Seidl-2002-Journal of Economic Surveys

PREFERENCE REVERSAL

Christian Seidl

Christian-Albrechts-Universitat zu Kiel

Abstract. Preference reversal concerns a systematic divergence between prices oflotteries and subjects’ expressed preferences for playing the respective lotteries.This article surveys the discovery of preference reversal by psychologists, its re-examination and corroboration both by psychologists and later on by (firstsceptical) economists, as well as the causes of preference reversal. The preferencereversal phenomenon has been explained to be caused by four determinants, viz.by the mode of elicitation of certainty equivalents, by intransitivity of preferences,by overpricing and=or underpricing of lotteries, and by nonlinear probabilities.JEL classification: D81

Keywords. Preference reversal; decision theory; procedure invariance; descriptioninvariance.

1. Introduction

Subjects’ attitudes towards lotteries may be elicited in various ways. They may,for instance, be directly asked for their preferences to play various lotteries, ortheir choice of playing a lottery taken from several equally available lotteries maybe observed. Alternatively, subjects may be asked to state prices for lotteries orexpress their judgments of lotteries. Rationality considerations would demand thatthe results of different elicitation methods of preferences yield the same results.This is a consequence of procedure invariance: ‘Procedure invariance demands thatstrategically equivalent methods of elicitation will give rise to the same preferenceorder’. [Tversky, 1996, p. 185. Cf. also Tversky et al., 1988, p. 371; Tversky et al.,1990, p. 204; Johnson and Schkade, 1989, p. 406.]

Yet there is stunning empirical evidence of systematic divergence of subjects’attitudes towards lotteries derived in alternative ways. This phenomenon is calledpreference reversal. It is one of the most spectacular violations of procedureinvariance. It was first studied by psychologists and later on also by economists.Whereas psychologists have focused on juxtapositions of subjects’ judgments andchoices, economists were more concerned with comparisons of subjects’ prices andpreferences [Goldstein and Einhorn, 1987, p. 237].

2. The discovery of preference reversal

The preference reversal phenomenon is an empirical regularity such that thereexists a robust experimental design of lotteries for which substantial fractions of

0950-0804/02/05 0621–35 JOURNAL OF ECONOMIC SURVEYS Vol. 16, No. 5# Blackwell Publishers Ltd. 2002, 108 Cowley Rd., Oxford OX4 1JF, UK and 350 Main St., Malden,MA 02148, USA.

subjects state prices or judgments which are opposite to the preferences expressedfor or the choices made out of the respective lotteries. [Unless stated otherwise, weshall henceforth abridge judgments and prices to ‘prices’, and choices andpreferences to ‘preferences’.]

More precisely, let (x; p; y) and (�; q; w) be lotteries where p and q denote theprobability of winning the payoff x or �, respectively, and (1� p) and (1� q)denote the probability of winning the payoff y or w, respectively. Let � (�) denotethe price of a lottery and u a preference relation. Then preference reversalprevails if

(x; p; y) u (�; q; w) and � (�; q; w) > � (x; p; y): (2:1)

The preference reversal phenomenon seems to have been discovered byLindman1 and was later noticed by Slovic and Lichtenstein (1968) in a studywhich was mainly directed at investigating the influence of probabilities andpayoffs on risk taking. Conceiving of lotteries as multidimensional stimuli [seealso Payne, 1973] and working with prices, they perceived that ratings of lotteries[in the sense of asking subjects to indicate their strength of preference for playing alottery on a bipolar rating scale extending from �5 to þ5] are more highlycorrelated with the probabilities involved, whereas prices are more highlycorrelated with the payoffs of the lotteries [Slovic and Lichtenstein, 1968,pp. 10–1; in particular Table 6 on p. 11]. Working further along these lines,Lichtenstein and Slovic (1971) were able to present formidable evidence ofpreference reversals. Administering three experiments to 173, 74, and 14 subjects,respectively, involving scores of comparisons of lotteries, they detected the basicstructure of the critical experimental design which elicits preference reversals. Thecritical pair of lotteries involves ‘a ‘P-bet’, i.e., a bet with a high probability ofwinning a modest amount and a low probability of losing an even more modestamount, and a ‘$-bet’, i.e. a bet with a modest probability of winning a largeamount and a large probability of losing a modest amount’. [Lichtenstein andSlovic, 1971, p. 47.]

