann–Whitney U test; ps > 0.10). 6. Discussion The results analysed by task show with high confidence that the preference reversal phenomenon is not an artefact of the transparency of prospects (Section 5.1); that is, hypothesis 1 is rejected. Even for gambles which were vaguely specified in both probabilities and outcomes, a significant asymmetry between choice and valuation results occurred, conforming to the classic preference reversal pattern. We found no evidence of a consistent relationship between reversals and the degree of ambiguity of prospects. Fully transparent gambles may nonetheless encourage reversals. There is a degree of support for hypothesis 2, since the fully transparent gamble pair, a, produced significantly more reversals than the average across the other pairs. This result is directionally consistent within each subsample, but only significant at the 5% level for the selling treatment (2-tailed Z-test, p = 0.01 and p = 0.11). The (limited) support for hypothesis 2 is tentative, since there is considerable variation in the frequency of reversals between gamble pairs, and it may be the case that differently-specified vague gambles could match the frequency of reversals in transparent gambles. However, it seems to us unlikely that the difference is down to the specific gamble used, since: (i) the rate of standard reversals observed for pair a was not particularly high relative to other designs; and (ii) at least one gamble pair, c, produced a similar split to pair a between the P- and $-bets in the choice task, but fewer observed standard reversals.
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Essentially the same pattern of results obtains in both selling and gift treatments (Section 5.2). This is evidence against loss aversion explanations of preference reversal, since we ought to have found a marked decrease in reversals in the gift treatment. Hypotheses 3 and 4 are therefore rejected. Given that we found null results across the selling and gift treatments, one might question whether our treatment manipulation was actually effective. However, in the present context it is important to note that we are testing the loss aversion explanation of preference reversals. The factor being manipulated is the specification of the reference point, this specification being constituted by the instructions exactly as in standard designs. These instructions are entirely at the discretion of the experimenter, and the loss aversion explanation assumes the efficacy of the instructions to implement different reference points. It may be the case that the selling treatment did not give rise to a marked differential in the perceived reference point, or experienced loss aversion, but either possibility counts against the loss aversion explanation of preference reversals. The valuations data shown in Table 5 are consistent with a small difference in loss aversion across treatments, but the observed difference is not statistically significant. Another explanation for this, assuming for the sake of argument that it is a real difference, is that subjects are applying a heuristic adapted to real life bargaining situations of asking for a higher price than one’s actual reserve price. The logic of the loss aversion account of preference reversals, as formalised in PT3, is impeccable. However, it seems to us that it may be psychologically implausible. This is because it requires that subjects view selling a gamble as risking a loss, even if its resolution is to remain unknown. That is, they are supposed to regard the abstract likelihood that they sell a gamble which would have paid off in the same way as the likelihood that they experience a departure from a concrete asset that they actually hold. This is, we suspect, especially unlikely given that the subject is only fleetingly endowed with each gamble, for the purposes of one task. An experiment could no doubt be designed which makes the endowment of a gamble more meaningful. But the loss aversion/PT3 account is supposed to explain reversals in the standard design, where it is operationalised exactly as in the experiment reported here. Our design was not optimised for systematic exploration of scale compatibility effects. However, it is worth noting that pair f produced standard reversals where scale compatibility seems not to apply. This is because of the non-monetary prize here. Cubitt et al. (2004) also find reversals where scale compatibility is not a plausible explanation. Concerning pair f, scale compatibility is certainly suppressed relative to the other pairs, but cannot be entirely excluded because we had informed subjects that each gamble is expected to cost the experimenter the same amount in the event it is selected to be played. Subjects could therefore deduce that $6 had a more expensive prize than P6 even though they were told it was not a cash prize. Another aspect of our results worth noting, given the general robustness of the preference reversal phenomenon, is the absence of systematic reversals for pair e. There, standard and nonstandard reversals became equally frequent. Thus, for pair e and pair e only, the evidence is consistent with a symmetric stochastic process of preference reversal production, whereby subjects deviate with some probability from an underlying disposition to favour either the P- or $-bet. One possibility is that subjects were simply confused and chose at random. This does not seem plausible because the observations are not uniformly distributed, as the P-bet was chosen twice as often here as the $-bet. A second possibility is that the use of frequency information underlies this result. For the other tasks, no frequency information was given and it would have been unclear to subjects how the prospects were to be resolved. However, frequency presentations are common in preference reversal studies and do not normally dispel systematic reversals. We find it likely that pair e involved more cognitive difficulty than the other gamble pairs. Pair e was designed so that subjects are in possession of good information, qualitatively speaking, but deprived of transparency. That is, it is deducible from the information provided here that, although the P-bet is far more likely to pay out, the two bets have exactly the same expected value, although that expected value is not specified. Proximity in expected value is a requirement for reversals, as we noted earlier, since one gamble should not be overwhelmingly attractive. However, the task does require subjects to reason. This may have caused the difference observed, since it may have undermined the clarity or intensity of preferences between the bets, resulting in a greater stochastic element in decision making. The reason for constructing the task in this way is that many real world decisions are opaque in a similar way; we wished to see if the results of transparent environments extend to them. 7. Conclusions Our results clearly indicate that preference reversals are not dependent on the gamble transparency, nor on the shifting reference point, which feature in almost the entire body of previous preference reversal studies. Firstly, classic preference reversal results were obtained with prospects that are ambiguous in both probabilities and consequences. Explanations of reversals which have been stated with reference to precise valuations, including anchoring and adjustment, and preference imprecision, may need to be revised to accommodate this case. We find the normal pattern of reversals even where the notion of scale compatibility seems inapplicable because a non-monetary prize is specified. We do find some evidence, however, that reversals are encouraged by transparency. Secondly, reversals were undiminished when choice and valuation tasks deployed the same reference point. This contradicts any explanation of preference reversals based on loss aversion, including the loss aversion explanation recently offered under PT3. Thus, the latest attempt to reconcile preference reversals with decision theory seems to be rejected by the lab. In
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the preference reversal context, loss aversion may be psychologically implausible because the loss concerned is a departure from a potential, rather than an experienced, monetary holding. The failure of the reference point manipulation to impact on reversals might be considered further good news for the robustness of experimental results on choice under risk. For what is arguably a rather artificial aspect of the standard design is found to have no discernible impact on reversals. A question our results raise is whether the rejection of the loss aversion account of preference reversals is a serious problem for PT3 itself. This might be explored by strengthening the reference point manipulation compared to what is done in the standard preference reversal design. Our results seem consistent with an interpretation of preference reversal offered by Cubitt et al. (2004), in terms of a conjunction of psychological effects, rather than an economic rationalisation. The data suggest that fully transparent prospects encourage reversals, which is consonant with compatibility effects, but that they also occur where scale compatibility is suppressed. A hypothesis we form for future investigation is that systematic reversals may be diminished by cognitive difficulty of decision tasks. Acknowledgments Research for this paper was supported by a Collaborative Fund grant from the ESRC National Centre for Research Methods (NCRM), grant RES-576-47-5001, and was initiated whilst Bardsley was affiliated to NCRM at the University of Southampton. We are grateful to Robert Sugden and Chris Starmer for comments received. The authors thank Caroline Wade for assistance with running the experiments reported. Ordering of authors is alphabetical. Appendix A A.1. Detecting preference reversals Preference reversals generate results of the form shown in the table below: Task