The subjects were asked to rank the lotteries of the presented pairs, and, at alater stage, to indicate their minimum selling price (Experiment I), their highestbid price (Experiment II), or their minimum selling price controlled, however, bya buying price as a counter-offer of the experimenter (Experiment III). ForExperiment III, the counter-offer meant nothing else than determining subjects’selling prices according to the Becker-DeGroot-Marschak (henceforth BDM)elicitation scheme.2 In Experiment I, payoffs were hypothetical, in ExperimentII, subjects were paid by the hour, and in Experiment III, the lotteries wereactually played, and subjects were paid their winnings by way of convertingtokens into money. The stimulus of all three experiments was six lottery pairs,three of which had equal expected values, and three had roughly equal expectedvalues.

Table 1 contains the proportions of observed conditional preference reversalstaken over all six lottery pairs. These are the proportions of times that the relationof the prices of the lotteries contradict the stated preferences. In this figure,

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# Blackwell Publishers Ltd. 2002

‘� ($) > � (P ) given P u $’ refers to predicted conditional preference reversals,‘� (P) > � ($) given $ u P’ refers to unpredicted conditional preference reversals.

Recall that all entries in Table 1 ought to be zero if procedure invarianceobtained. However, Table 1 shows severe violations of procedure invariance.More than half of the subjects who prefer lottery P to lottery $ indicate a higherprice of lottery $ than of lottery P. Among the three experiments, predictedpreference reversals are highest for Experiment I,3 markedly lower for ExperimentIII and lowest for Experiment II, which shows that monetary payoffs and morescrutiny in the conduct of experiments reduce predicted preference reversals. Thistendency is reinforced if subjects’ bid prices instead of their selling prices are takenas their lottery values. Yet even in this case predicted conditional preferencereversals amount to more than 50%. Unpredicted preference reversals developcontrary to predicted preference reversals for these three experiments, althoughtheir proportion remains distinctly below the proportion of predicted preferencereversals. Lichtenstein and Slovic (1971, pp. 52–3) try to explain subjects’behaviour using an error model, but found predicted preference reversals to be arather robust and statistically significant phenomenon provided the basicstructure of the lotteries to be compared (a P-bet juxtaposed to a $-bet) ismaintained.

3. Re-examinations of preference reversal

The preference reversal phenomenon has perplexed first the psychologyprofession and later on also the economics profession. Lichtenstein and Slovic’sexperiments have been amply repeated.4

Lindman (1971), working at the same time, but independently of Lichtensteinand Slovic, carried out five experiments. In Experiment I, choice took placeamong triples of lotteries. For Experiments II–V, subjects were asked to choosebetween pairs of lotteries. All experimental stimuli (indication of prices and lotterychoices) were repeated three times for Experiments I–IV, and six times forExperiment V. Lindman (1971, p. 396) observed similar results as Lichtensteinand Slovic:

In general, choices–whether among pairs or triples of gambles–tend to revealpreferences for gambles in which the more favorable outcome is most likely.Conversely, selling prices tend to reveal a preference for the gamble for whichthe more favorable outcome has the largest value.

Table 1. Proportions of Conditional Preference Reversals [own computations fromLichtenstein and Slovic 1971, p. 52, Table 4].

Cond. Pref. Reversal Experiment I Experiment II Experiment III

�($) > � (P ), given P u $ 83.33% 50.94% 56.22%� (P ) > �($), given $ u P 6.33% 27.14% 11.19%

PREFERENCE REVERSAL 623


Lichtenstein and Slovic (1973) replicated their former experiment with 44 players inthe balcony of the Four Queen Casino in Las Vegas. Subjects played for chips whichthey had to buy for their own money (and which were later redeemed in cash for theirface value). They chose between 10 stimulus lottery pairs (consisting of a P-bet and a$-bet each) with positive expected value (the ‘good’ lotteries), and 10 stimulus lotterypairs (also consisting of a P-bet and a $-bet each) with negative expected value (the‘bad’ lotteries). Each lottery consisted of a gain and a loss. ‘The highest net win was$83.50, the largest loss was $82.75, the mean outcome was �$2.36, and theinterquartile range was �$8.40 to þ$5.50’. [Lichtenstein and Slovic, 1973, p. 18.]

The results of the Las Vegas experiment confirmed the earlier observations. Forthe ‘good’ lotteries the proportions of conditional preference reversals were [owncalculations from Lichtenstein and Slovic, 1973, p. 19, Table 2]:

Predicted [� ($) > � (P ); given P u $]: 80:79%;

Unpredicted [� (P ) > � ($); given $ u P ]: 9:80%:

For the ‘bad’ lotteries the proportions of conditional preference reversals were[notice that theory predicts now the opposite pattern],

Predicted [� (P ) > � ($); given $ u P ]: 75:89%;

Unpredicted [� ($) > � (P ); given P u $]: 19:49%:

The pattern of preference reversals for lotteries with negative expected value (the‘bad’ lotteries) was some two decades later confirmed for the pure loss domain ofpayoffs by work of MacDonald et al. (1992).

A related phenomenon was observed by Zagorski (1975). Paying his subjects$1.50 per hour plus winnings, he asked them for their selling prices (using a BDMelicitation scheme) they would demand to exchange the better one of pairs oflotteries taken from the set ($8, 0.8, 0), ($5, 0.8, 0), ($8, 0.5, 0), and ($5, 0.5, 0).Denote this price by �(�) and 2 to indicate ‘move to’. He found that5

�[($8; 0:8; 0) 2 ($5; 0:8; 0)]þ �[($5; 0:8; 0) 2 ($5; 0:5; 0)]

> �[($8; 0:8; 0) 2 ($8; 0:5; 0)]þ �[($8; 0:5; 0) 2 ($5; 0:5; 0)]; (3:1)

which means that the sum of the selling prices for exchanging ($8, 0.8, 0) for($ 5, 0.5, 0) was greater if the first pair of lotteries involved only a differencein payoffs, maintaining the probabilities. Thus, Zagorsky (1975) demonstrateda phenomenon closely related to preference reversal, viz. a systematic pathdependence of selling prices of exchanging better for worse lotteries.

Hamm (1979) used a rich experimental design to examine the preferencereversal phenomenon. In addition to selling prices, he included also bid prices asprices for lotteries, had ample discussions with his subjects on decision strategies,and studied the stability of reversals over repeated experiments. He found a robustincidence of preference reversals.

By the end of the 1970’s, economists took cognizance of preference reversals.Grether and Plott (1979, p. 634), as they frankly admitted, set out to refute whatthey considered psychologists’ humbug, but had to give in eventually:

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Needless to say the results we obtained were not those expected when weinitiated this study. Our design controlled for all the economic-theoreticexplanations of the phenomenon which we could find. The preference reversalphenomenon which is inconsistent with the traditional statement ofpreference theory remains.

The glimpse of an intriguing psychological phenomenon perplexed theeconomics profession so much that the American Economic Review readily servedas a showcase for some follow-up studies. The work of Reilly (1982) and ofPommerehne et al. (1982) by and large confirmed the earlier results. Pommerehneet al. (1982) found that the frequency of preference reversals decreases for moresizable payoffs.6

Mowen and Gentry (1980) presented the preference-reversal experimentaldesign to their 97 subjects in the garb of the market introduction of new products.First, they observed a confirmation of the preference reversal phenomenon in thissetting, too. Subjects tend to introduce the product with the higher probability ofsuccess. However, in the pricing task, they are influenced by the higher possibleprofit. Second, Mowen and Gentry (1980, p. 721), who provided also for groupdecision making in their experimental design, observed ‘a general tendency forgroups to reveal the preference-reversal phenomenon more frequently thanindividuals’.

Berg et al. (1985) investigated the preference reversal phenomenon using afourfold experimental design. First, they used two elicitation methods for lotteryprices, viz. the BDM elicitation scheme, and the O’Brien elicitation scheme. Theyargue against the BDM elicitation scheme that it only reveals a subject’s minimumselling price unless the subject’s utility function demonstrates constant absoluterisk aversion in the Pratt (1964)7 sense.8 Only for constant absolute risk aversiondoes a subject’s selling price equal his or her bid price. In order to achieve that ingeneral, they also used a procedure developed by O’Brien. Under this procedure, asubject must state a single price for his or her lottery at which he or she is preparedto enter into an exchange transaction as either a buyer or a seller. Then an offerprice is drawn from a probability distribution. If this offer price is greater or equalto the subject’s quoted price, the subject must exchange his or her lottery for theoffer price. If the offer price is less than the subject’s quoted price, then he or shemust purchase an additional lottery of the same sort. This procedure shouldwarrant equality of a subject’s selling and bid price of a lottery.