Valuation
Outcome
P>$ $>P
Choice P
$
A C
B D
The preference reversal literature sometimes cites a difference between the proportions B/(A + B) and C/(C + D) as evidence of an asymmetry in reversals. However, this is a mistake, from a stochastic point of view. Suppose that all subjects prefer the P to $, but choose $ with probability q. Then A, B, C, and D expressed as a proportion of total choices would become (1 q)2, q(1 q), q(1 q) and q2 respectively. It follows that B/(A + B) and C/(C + D) would equal q and 1 q respectively. Thus, although choices and valuations would be consistent in the sense that under either procedure P is selected with probability 1 q, the measure would report an ‘asymmetry in reversals.’ The binomial test, or an approximate version such as McNemar’s (1947) test of marginal homogeneity, is suitable to detect an imbalance in the off diagonal elements of such a matrix. A.2. Instructions Welcome to this experiment. Each of you has been given a show-up fee of £4 for attending today. In addition to this, you each have the chance to win a further prize. The details of how this will happen are as follows. The experiment consists of two parts, a ‘‘choice’’ part and a ‘‘valuation’’ part. In each case you have to judge the attractiveness of possible lotteries. At the end of the experiment a coin will be tossed to determine which part is played out for real, and one of the tasks will be chosen at random. Which prize you receive depends on which task is chosen and which decision you made in that task, plus a purely random element. The tasks in both parts involve lotteries. Each lottery refers to a chance of winning a prize. The chances and prizes involved are different in each lottery. You have to say what you would choose to do in each situation below, based on how attractive you think the lotteries are. All the lotteries would cost the experimenter a similar amount of money if everyone chose to play them. All the lotteries will be revealed at the end, after your decisions have been recorded. A.2.1. Choice part In this part of the experiment you have to choose between two lottery tickets in each task. Indicate your choice by circling letter A or B. What you decide here will affect your potential prize from the experiment if a task from the choice part is paid out at the end. (Recall that one task from the whole experiment is selected at random and played out for real.) [A practice choice task using transparent gambles was played out before subjects received their answer booklets, but not paid. The booklets presented the tasks in the following form:]
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1. Choose between A. A 90% chance of winning. The prize is £6.67, and B.A 20% chance of winning. The prize is £30.00. 2. Choose between A.A fairly good chance of winning £10, and B.A pretty low chance of winning £60. [And so on. The order of gamble pairs was counterbalanced across subjects.] A.2.2. Valuation part Selling treatment: In this part of the experiment we would like you to tell us how much you would be willing to sell a lottery ticket for. We appreciate that for some of the lotteries this may be quite difficult. However, we think that it is in your interests to take this seriously. What you decide here will affect your potential prize from the experiment if a task from the valuation part is paid out at the end. (Recall that one task from the whole experiment is selected at random and played out for real.) In each task you are given a lottery ticket. For each ticket we would like you to state a price for which you would be just willing to sell it. We’ve constructed this part of the experiment to give you a reason to do just that – though you are quite free to do what you like! The envelope which was selected (at random, by a participant) at the start of this experiment contains a number of pounds (and pence), which we will call the ‘‘offer.’’ This number, which is different for each ticket, will only be revealed after all decisions have been made. If the number you gave as the price of the ticket is less than or equal to the offer number, you will be given that offer in pounds if this task is paid out. Otherwise you will keep the lottery ticket, and that will determine your earnings by a play of the lottery. The higher the price you specify, the more likely it is that you will keep the lottery ticket. The lower the price, the more likely it is that you get the money instead. Suppose you think the ticket is worth £x. If you say that your price is more than £x, then you risk that the offer is higher than £x, but less than your price. In that case you will keep the lottery ticket, whereas you could have received more than £x by stating the price as £x. If, alternatively, you say that your price is less than £x, you risk that the offer is less than £x but more than your price. In that case you will receive a sum of money which you think is inferior to the ticket, whereas you could have kept the ticket by stating a price of £x. (Verbal) Write down any number, large or small. Imagine someone who owns an antique. He’s not sure what the market value is but probably it’s not worth much. But there’s a small chance it’s worth a lot. He decides, taking this into account, that it’s worth £30 to him. Now imagine he’s playing the same game described in the instructions, with your number as the offer. Suppose he said £1000 as the price. What could go wrong? . . . (answer elicited) Suppose he said £100 as the price. What could go wrong? . . . (answer elicited) It’s the same problem for any price higher than £30. Now suppose he said £2 as the price. What could go wrong? . . . (answer elicited) Suppose he said £20 as the price. What could go wrong? . . . (answer elicited) It’s the same problem for any price lower than £30. This shows that the best thing you can do is state your true price. Gift treatment: In this part of the experiment we would like you to tell us how valuable a lottery ticket is to you. We appreciate that for some of the lotteries this may be quite difficult. However, we think that it is in your interests to take this seriously. What you decide here will affect your potential prize from the experiment if a task from the valuation part is paid out at the end. (Recall that one task from the whole experiment is selected at random and played out for real.) In each task you are required to consider a lottery ticket. For each ticket we would like you to state a sum of money which you think is as good as it but no better. We’ve constructed this part of the experiment to give you a reason to do just that – though you are quite free to do what you like! The envelope which was selected (at random, by a participant) at the start of this experiment contains a number of pounds (and pence), which we will call the ‘‘deciding number.’’ This number, which is different for each ticket, will only be revealed after all decisions have been made. If the number you gave as the value of the ticket is less than or equal to the deciding number, you will be given that number in pounds if this task is paid out. Otherwise you will be given the lottery ticket, and that will determine your earnings by a play of the lottery. The higher the value you specify, the more likely it is that you will get the lottery ticket. The lower the value, the more likely it is that you get the money instead. Suppose you think the ticket is worth £x. If you say that the value is more than £x, then you risk that the deciding number is higher than £x, but less than your stated value. In that case you will be given the lottery ticket, whereas you could have received more than £x by stating the value as £x. If, alternatively, you say that the value is less than £x, you risk that the
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deciding number is less than £x but more than your stated value. In that case you will receive a sum of money which you think is inferior to the ticket, whereas you could have received the ticket by stating a value of £x. (Verbal) Write down any number, large or small. Imagine someone who owns an antique. He’s not sure what the market value is but probably it’s not worth much. But there’s a small chance it’s worth a lot. He decides, taking this into account, that it’s worth £30 to him. Now imagine he’s playing the same game described in the instructions, with your number as the deciding number. Suppose he said £1000 as the value. What could go wrong? . . . (answer elicited) Suppose he said £100 as the value. What could go wrong? . . . (answer elicited) It’s the same problem for any value higher than £30. Now suppose he said £2 as the value. What could go wrong? . . . (answer elicited) Suppose he said £20 as the value. What could go wrong? . . . (answer elicited) It’s the same problem for any value lower than £30. This shows that the best thing you can do is state your true value. [After the valuation instructions, subjects valued two practice lotteries with transparent prospects, which were not paid out. They then valued the 12 lotteries, the order being counterbalanced across subjects.] References Bardsley, N., Cubitt, R., Loomes, G., Moffatt, P., Starmer, C., & Sugden, R. (2010). Experimental economics: Rethinking the