Second, they provided for two roles of the experimenter, either to act as aneutral instance, or to exercise arbitrage in the sense of buying lotteries from andselling lotteries to the subjects at their stated prices such as to extract profits fromthese transactions.

Surprisingly, these experimental variations had hardly any effect on the results.The elicitation scheme for lottery prices produced the same fraction of preferencereversals (31.5% for the BDM scheme and 31.83% for the O’Brien scheme), andthe incidence of arbitrage even increased the occurrence of preference reversals(from 29.67% to 33.5%), although the increase was not statistically significant.9



However, the presence of arbitration exerted strong influence on the lottery pricesas indicated by the subjects. They were much more cautious in the arbitrage mode.The mean Dollar magnitude of reversals per subject decreased from $4.10 to $2.52for the arbitrage mode [Berg et al., 1985, pp. 43–4].

Other studies examined the preference reversal phenomenon using some slightvariations of experimental design. Bohm and Lind (1993) used real-world Swedishlottery as stimuli in their experiment. They observed reduced incidence ofpreference reversals: The rate of predicted preference reversals dropped to 23% ascompared to 73% in a control experiment. Knez and Smith (1987) and Cox andGrether (1991) observed reductions in preference reversals in successive rounds ofexperiments, for instance when simulating market contexts. Cox and Epstein(1989) found that preference reversals were not reduced even for large differencesin the expected values of P-bets and $-bets. However, Cox and Epstein (1989,p. 422) observed only about half of the preference reversals to be of the predictedtype, the other half being unpredicted preference reversals (i.e. subjects stated apreference for the $-bet, but a higher selling price for the P-bet).

An even stranger result was observed by Casey (1991). He combined bid priceswith high (but only hypothetical) positive payoffs of the lotteries and observed anew preference reversal pattern: The $-bet was preferred, and the bid price of theP-bet exceeded the bid price of the $-bet. He found the following proportions ofconditional preference reversals [own calculations from Casey, 1991, p. 236,Table 5]: New preference reversals [� (P ) > �($), given $ u P]: 70.93%; traditionalpreference reversals [�($)> � (P ), given P u $]: 20.58%. For small payoffs, Casey(1991, p. 240, Table 5) evidenced the traditional pattern of preference reversals.Casey (1991, p. 234) made also a significant step further. Using an experimentaldesign akin to that developed by Lopes (1987, p. 265; 1990, p. 282; 1993, pp. 30and 41; 1995, p. 37), he generalized the P-bet and the $-bet to multi-outcomelotteries. Basically, he devised P-bets as negatively skewed distributions and $-betsas positively skewed distributions. However, as Casey (191, p. 237) did notobserve major differences in results for the traditional binary lotteries of thepreference-reversal experimental design and the generalized multi-outcomelotteries in his Experiment I, he might have lost interest in the generalizedexperimental design, as he did not resume working with multi-outcome lotteries inhis Experiment II.

However, the challenge to explore also other shapes of multi-outcome lotteriesfor preference reversals remains an unexplored field of future research. Other payoffdistributions within the Lopes framework, in particular bimodal and symmetricunimodal distributions, may emerge as promising candidates to examine effects ofpreference reversal [see recent work by Camacho-Cuena et al., 2002].

Although both psychologists and economists investigated the preferencereversal phenomenon, Goldstein and Einhorn (1987, p. 237) noticed that theiranalytical approaches differed: ‘ ... psychologists have tended to view reversals asa discrepancy between judgment and choice’, whereas economists perceivedpreference reversals as a discrepancy between price and preference. Combiningjudgment and choice with price and preference gives four combinations of

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response mode (judgment and choice) and worth scale (price and preference).Goldstein and Einhorn (1987, pp. 238–9) use this pattern to define six (3� 4

2 ) typesof preference reversal. Their experiments basically confirmed the incidence ofpreference reversals. Moreover, they used their framework to develop expressiontheory to explain the occurrence of preference reversals. This will be dealt with inSection 4.3 below.