rules. Princeton, NJ: Princeton University Press. Blavatskyy, P. (2009). Preference reversals and probabilistic decisions. Journal of Risk and Uncertainty, 39, 237–250. Bohm, P. (1994). Behaviour under uncertainty without preference reversal: A field experiment. Empirical Economics, 19, 185–200. Bohm, P., & Lind, H. (1993). Preference reversal, real-world lotteries and lottery-interested subjects. Journal of Economic Behavior and Organization, 22, 327–348. Butler, D. J., & Loomes, G. (2007). Imprecision as an account of the preference reversal phenomenon. American Economic Review, 97, 277–297. Cox, J. C. (2008). Preference reversals. In C. R. Plott & V. L. Smith (Eds.). Handbook of experimental economics results (Vol. 1). New York: Elsevier Press. Cubitt, R. P., Munro, A., & Starmer, C. (2004). Testing explanations of preference reversal. Economic Journal, 114, 709–726. Fischer, G. W., & Hawkins, S. A. (1993). Strategy compatibility, scale compatibility, and the prominence effect. Journal of Experimental Psychology: Human Perception and Performance, 19, 580–597. Grether, D., & Plott, C. (1979). Economic theory of choice and the preference reversal phenomenon. American Economic Review, 69, 623–638. Hausman, D. (1992). The inexact and separate science of economics. Cambridge and New York: Cambridge University Press. Lichtenstein, S., & Slovic, P. (1971). Reversals of preference between bids and choices in gambling situations. Journal of Experimental Psychology, 89, 46–55. Lichtenstein, S., & Slovic, P. (1973). Response-induced reversals of preference in gambling: An extended replication in Las Vegas. Journal of Experimental Psychology, 101, 16–20. Maafi, A. (2011). Preference Reversals under Ambiguity. Management Science, in press. McNemar, Q. (1947). Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika, 12, 153–157. Schkade, D., & Johnson, E. J. (1989). Cognitive processes in preference reversals. Organizational Behavior and Human Decision Processes, 44, 203–231. Schmidt, U., Starmer, C., & Sugden, R. (2008). Third generation prospect theory. Journal of Risk and Uncertainty, 36, 203–223. Schram, A. (2005). Artificiality: The tension between internal and external validity in experimental economics. Journal of Economic Methodology, 12, 225–238. Seidl, C. (2002). Preference reversal. Journal of Economic Surveys, 16, 621–655. Sugden, R. (2003). Reference dependent subjective expected utility theory. Journal of Economic Theory, 111, 172–191. Trautmann, S. T., Vieider, F. M., & Wakker, P. P. (2011). Preference reversals for ambiguity aversion. Management Science, 57, 1320–1333. Tversky, A., Sattath, S., & Slovic, P. (1988). Contingent weighting in judgement and choice. Psychological Review, 95, 371–384.
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Journal of Economic Psychology journal homepage: www.elsevier.com/locate/joep
Do preference reversals generalise? Results on ambiguity and loss aversion Linden J. Ball a, Nicholas Bardsley b,⇑, Tom Ormerod a a b
University of Lancaster, Department of Psychology, United Kingdom University of Reading, Department of Food Economics and Marketing, School of Agriculture, Policy and Development, United Kingdom
a r t i c l e
i n f o
Article history: Received 10 February 2011 Received in revised form 31 August 2011 Accepted 2 September 2011 Available online 14 September 2011 JEL classification: C91 D03 PsychINFO classification: 2260 2340
a b s t r a c t Preference reversals are frequently observed in the lab, but almost all designs use completely transparent prospects, which are rarely features of decision making elsewhere. This raises questions of external validity. We test the robustness of the phenomenon to gambles that incorporate realistic ambiguity in both payoffs and probabilities. In addition, we test a recent explanation of preference reversals by loss aversion, which would also restrict the incidence of reversals outside the lab. According to this account, reversals occur largely because the valuation task endows subject with a gamble, activating loss aversion. This contrasts with the choice task, where the reference point is pre-experiment wealth. We test this explanation by holding the reference point constant. Our evidence suggests that reversals are only slightly diminished with ambiguity. We find no evidence supporting their explanation by loss aversion. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Preference reversals External validity Ambiguity Loss aversion
1. Introduction Psychologists often, perhaps even typically, interpret the experimental literature on decision making as providing copious evidence that people do not exhibit the rational consistency in behaviour that mainstream economic theory attributes to them. But many researchers, especially in economics, have been reluctant to endorse this interpretation. It is common instead to question aspects of the lab settings, to run designs which attempt to make the troublesome results disappear, and to devise modifications of decision theory which might rationalise them. This pattern is particularly evident regarding the ‘‘preference reversal’’ phenomenon, first reported by Lichtenstein and Slovic (1971). A preference reversal occurs when a preference is evidenced both for one option over another and vice versa. The most common pattern of reversal, a ‘‘standard reversal’’, occurs as follows. A subject in an experiment chooses a bet which offers a high probability of a small cash prize, denoted the ‘‘P-bet’’, signifying probability, in preference to a bet which offers a lower probability of a larger sum, denoted the ‘‘$-bet’’, signifying money. When asked to value each bet, however, typically by specifying a price for which they would be willing to sell it, the subject attaches a higher monetary value to the $-bet. For example, a majority of subjects might choose a 75% chance of winning £4 (the P-bet) over a 25% chance of winning £12 (the $-bet), but express a higher selling price for the latter than for the former. In doing so, they attach greater
⇑ Corresponding author. E-mail address: [email protected] (N. Bardsley). 0167-4870/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.joep.2011.09.001
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value to the option which was not chosen. The opposite pattern of reversal ($-bet chosen but P-bet valued higher) is also observed but is relatively uncommon. Preference reversals are especially puzzling to economists as they appear to contradict basic tenets of rationality that are routinely applied in economics. Grether and Plott (1979), amongst the first economists to investigate the phenomenon, complained that ‘‘Taken at face value the data are simply inconsistent with preference theory and have broad implications about research priorities within economics. The inconsistency . . . suggests that no optimisation principles of any sort lie behind even the simplest of human choices . . .’’ Further, the experiments have been heavily replicated, also in designs using monetary incentives, and almost all attempts to make reversals disappear have failed. (For a review of preference reversal studies see Seidl, 2002.) Notwithstanding this, there has only ever been very localised impact of the results on the practice of economics, causing at least one influential methodologist, over 20 years on from the discovery of the phenomenon, to complain of dogmatism (Hausman, 1992). It is plausible that doubts that the results of the experiments generalise to naturally-occurring settings contribute to such resistance. We aim to shed light on the extent to which the findings generalise and to test an apparently compelling, recent attempt to rationalise them. Psychologists’ interpretations of the phenomenon have tended to be in terms of ‘‘compatibility’’ between gamble attributes and the tasks involved in the design (Fischer & Hawkins, 1993; Tversky, Sattath, & Slovic, 1988). This has been specified in terms of the scale of the valuation task being the same as the gamble’s prize dimension (‘‘scale compatibility’’), and the valuation task triggering quantitative rather than qualitative reasoning (‘‘strategy compatibility’’). Either factor supposedly leads to greater weight being placed on the prize dimension of the gamble in the valuation task than in the choice task. Whilst this is an explanation of the phenomenon that is still standing after testing (for example by Cubitt, Munro, & Starmer, 2004), one might question the extent to which the phenomenon is informative about decisions in general under this account. Mechanisms that have been suggested to be consistent with a compatibility hypothesis, including anchoring and adjustment, and imprecision of preferences, are typically expounded with reference to precise monetary values for the prizes (see, for example, Blavatskyy, 2009; Butler & Loomes, 2007; Cox, 2008; Schkade & Johnson, 1989). In the real world, one might argue, we do not often encounter chances of specific sums of money; objects of choice and valuation are either non-monetary or constitute more complex and less well-defined financial prospects. It is debatable philosophically whether an experiment is always undermined by a lack of external validity, and the contrary view expressed by Schram (2005), that theory testing is a sufficient and distinct design justification, is perhaps prevalent amongst experimental economists. Independently of this issue, though, we find it natural to probe how the results might extend to other contexts than the lab, given the large amount of research attention they have attracted over recent decades.1 Even if designs do not have to have external validity, for example because they are testing theories, we presumably learn more from them if they do, so it is important to ask the question. As the quotation from Grether and Plott (1979) suggests, it is possible that preference reversal portends a high degree of variability and/or malleability in decision making quite generally, but given the specific explanations psychologists have proposed it seems debatable how the results might generalise. It is this uncertainty that we seek to address. Bohm’s (1994) field experiment is perhaps the best known test of whether preference reversals generalise to tasks which are more realistic, in that the probabilities and consequences are not transparent. Bohm used real goods in an auction, namely two used cars, to conduct the valuation task of his experiment, finding no evidence of preference reversals. However, although this design clearly departed from transparency, it is perhaps too uncertain how subjects interpreted the options available. A possible interpretation of the results is that subjects had very firm preferences over the cars, which were reflected in both choice and valuation procedures. Even using transparent prospects, it is easy to prevent reversals by making one of the gambles overwhelmingly attractive, either by making its expected value much greater or making it stochastically dominant. But that does not alter the fact that reversals are common when the two gambles are closer in attractiveness. We adopt a different strategy to Bohm, suggested by Bardsley et al. (2010). This is to probe external validity in the lab. We do so by modifying the attributes of the gambles that can be regarded as most artificial, but retaining a degree of control over the gamble characteristics that is difficult to achieve in a field setting. We retain the essential characteristics of the P- and $bets, described above, but explore the introduction of ambiguity in both payoffs and probabilities. Analytically, it seems plausible that ambiguity reduces to uncertainty over probabilities.2 However, it does not follow that experimenters should only ever use tasks that are transparent in consequences. For decision processes, and outcomes, may be different when subjects have to construct for themselves the set of consequences over which probabilities range. We vary the degree of vagueness of the tasks to explore the robustness of preference reversals to departures from transparency. Our use of prospects which are nontransparent in both probabilities and consequences extends work by Trautmann, Vieider, and Wakker (2011) and Maafi (2011), which found reversals for gambles which were ambiguous in probabilities only. We include as a limiting case a gamble pair where even the type of prize is unknown. A second aim of the experiment was to test the loss aversion explanation of preference reversal, first stated by Sugden (2003), and which has been formally derived from Schmidt et al.’s (2008) ‘‘Third Generation Prospect Theory’’ (PT3). The idea
1 Lichtenstein and Slovic (1973) reproduced the phenomenon in a casino. This is somewhat encouraging for external validity, but only concerning generalisation to a (non-student) subject pool and a naturally-occurring setting. Casino gambling has in common with the lab an unusual degree of prospect transparency compared to the generality of normal decisions. The gambles used in the casino replication were regular lab gambles, in that they were fully transparent in both probabilities and consequences. 2 We owe this point to professor Peter Wakker (personal correspondence).
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Table 1 Hypotheses and predictions. Hypothesis
Prediction
1. Preference reversals are dependent on transparent prospects 2. Preference reversals are encouraged by transparency 3. Preference reversals are dependent on the shifting reference point in the standard design 4. Preference reversals are encouraged by the shifting reference point in the standard design
No systematic reversals in gamble pairs b–f More reversals the lower the – rating of the gamble pair No systematic reversals in the gift treatment More reversals in the gift treatment than in the selling treatment
is that, in the choice task, the reference point from which gambles are evaluated is pre-experiment wealth, so that both gambles are regarded as improvements on the status quo. Since the valuation task is generally conducted under a selling protocol, subjects are, in contrast, placed in the position of already possessing the gamble in question. The theoretical innovation of PT3 is a reworking of prospect theory to accommodate changes in the reference point, including situations where the reference point is a lottery. A key motivation of the authors is to provide an improved explanation of preference reversals. Using this framework, Schmidt et al. (2008) demonstrate that the standard pattern of reversals might be attributed largely to loss aversion. That is, sellers of gambles may be averse to the prospect of having sold a $-bet in the event that it would have paid out, thus ‘‘losing’’ the high payoff. In the choice task, since subjects choose between potential gains, such an eventuality would not be evaluated there as a loss. Given the asymmetry of the valuation function specified in Prospect Theory, one would expect standard preference reversals to be encouraged if the reference point shifts from pre-experiment wealth (in the choice task) to the gamble endowment (in the valuation task). Testing the loss aversion explanation of preference reversals is of interest in its own right, but our two research questions are not entirely unrelated. For if a reference point shift is responsible for the results, one might expect behaviour outside the lab to have greater consistency than it would under the compatibility account.3 The reason for this is related to Bohm’s (1994) reasoning. Bohm notes that the prospects used in preference reversal designs are unusual in that they are such good bets. If a casino used such gambles in its daily business, it would be showering its pundits with money; thus, such bets cannot exist in a market. Be that as it may, for Bohm’s criticism to hold a reason still needs to be found why reversals are especially likely with unusually good bets, since the choice and valuation tasks use the same bets. We share Bohm’s intuition that the standard design might be unrepresentative, but we think that the more likely culprit is the choice task. The reference point implemented by the choice task seems unusual, because in normal economic life choices between free gifts are, sadly, relatively rare. Presents aside, we tend to be giving something up to gain something else, as in buying or selling. The loss aversion account of reversals is tested by holding the reference point constant between choice and valuation tasks. We run both tasks with a free gift protocol comparing this with what happens in the usual protocol where there is gift selection in choice tasks and selling in the valuation task. We use the terms ‘‘gift treatment’’ and ‘‘selling treatment’’ respectively to denote these conditions. This terminology is not used in the instructions however, where we use exactly the kind of language that is standard in the literature. There should be no systematic reversals in the gift treatment if preference reversal depends on loss aversion. Standard reversals should be less frequent in the gift treatment than in the selling treatment if loss aversion is at least a substantial cause. We summarise the foregoing discussion in Table 1 which sets out the hypotheses to be tested. The tasks referred to in the table are described in the next section (Table 2). 2. Tasks The task set consisted of 12 gambles, divided into 6 pairs (a–f) for the choice tasks, but presented in randomised order for the valuation tasks. The information about payoffs and probabilities were chosen, on the basis of a pilot study, to produce a mixture of P-bet and $-bet choosers. This was necessary because if either bet were too attractive it would be chosen in both choice and valuation tasks and the preference reversal phenomenon would be excluded merely by imbalanced bet parameters. The paired gambles were as shown in Table 2 below, with a ⁄ rating to indicate the intended degree of ambiguity. A zerostar rating means nothing is ambiguous, so that the gamble is an ‘industry-standard’ prospect, with transparency in both probabilities and payoffs. A one-star rating implies that there is one attribute of the gamble that is ambiguous, a two-star rating that there are two ambiguous attributes, and so on. For the three-star gamble of pair f, neither the probability, type,
3 It should be noted, however, that there is more than one respect in which preference reversal experiments may have or lack external validity. One is the extent to which reversals are potential features of the world outside the lab, in the sense of decisions being influenced by seemingly irrelevant features of framing or response mode. This matter motivates Bohm and Lind (1993) and Bohm (1994). Another is the extent to which the factors responsible for preference reversals also underlie behaviour elsewhere. If preference reversals do stem from loss aversion, for example, and loss aversion is important in other domains, this would be an important kind of external validity even if there were very few naturally-occurring preference reversals, or even none at all. We are concerned here exclusively with the former matter. We owe this point to professor Chris Starmer (personal correspondence).