The methodological specialty of the Goldstein and Einhorn paper was,however, their resumption of the Slovic and Lichtenstein (1968) method ofratings to express the evaluation of lotteries. In rating exercises subjects are askedto mark attractiveness of a choice alternative (a lottery or a pattern of delayedpayments) on a point scale, for instance, reaching from 0 to 20, [Goldstein andEinhorn, 1987, p. 239; Tversky et al., 1990, p. 213], or from 0 to 100 [Schkade andJohnson, 1989, p. 219]. A related method to elicit attractiveness of a lottery ismatching. It can be used to analyze preference reversals in the form of probabilitymatching or payoff matching.10 Consider, for example, a P-bet ($10, 0.9, 0) and a$-bet ($120, 0.08, 0). Replace now the probability 0.08 in the $-bet by a questionmark and ask the subject for the probability p which would render him or herindifferent between both lotteries. If p > 0:08, we may well infer that the subjectprefers the P-bet to the $-bet; if p < 0:08, we infer that the subject prefers the $-betto the P-bet. Payoff matching works analogously by way of payoffs.

Using ratings in place of choice as proxies for preferences among lotteries stillincreases the incidence of preference reversals.11 Concerning the matchingexercise, probability matching provides roughly the same incidence of preferencereversals as choice data. Payoff matching reduces the incidence of preferencereversals considerably without, however, eliminating it altogether; it is stillmanifest. Considerably higher payoffs of the lotteries involved have onlynegligible effects upon this pattern [Slovic et al., 1990, p. 21, Table 1.5].

This finding sheds again light on the role of financial incentives for thepreference reversal phenomenon. Following Smith’s (1982, p. 934) precept ofreward dominance, i.e. that the reward structure in an experiment has to dominateany subjective costs associated with participation in an experiment, Harrison(1989, 1990, 1992, 1994) has ever blamed experimentalists of having violatedreward dominance. Harrison (1992, p. 1430) explicitly mentioned also experi-mental work on preference reversal of having violated this precept.12 Reviewing 74experiments, Camerer and Hogarth (1999), on the other hand, did not reportoverwhelming effects of financial incentives. After all, the role of financialincentives on the preference reversal phenomenon remains ambiguous. Financialincentives seem to somewhat weaken the incidence of preference reversals,although this can be stated only very vaguely.

In Experiment II of their pioneering study, Lichtenstein and Slovic (1971)observed markedly less preference reversals for using financial incentives [cf.Table 1 above]. However, when Las Vegas subjects played with their own money,Lichtenstein and Slovic (1973) observed rather high rates of preference reversal.Pommerehne et al. (1982) and Casey (1991) observed different behaviour for highpayoffs, alas, their payoffs were only hypothetical and, thus, did not satisfy



reward dominance. When working with claims to real-world Swedish lotteries,Bohm and Lind (1993) observed a drop in predicted preference reversals to 23%.

For a somewhat different task [reception of a smaller payment in three months or ahigher payment in fifteen months], Bohm (1994a) observed a lower rate of preferencereversal for financial rewards and bid prices for lotteries. However, subjects who hadchosen the higher and later payment exhibited still less preference reversals for thehypothetical-rewards cases [Bohm, 1994a, pp. 1374–5, Tables 1 and 2].

Wilcox (1993) found clear evidence that high financial rewards induce subjectsto use more time for problem analysis, but this had no effect on lottery choices forthe simple-treatment cases. Only for the complex-treatment cases did he observe atendency to choose the lottery with the higher expected value more often for thehigh-reward designs.

Thus, it seems that the role of financial incentives for the reduction ofpreference reversals is tenuous. Other components of the experimental design,such as selling prices or bid prices, the mode of lottery price elicitation, etc., maywell be more influential than financial incentives.

We close this section by noting that the preference reversal phenomenon hasnot stopped at lotteries. It is, as Tversky and Thaler (1990, p. 208) have remarked,‘an example of a general pattern, rather than a peculiar characteristic of choicebetween bets’. A more general framework of preference reversal is no longerconfined to procedure invariance, but encompasses also violations of descriptioninvariance, which means that different presentations of the very same problemcause subjects to exhibit different preferences.

For instance, phenomena of preference reversal have also been observed for timepreferences,13 where they described a discount reversal effect. As discount reversal isformally equivalent to nonlinear preferences, we shall dwell on it in Section 4.4 below.

Preference reversals occur also with respect to perceived fairness. Subjects tendto accept jobs with fair, but lower payment, although they appreciate the higher,but unfairly paid job more. Fair payment means in this context equal pay forequal effort and equal skill. [Cf., e.g. Tversky and Griffin (1991); Blount andBazerman (1996); Clark and Oswald (1996)].

With respect to equitable taxation, Seidl and Traub (2001, p. 260–1) observedthat between one and two thirds of subjects who complained that the goingincome tax is either ‘far too high’ or ‘too high’ actually state equitable taxes whichare as high or higher than the prevailing taxes. Moreover, 82.6% of those subjectswho perceive a proportional income tax as the most equitable tax scale, actuallystate equitable numerical taxes which form a progressive tax scale.

Another instance of preference reversal, which has been observed in marketingcontexts, is the asymmetric dominance effect. Suppose two brands of a certain goodare on the market whose characteristics do not dominate each other, which meansthat one brand is superior with respect to some characteristics, the other withrespect to other characteristics. When now another brand, possibly a decoy, whichis dominated by the first, but not by the second brand, is introduced into themarket, this increases the sales of the first brand. [Cf. Huber et al., 1982; Huber andPuto, 1983; Wedell, 1991; Simonson and Tversky, 1992; Simonson, 1993.]

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Bohm (1994a) conducted a preference-reversal experiment for two used carsoffered to students of the University of Stockholm. Subjects were told that thecars could be acquired in two ways, either as prizes of lotteries (with a two thirdschance of winning the preferred car), or by way of a second price sealed-bidauction (cf. Vickrey, 1961). They were asked both for their preferences and theirprices for these cars. For 26 subjects, Bohm observed only four weak preferencereversals [i.e. preference for one specific make while indicating the same bid pricesfor both cars]. Unfortunately, the gear box of the Opel car broke down when testdriven, which seems to have directed subjects’ preferences toward the Volvo car(only 5 out of 26 subjects indicated preferences for the Opel car). Therefore, it isnot clear how much this mishap contributed to the absence of strict preferencereversals. However, given experience with other preference reversals, two morefactors may have contributed to this result, to wit, the use of bid prices instead ofselling prices, and the high valuation of the objects at stake, which inducessubjects to very careful pondering.

In a recent comprehensive experimental study, Camacho-Cuena et al. (2002)have shown that the preference reversal phenomenon is still more prevalent forincome distributions than for lotteries.

These instances afford but a short glimpse of the plethora of preference reversalphenomena. Many of them would ask for a separate survey. Therefore, I confinemyself to just mentioning that preference reversal phenomena extend to muchwider fields than can be covered in this survey. I had to concentrate on the purepreference reversal phenomenon which deals with lottery preferences which areinconsistent with the relation of the respective lottery prices.

4. The causes of preference reversal

Of course, scholars have brooded over explanations of the preference reversalphenomenon. Several hypotheses have been brought forward. They can begrouped according to whether they locate preference reversal as being caused bythe elicitation mode of certainty equivalents, by intransitive preferences, bytheories explaining overpricing the $-bet and=or underpricing the P-bet, or bynonlinear probabilities.

4.1. Preference reversal caused by the elicitation mode

The first application of incentive-compatible evaluation of outcomes seems to bedue to Becker et al. (1964, p. 227) [BDM], who determined a von Neumann-Morgenstern utility function by means of a sequential experimental method usingcertainty equivalents estimated as selling prices of lotteries. Thus, an incentive-compatible derivation of a von Neumann-Morgenstern utility function requiresan incentive-compatible evaluation of certainty equivalents. To accomplish that,Becker et al. (1964, p. 228) proposed an ingenious procedure: Subjects are askedfor a minimum selling price for a binary lottery, and then a ‘buying price’ israndomly chosen. If it exceeds the stated selling price, then the subject gets the



buying price; if not, then the respective lottery is played out. It is easy to show that— at least if expected utility is the subject’s correct decision model14 — it neverpays for the subject to cheat with respect to the revelation of his or her truecertainty equivalent:15 If the stated certainty equivalent exceeds the true one, thenthe subject may be forced to play out the lottery, whereas he or she would havepreferred the buying price; if the subject understates his or her certaintyequivalent, then the subject may be forced to accept the buying price, whereas heor she would have preferred to play out the lottery. Both sources of loss can onlybe avoided by stating the true certainty equivalent.