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L.J. Ball et al. / Journal of Economic Psychology 33 (2012) 48–57 Table 2 Gambles as presented to subjects. (a) P1 A 90% chance of winning. The prize is £6.67 $1 A 20% chance of winning. The prize is £30.00 (b) P2 A fairly good chance of winning £10 $2 A pretty low chance of winning £60 (c) P3 A very high probability of winning. The prize is a modest sum of money $3 A low probability of winning. The prize is a very generous sum of money (d) P4 A reasonable probability of winning a bag of coins $4 A low probability of winning a roll of notes (e) P5 A large sum of money has been divided equally into a large number of equal shares, which are represented by scrabble tiles in a bag. If you draw any of these tiles you will get one of these shares. There are six tiles that do not win: J, K, Q, X and Z (there are two Ks) $5 A large sum of money has been divided into six equal shares, which are represented by scrabble chips in a bag. If you draw any of these tiles you will get one of these shares. The six tiles that win are: J, K, Q, X and Z (there are two Ks) (f) P6 A high probability of winning. The prize (not cash) is modest, but we are confident that most people would be happy to get this $6 A low probability of winning. The prize (not cash) is something extremely desirable, we are confident that most people would be very happy indeed to get this Note: For gamble pair (d) the actual bag of coins and roll of notes were used as stimuli.
Table 3 Underlying gamble payoffs and probabilities. Pair
Gamble
Prize
Probability
a
P1 $1 P2 $2 P3 $3 P4 $4 P5 $5 P6 $6
£6.67 £30 £10 £60 £7 £100 £14 £35 £7.95 £58.30 £10 store voucher £50 store voucher
0.90 0.20 0.70 0.10 0.95 0.05 0.45 0.18 0.80 0.12 0.70 0.10
b c d e f
Note: The store vouchers used were for HMV stores, which are exchangeable for CDs, DVDs, computer games and so on.
nor monetary value of the prize is specified. Subjects were only informed about the desirability of the prize here and that it was not a sum of money. Thus, only for P1 and $1 were the gambles transparent. The underlying probabilities and payoffs for the other gambles were not revealed to subjects until after the experiment. Payoffs and probabilities for each gamble were in fact as shown in Table 3 below. 3. Procedures 3.1. General Experiments were run in May and October 2009 in the Psychology department at the University of Lancaster, UK. 104 subjects took part, 52 in the selling treatment and 52 in the gift treatment. Subjects were undergraduates and postgraduates drawn from across the University. Subjects received a show-up fee of £4 in addition to their earnings from the preference reversal tasks, one of which was paid out for real in accordance with standard experimental economics practice. Expected earnings were approximately £10 in total, consisting of the show-up fee and around £6 from the preference reversal tasks. The experiment lasted around 40 min in total, resulting in a wage rate comparable to that obtainable for ad hoc postgraduate research assistance. Instructions were administered both in written form and verbally, and the procedure determining payouts was demonstrated, both for choice and valuation tasks. Tasks were completed by subjects working through a booklet of answers, recording the choices and valuations in each task. Separate booklets were completed for choices and valuations, and tasks were counterbalanced across sessions. The information for the ambiguous gambles was necessarily incomplete, but the instructions emphasised that the expected payouts were similar in terms of their monetary value. This was explained in terms of the experiment costing roughly the same amount regardless of which task was to be paid out. This served two purposes: to make the comparison between the transparent and ambiguous gambles meaningful, and to ensure that all tasks were taken with roughly equal seriousness. The instructions are given in Appendix.
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3.2. Gamble realisation and payments The sequence proceeded as follows. Before the valuation (choice) tasks were completed, a valuation (choice) task was demonstrated, then the subjects completed the valuation (choice) tasks. At that point the answers to the valuation (choice) tasks were collected. A choice (valuation) task was then demonstrated, and the subjects went on to complete the choice (valuation) tasks. After these answers had been collected, the paid task was selected and played out: a coin flip decided whether this would be a choice or valuation task, then a separate die roll determined which gamble would be paid out. The same gamble (or gamble pair, if a choice task paid out) was selected for everyone. In the event that a gamble paid out, a separate die roll for each subject determined the outcome of the selected lottery, excepting P5 and $5, which were resolved by independent draws of a chip from a bag. Up to 2 rolls of a 10 sided-die, were used, giving a range of numbers from 0 to 99, with the first roll determining the first digit. Thus, for example, for P4, if a subject rolled higher than a 4 on the first roll she lost; if she rolled lower than 4 she won. If she rolled a 4, she rolled again, with numbers 0–4 paying out. In this way, numbers 0–44 paid out and 45–99 lost. For valuation tasks, 12 sealed envelopes were picked from a box by a subject at the beginning of the experiment and assigned as they chose to numbers 1–12, which represented valuation tasks. These envelopes were stuck to a board at the front of the room. Each envelope specified a sum of money, which was to serve as an offer that could be exchanged for a lottery depending on the subject’s reserve price. In the event that a valuation task paid out, a subject drew once from tickets numbered 1–12 to determine which task, for all subjects, was chosen. The number was taken from the envelope to determine whether each subject played the gamble. If their reserve price was lower than the offer, they received the offer in pounds. If their reserve price was higher than the offer they played the gamble. 3.3. Selling versus gift manipulation In the selling treatment, instructions for the valuation tasks were couched in the standard language of prices and offers. That is, subjects were told they had been given a gamble and were asked to specify the price for which they would be willing to sell it. It was explained that an offer had been made for each gamble, contained in the sealed envelopes at the front of the room. If a particular gamble was selected to be played out, then if the offer exceeded their price they would exchange the gamble opportunity for the cash, otherwise they would ‘‘keep’’ and play the gamble. The logic underpinning this procedure is exactly the logic of the more usual Becker Degroot Marshak mechanism, but without the complicating factor of a randomising device to determine the offers. In the gift treatment, both the gamble and the amount of money in the envelope were described as gains, and the instructions avoided the language of buying, selling, offers, keeping or exchanging. Thus, subjects were asked to consider a gamble, the offer was described simply as a number and they were asked to say how much they thought the gamble was worth. In other words, the task was to specify an equivalent gain. Both outcomes were described as prizes – either a subject would receive a gamble to play or they would receive some cash, depending on whether the number in the envelope was less or greater than their valuation respectively. In the gift treatment, therefore, the reference point against which possible outcomes are judged is the same in the choice and valuation tasks, whilst in the selling treatment it differs. There, subjects in the valuation task are asked to put a selling price on something they have already been given, whilst in the choice tasks they choose to receive one of two gambles. 4. Results Results are set out in Table 4 below. We report the result of a two-sided binomial test, where the null hypothesis is that standard and nonstandard reversals are equiprobable. This allows us to distinguish between random and systematic reversals. An exposition of why this is necessary is given in Appendix.4 The proportion of responses that constitute standard preference reversals, that is, the percentage of subjects in each task choosing the P-bet but valuing the $-bet more highly, is shown in the ‘‘reversals’’ entry. This is calculated excluding equal valuations. Summary statistics on valuations are shown in Table 5 below. 5. Analysis 5.1. By task For all gamble pairs, the binomial test is highly significant except for pair e, for which the test statistic is non-significant as reversals are symmetric. In every case where the statistic is significant, standard reversals outnumber nonstandard reversals. Thus, in all cases except pair e there is a statistically significant asymmetry between behaviour in the choice and valuation tasks, which is definitive of the preference reversal phenomenon.