As the BDM elicitation scheme must hold under expected utility, its violationmeans a violation of expected utility. Several authors have blamed the violation ofspecific axioms of expected utility for preference reversals and thus for the failureof the BDM elicitation scheme.

In order to avoid wealth effects, Becker et al. (1964, p. 229) had provided foronly one lottery to be paid to the subjects. At the end of the experiment, onelottery should be drawn at random and the subject should receive his or her payoffaccording to the BDM elicitation scheme with respect to the drawn lottery only.This procedure has later been called the random-lottery incentive system or therandom-lottery procedure [Starmer and Sugden, 1991].

Holt (1986) has argued that a subject’s decision problem under this procedure ischanged to a choice between the compound lotteries

(P; 13; Z; 2

3); and (Q; 13; Z; 2

3); (4:1)

where Z: ( ~PP; 12;

~QQ; 12), and

~PP is a lottery with P as a payoff if the selling price isgreater or equal to the buying price and the buying price if it exceeds the sellingprice (with equal probability of the buying price over the respective support). Thesame applies to ~QQ.

Suppose P u Q, i.e. the subject prefers P to Q. Then the left hand side in (4.1) isalways preferred to the right hand side iff the independence axiom holds [Holt,1986, p. 511]. If the independence axiom is violated, then instances are possiblesuch that P u Q and (P; 1

3; Z; 23) e (Q; 1

3; Z; 23). But, as the certainty equivalents,

and thus the prices of the lotteries, under the BDM elicitation scheme are takenfrom the lotteries in (4.1), they contradict the true preference P u Q in theseinstances; this is the preference-reversal phenomenon. Moreover, they aredifferent from the certainty equivalents of P and Q, respectively; this demonstratesviolation of the BDM elicitation scheme. Therefore, it cannot provide the truecertainty equivalents of P and Q, and the experimenter gets wrong results.

Karni and Safra (1987) argue also using the independence axiom. Suppose � (P )is a subject’s true certainty equivalent of the lottery P, and ~�� 6¼ � (P ) for somefalsely announced certainty equivalent. Let R(~��) denote the lottery for the buyingprices which exceed ~��, and let q(~��) denote the probability that the buying priceexceeds ~��. Then a subject compares

[P; (1� q(~��)); R(~��); q(~��)] and [~��; (1� q(~��));R(~��); q(~��)]: (4:2)

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If the subject maximizes expected utility and if the independence axiom holds,then his or her optimum strategy is to announce ~�� ¼ � (P ), because otherwise thesubject may end up with the worse lottery in (4.2), which is assigned to him or herunder the BDM elicitation scheme. However, if the independence axiom isviolated, there is no reason for the subject to announce ~�� ¼ � (P ). This may thenalso give rise to preference reversals.

Violation of the independence axiom is not the only cause of failure of theBDM elicitation scheme and thus for preference reversal. Using a numericalexample, Segal (1988) showed that failure of the BDM elicitation scheme — and afortiori preference reversal — may also occur as a consequence of violations ofthe reduction-of-compound-lotteries axiom [reduction axiom for short], even if theindependence axiom, transitivity, continuity of the utility function, and stochasticdominance are observed. To see this, suppose that the subject inserts his or hertrue certainty equivalent � (P ) in both lotteries in (4.2). Then a rational subjectought to be indifferent between these two lotteries. However, if the reductionaxiom is violated, then substitution of a lottery by its certainty equivalent mayviolate indifference of the master lotteries in (4.2) and may produce preferencereversals.