4
We owe this point to professor Robert Sugden (personal correspondence).
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L.J. Ball et al. / Journal of Economic Psychology 33 (2012) 48–57 Table 4 Results. Pair
a
Treatment
Selling
Gift
b Selling
Gift
c Selling
Gift
d Selling
Gift
e Selling
Gift
f Selling
Gift
Sample size Choice % Choosing P % Choosing $ Choice and Valuation Consistent, prefer P Consistent, prefer $ Standard Reversal Counter Reversal Binomial p-value Standard reversals
52
52
52
51
51
52
51
52
51
52
52
51
0.85 0.15
0.81 0.19
0.71 0.29
0.73 0.27
0.84 0.16
0.88 0.12
0.53 0.47
0.40 0.60
0.65 0.35
0.60 0.40
0.73 0.27
0.73 0.27
19 5 25 3 0 0.48
15 8 27 2 0 0.52
24 13 13 2 0.01 0.25
19 7 18 7 0.04 0.35
23 4 20 4 0 0.38
30 6 16 0 0 0.31
10 20 17 4 0.01 0.33
9 26 12 5 0.14 0.23
21 9 12 9 0.66 0.23
20 10 11 11 1 0.21
16 9 22 5 0 0.42
20 10 17 4 0.01 0.32
Notes: (i) ‘Consistent, prefer P ($)’ denotes P ($) was chosen and valued at least as highly as $ (P). (ii) Different sample sizes reflect missing data, from one subject in each treatment.
Table 5 Valuations. Selling
Mean Median Sample standard deviation
Gift
P
$
P
$
£7.43 £6.58 £2.89
£9.39 £8.38 £5.39
£7.06 £6.25 £3.20
£8.64 £7.29 £4.87
Note: Median and standard deviation are shown for subjects’ mean valuations over the 12 gambles.
By inspection of Table 4 there is no evidence of a straightforward relationship between the degree of ambiguity of the tasks and the incidence of reversals as defined by their rating in Table 2. There is a significant difference, however, between the percentage of standard reversals for the transparent gambles and across the other tasks (50% versus a mean of 31%; 2tailed Z-test, p < 0.01). This comparison uses subjects’ decisions averaged over the other gambles; on average standard preference reversals made up 31% of observations for pairs b–f. 5.2. By treatment Asymmetric preference reversals are a prominent feature of the results in both treatments, as indicated by the binomial test. The standard pattern of preference reversal results is generally reproduced in both treatments, meaning that there is a greater propensity for subjects to favour P in the choice task than in the valuation task. Differences in the proportions of standard reversals across treatments are not significant for any gamble pair (2-tailed Z-test, ps > 0.10). Differences in valuations across treatments are non-significant according to standard tests (2-tailed T-test; Mann–Whitney U test; ps > 0.10). 6. Discussion The results analysed by task show with high confidence that the preference reversal phenomenon is not an artefact of the transparency of prospects (Section 5.1); that is, hypothesis 1 is rejected. Even for gambles which were vaguely specified in both probabilities and outcomes, a significant asymmetry between choice and valuation results occurred, conforming to the classic preference reversal pattern. We found no evidence of a consistent relationship between reversals and the degree of ambiguity of prospects. Fully transparent gambles may nonetheless encourage reversals. There is a degree of support for hypothesis 2, since the fully transparent gamble pair, a, produced significantly more reversals than the average across the other pairs. This result is directionally consistent within each subsample, but only significant at the 5% level for the selling treatment (2-tailed Z-test, p = 0.01 and p = 0.11). The (limited) support for hypothesis 2 is tentative, since there is considerable variation in the frequency of reversals between gamble pairs, and it may be the case that differently-specified vague gambles could match the frequency of reversals in transparent gambles. However, it seems to us unlikely that the difference is down to the specific gamble used, since: (i) the rate of standard reversals observed for pair a was not particularly high relative to other designs; and (ii) at least one gamble pair, c, produced a similar split to pair a between the P- and $-bets in the choice task, but fewer observed standard reversals.
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Essentially the same pattern of results obtains in both selling and gift treatments (Section 5.2). This is evidence against loss aversion explanations of preference reversal, since we ought to have found a marked decrease in reversals in the gift treatment. Hypotheses 3 and 4 are therefore rejected. Given that we found null results across the selling and gift treatments, one might question whether our treatment manipulation was actually effective. However, in the present context it is important to note that we are testing the loss aversion explanation of preference reversals. The factor being manipulated is the specification of the reference point, this specification being constituted by the instructions exactly as in standard designs. These instructions are entirely at the discretion of the experimenter, and the loss aversion explanation assumes the efficacy of the instructions to implement different reference points. It may be the case that the selling treatment did not give rise to a marked differential in the perceived reference point, or experienced loss aversion, but either possibility counts against the loss aversion explanation of preference reversals. The valuations data shown in Table 5 are consistent with a small difference in loss aversion across treatments, but the observed difference is not statistically significant. Another explanation for this, assuming for the sake of argument that it is a real difference, is that subjects are applying a heuristic adapted to real life bargaining situations of asking for a higher price than one’s actual reserve price. The logic of the loss aversion account of preference reversals, as formalised in PT3, is impeccable. However, it seems to us that it may be psychologically implausible. This is because it requires that subjects view selling a gamble as risking a loss, even if its resolution is to remain unknown. That is, they are supposed to regard the abstract likelihood that they sell a gamble which would have paid off in the same way as the likelihood that they experience a departure from a concrete asset that they actually hold. This is, we suspect, especially unlikely given that the subject is only fleetingly endowed with each gamble, for the purposes of one task. An experiment could no doubt be designed which makes the endowment of a gamble more meaningful. But the loss aversion/PT3 account is supposed to explain reversals in the standard design, where it is operationalised exactly as in the experiment reported here. Our design was not optimised for systematic exploration of scale compatibility effects. However, it is worth noting that pair f produced standard reversals where scale compatibility seems not to apply. This is because of the non-monetary prize here. Cubitt et al. (2004) also find reversals where scale compatibility is not a plausible explanation. Concerning pair f, scale compatibility is certainly suppressed relative to the other pairs, but cannot be entirely excluded because we had informed subjects that each gamble is expected to cost the experimenter the same amount in the event it is selected to be played. Subjects could therefore deduce that $6 had a more expensive prize than P6 even though they were told it was not a cash prize. Another aspect of our results worth noting, given the general robustness of the preference reversal phenomenon, is the absence of systematic reversals for pair e. There, standard and nonstandard reversals became equally frequent. Thus, for pair e and pair e only, the evidence is consistent with a symmetric stochastic process of preference reversal production, whereby subjects deviate with some probability from an underlying disposition to favour either the P- or $-bet. One possibility is that subjects were simply confused and chose at random. This does not seem plausible because the observations are not uniformly distributed, as the P-bet was chosen twice as often here as the $-bet. A second possibility is that the use of frequency information underlies this result. For the other tasks, no frequency information was given and it would have been unclear to subjects how the prospects were to be resolved. However, frequency presentations are common in preference reversal studies and do not normally dispel systematic reversals. We find it likely that pair e involved more cognitive difficulty than the other gamble pairs. Pair e was designed so that subjects are in possession of good information, qualitatively speaking, but deprived of transparency. That is, it is deducible from the information provided here that, although the P-bet is far more likely to pay out, the two bets have exactly the same expected value, although that expected value is not specified. Proximity in expected value is a requirement for reversals, as we noted earlier, since one gamble should not be overwhelmingly attractive. However, the task does require subjects to reason. This may have caused the difference observed, since it may have undermined the clarity or intensity of preferences between the bets, resulting in a greater stochastic element in decision making. The reason for constructing the task in this way is that many real world decisions are opaque in a similar way; we wished to see if the results of transparent environments extend to them. 7. Conclusions Our results clearly indicate that preference reversals are not dependent on the gamble transparency, nor on the shifting reference point, which feature in almost the entire body of previous preference reversal studies. Firstly, classic preference reversal results were obtained with prospects that are ambiguous in both probabilities and consequences. Explanations of reversals which have been stated with reference to precise valuations, including anchoring and adjustment, and preference imprecision, may need to be revised to accommodate this case. We find the normal pattern of reversals even where the notion of scale compatibility seems inapplicable because a non-monetary prize is specified. We do find some evidence, however, that reversals are encouraged by transparency. Secondly, reversals were undiminished when choice and valuation tasks deployed the same reference point. This contradicts any explanation of preference reversals based on loss aversion, including the loss aversion explanation recently offered under PT3. Thus, the latest attempt to reconcile preference reversals with decision theory seems to be rejected by the lab. In
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the preference reversal context, loss aversion may be psychologically implausible because the loss concerned is a departure from a potential, rather than an experienced, monetary holding. The failure of the reference point manipulation to impact on reversals might be considered further good news for the robustness of experimental results on choice under risk. For what is arguably a rather artificial aspect of the standard design is found to have no discernible impact on reversals. A question our results raise is whether the rejection of the loss aversion account of preference reversals is a serious problem for PT3 itself. This might be explored by strengthening the reference point manipulation compared to what is done in the standard preference reversal design. Our results seem consistent with an interpretation of preference reversal offered by Cubitt et al. (2004), in terms of a conjunction of psychological effects, rather than an economic rationalisation. The data suggest that fully transparent prospects encourage reversals, which is consonant with compatibility effects, but that they also occur where scale compatibility is suppressed. A hypothesis we form for future investigation is that systematic reversals may be diminished by cognitive difficulty of decision tasks. Acknowledgments Research for this paper was supported by a Collaborative Fund grant from the ESRC National Centre for Research Methods (NCRM), grant RES-576-47-5001, and was initiated whilst Bardsley was affiliated to NCRM at the University of Southampton. We are grateful to Robert Sugden and Chris Starmer for comments received. The authors thank Caroline Wade for assistance with running the experiments reported. Ordering of authors is alphabetical. Appendix A A.1. Detecting preference reversals Preference reversals generate results of the form shown in the table below: Task
Valuation
Outcome
P>$ $>P
Choice P
$
A C
B D
The preference reversal literature sometimes cites a difference between the proportions B/(A + B) and C/(C + D) as evidence of an asymmetry in reversals. However, this is a mistake, from a stochastic point of view. Suppose that all subjects prefer the P to $, but choose $ with probability q. Then A, B, C, and D expressed as a proportion of total choices would become (1 q)2, q(1 q), q(1 q) and q2 respectively. It follows that B/(A + B) and C/(C + D) would equal q and 1 q respectively. Thus, although choices and valuations would be consistent in the sense that under either procedure P is selected with probability 1 q, the measure would report an ‘asymmetry in reversals.’ The binomial test, or an approximate version such as McNemar’s (1947) test of marginal homogeneity, is suitable to detect an imbalance in the off diagonal elements of such a matrix. A.2. Instructions Welcome to this experiment. Each of you has been given a show-up fee of £4 for attending today. In addition to this, you each have the chance to win a further prize. The details of how this will happen are as follows. The experiment consists of two parts, a ‘‘choice’’ part and a ‘‘valuation’’ part. In each case you have to judge the attractiveness of possible lotteries. At the end of the experiment a coin will be tossed to determine which part is played out for real, and one of the tasks will be chosen at random. Which prize you receive depends on which task is chosen and which decision you made in that task, plus a purely random element. The tasks in both parts involve lotteries. Each lottery refers to a chance of winning a prize. The chances and prizes involved are different in each lottery. You have to say what you would choose to do in each situation below, based on how attractive you think the lotteries are. All the lotteries would cost the experimenter a similar amount of money if everyone chose to play them. All the lotteries will be revealed at the end, after your decisions have been recorded. A.2.1. Choice part In this part of the experiment you have to choose between two lottery tickets in each task. Indicate your choice by circling letter A or B. What you decide here will affect your potential prize from the experiment if a task from the choice part is paid out at the end. (Recall that one task from the whole experiment is selected at random and played out for real.) [A practice choice task using transparent gambles was played out before subjects received their answer booklets, but not paid. The booklets presented the tasks in the following form:]
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1. Choose between A. A 90% chance of winning. The prize is £6.67, and B.A 20% chance of winning. The prize is £30.00. 2. Choose between A.A fairly good chance of winning £10, and B.A pretty low chance of winning £60. [And so on. The order of gamble pairs was counterbalanced across subjects.] A.2.2. Valuation part Selling treatment: In this part of the experiment we would like you to tell us how much you would be willing to sell a lottery ticket for. We appreciate that for some of the lotteries this may be quite difficult. However, we think that it is in your interests to take this seriously. What you decide here will affect your potential prize from the experiment if a task from the valuation part is paid out at the end. (Recall that one task from the whole experiment is selected at random and played out for real.) In each task you are given a lottery ticket. For each ticket we would like you to state a price for which you would be just willing to sell it. We’ve constructed this part of the experiment to give you a reason to do just that – though you are quite free to do what you like! The envelope which was selected (at random, by a participant) at the start of this experiment contains a number of pounds (and pence), which we will call the ‘‘offer.’’ This number, which is different for each ticket, will only be revealed after all decisions have been made. If the number you gave as the price of the ticket is less than or equal to the offer number, you will be given that offer in pounds if this task is paid out. Otherwise you will keep the lottery ticket, and that will determine your earnings by a play of the lottery. The higher the price you specify, the more likely it is that you will keep the lottery ticket. The lower the price, the more likely it is that you get the money instead. Suppose you think the ticket is worth £x. If you say that your price is more than £x, then you risk that the offer is higher than £x, but less than your price. In that case you will keep the lottery ticket, whereas you could have received more than £x by stating the price as £x. If, alternatively, you say that your price is less than £x, you risk that the offer is less than £x but more than your price. In that case you will receive a sum of money which you think is inferior to the ticket, whereas you could have kept the ticket by stating a price of £x. (Verbal) Write down any number, large or small. Imagine someone who owns an antique. He’s not sure what the market value is but probably it’s not worth much. But there’s a small chance it’s worth a lot. He decides, taking this into account, that it’s worth £30 to him. Now imagine he’s playing the same game described in the instructions, with your number as the offer. Suppose he said £1000 as the price. What could go wrong? . . . (answer elicited) Suppose he said £100 as the price. What could go wrong? . . . (answer elicited) It’s the same problem for any price higher than £30. Now suppose he said £2 as the price. What could go wrong? . . . (answer elicited) Suppose he said £20 as the price. What could go wrong? . . . (answer elicited) It’s the same problem for any price lower than £30. This shows that the best thing you can do is state your true price. Gift treatment: In this part of the experiment we would like you to tell us how valuable a lottery ticket is to you. We appreciate that for some of the lotteries this may be quite difficult. However, we think that it is in your interests to take this seriously. What you decide here will affect your potential prize from the experiment if a task from the valuation part is paid out at the end. (Recall that one task from the whole experiment is selected at random and played out for real.) In each task you are required to consider a lottery ticket. For each ticket we would like you to state a sum of money which you think is as good as it but no better. We’ve constructed this part of the experiment to give you a reason to do just that – though you are quite free to do what you like! The envelope which was selected (at random, by a participant) at the start of this experiment contains a number of pounds (and pence), which we will call the ‘‘deciding number.’’ This number, which is different for each ticket, will only be revealed after all decisions have been made. If the number you gave as the value of the ticket is less than or equal to the deciding number, you will be given that number in pounds if this task is paid out. Otherwise you will be given the lottery ticket, and that will determine your earnings by a play of the lottery. The higher the value you specify, the more likely it is that you will get the lottery ticket. The lower the value, the more likely it is that you get the money instead. Suppose you think the ticket is worth £x. If you say that the value is more than £x, then you risk that the deciding number is higher than £x, but less than your stated value. In that case you will be given the lottery ticket, whereas you could have received more than £x by stating the value as £x. If, alternatively, you say that the value is less than £x, you risk that the
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deciding number is less than £x but more than your stated value. In that case you will receive a sum of money which you think is inferior to the ticket, whereas you could have received the ticket by stating a value of £x. (Verbal) Write down any number, large or small. Imagine someone who owns an antique. He’s not sure what the market value is but probably it’s not worth much. But there’s a small chance it’s worth a lot. He decides, taking this into account, that it’s worth £30 to him. Now imagine he’s playing the same game described in the instructions, with your number as the deciding number. Suppose he said £1000 as the value. What could go wrong? . . . (answer elicited) Suppose he said £100 as the value. What could go wrong? . . . (answer elicited) It’s the same problem for any value higher than £30. Now suppose he said £2 as the value. What could go wrong? . . . (answer elicited) Suppose he said £20 as the value. What could go wrong? . . . (answer elicited) It’s the same problem for any value lower than £30. This shows that the best thing you can do is state your true value. [After the valuation instructions, subjects valued two practice lotteries with transparent prospects, which were not paid out. They then valued the 12 lotteries, the order being counterbalanced across subjects.] References Bardsley, N., Cubitt, R., Loomes, G., Moffatt, P., Starmer, C., & Sugden, R. (2010). Experimental economics: Rethinking the rules. Princeton, NJ: Princeton University Press. Blavatskyy, P. (2009). Preference reversals and probabilistic decisions. Journal of Risk and Uncertainty, 39, 237–250. Bohm, P. (1994). Behaviour under uncertainty without preference reversal: A field experiment. Empirical Economics, 19, 185–200. Bohm, P., & Lind, H. (1993). Preference reversal, real-world lotteries and lottery-interested subjects. Journal of Economic Behavior and Organization, 22, 327–348. Butler, D. J., & Loomes, G. (2007). Imprecision as an account of the preference reversal phenomenon. American Economic Review, 97, 277–297. Cox, J. C. (2008). Preference reversals. In C. R. Plott & V. L. Smith (Eds.). Handbook of experimental economics results (Vol. 1). New York: Elsevier Press. Cubitt, R. P., Munro, A., & Starmer, C. (2004). Testing explanations of preference reversal. Economic Journal, 114, 709–726. Fischer, G. W., & Hawkins, S. A. (1993). Strategy compatibility, scale compatibility, and the prominence effect. Journal of Experimental Psychology: Human Perception and Performance, 19, 580–597. Grether, D., & Plott, C. (1979). Economic theory of choice and the preference reversal phenomenon. American Economic Review, 69, 623–638. Hausman, D. (1992). The inexact and separate science of economics. Cambridge and New York: Cambridge University Press. Lichtenstein, S., & Slovic, P. (1971). Reversals of preference between bids and choices in gambling situations. Journal of Experimental Psychology, 89, 46–55. Lichtenstein, S., & Slovic, P. (1973). Response-induced reversals of preference in gambling: An extended replication in Las Vegas. Journal of Experimental Psychology, 101, 16–20. Maafi, A. (2011). Preference Reversals under Ambiguity. Management Science, in press. McNemar, Q. (1947). Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika, 12, 153–157. Schkade, D., & Johnson, E. J. (1989). Cognitive processes in preference reversals. Organizational Behavior and Human Decision Processes, 44, 203–231. Schmidt, U., Starmer, C., & Sugden, R. (2008). Third generation prospect theory. Journal of Risk and Uncertainty, 36, 203–223. Schram, A. (2005). Artificiality: The tension between internal and external validity in experimental economics. Journal of Economic Methodology, 12, 225–238. Seidl, C. (2002). Preference reversal. Journal of Economic Surveys, 16, 621–655. Sugden, R. (2003). Reference dependent subjective expected utility theory. Journal of Economic Theory, 111, 172–191. Trautmann, S. T., Vieider, F. M., & Wakker, P. P. (2011). Preference reversals for ambiguity aversion. Management Science, 57, 1320–1333. Tversky, A., Sattath, S., & Slovic, P. (1988). Contingent weighting in judgement and choice. Psychological Review, 95, 371–384.