Building on Karni and Safra’s (1987) assumption that the BDM elicitationscheme can be interpreted as two-stage lotteries and that the independenceaxiom is violated while the reduction axiom holds, Safra et al. (1990) developedeight testable hypotheses (which they called ‘propositions’) resulting fromthis assumption. Keller et al. (1993) investigated the empirical validityof Proposition 2, viz. that a subject’s optimal selling price and certaintyequivalent of a lottery are always on the same side of its expected value. Theyran an experiment with 149 subjects asking them for their certainty equivalentsin a direct way, and for their selling prices using the BDM elicitation scheme.They observed that for 30.3% of all lottery comparisons the certaintyequivalents and the selling prices were strictly on opposite sides of theexpected value, which contradicts the Karni and Safra (1987) approach atexplaining violations of the BDM elicitation scheme. As this pattern isconsistent with the Segal (1988) assumption that the independence axiomholds,16 while the reduction axiom is violated, they consider the Segal (1988)approach to be the better explanation for violations of the BDM elicitationscheme. Moreover, Keller et al. (1993) show that alternatives to expected utilitytheory, in particular anticipated utility, weighted utility, and quadraticutility,17 can be modelled as violations of the reduction axiom and are thusconsistent with the Segal (1988) approach.

However, the paper by Keller et al. (1993) does not contain an experimental testof the reduction axiom. They just infer that, as it is not violations of theindependence axiom which cause the failure of the BDM elicitation scheme, thevillain in the piece must be the reduction axiom.

A convincing direct test of the reduction axiom was accomplished by Starmerand Sugden (1991). They divided their 160 subjects into four groups of 40 subjects



each, and presented them two pairs of lotteries:

R 0 ¼ (£10; 0:2; £7; 0:75; 0; 0:05) and S 0 ¼ (£7; 1:0); (4:3)

R 00 ¼ (£10; 0:2; 0; 0:8) and S 00 ¼ (£7; 0:25; 0; 0:75): (4:4)

Group A was told that only the pair (4.4) was finally played out for real. Group Dwas informed that only the pair (4.3) was played out for real. Groups B and Cwere told that both (4.3) and (4.4) would be chosen with probability of 1

2 each.18

Notice that, if the reduction axiom holds, the latter case reduces both for (R 0; S 00)and (S 0; R 00) to the lottery

(£10; 0:1; £7; 0:5; 0; 0:4): (4:5)

If the reduction axiom holds, the choices (R 0; S 00) and (S 0; R 00) should have thesame expected frequency.

Yet from all 71 subjects, who chose either (R 0; S 00) or (S 0; R 00), 57, that is 80%,opted for (S 0; R 00). Among the B and C groups from all 38 subjects, who choseeither (R 0; S 00) or (S 0; R 00), 26, that is 68%, opted for (S 0; R 00). This documents arobust violation of the reduction principle.

Starmer and Sugden’s choice of the lotteries (4.3) and (4.4) is not fortuitous, asthey are in the spirit of the common consequence effect of Allais’ paradox,19 andthe general tendency of subjects to switch from S 0 u R 0 to R 00 u S 00 is welldocumented. This explains the prevalence of subjects’ (S 0; R 00) choices. Thus, thisimplies that subjects violate the independence axiom, too. We have, therefore, thejoint operation of the common consequence effect, of violations of the reductionaxiom, and of the isolation effect20 to generate Starmer and Sugden’s experimentalresult.

Starmer and Sugden (1991, p. 976) concede

that there is some tendency for random-lottery responses to differ from real-choice ones. We shall call such a tendency a contamination effect, becauseresponses to individual problems in a random-lottery experiment are beingcontaminated by the influence of other problems.

Pondering the pros and cons of their findings, Starmer and Sugden (1991,pp. 977–8) draw the conclusion:

Holt has shown that random-lottery experiments can fail to elicit truepreferences if the reduction principle holds and if the independence axiom isviolated. In showing that the reduction principle does not hold, our resultssuggest that experimental researchers need not be too concerned about thisparticular problem. Of course, this does not eliminate the possibility ... thatthe random-lottery design might be subject to some other sources of bias. Allwe can say is that for the choice problems used in our experiment, subjects’responses did not differ much between the random-lottery and real-choicedesigns. If there are any ‘contamination effects’ at work in the experiment,they seem to be fairly weak.

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Harrison (1992) objects against the BDM elicitation scheme that it violatesreward dominance. Using a numerical example of the costs to a subject ofmisrepresenting his or her true certainty equivalent under a BDM procedure,Harrison (1992, p. 1429) shows that misreporting of the true certainty equivalentsincurs but tiny costs, which are largely negligible for a subject. Thus, in Harrison’sview, subjects experience hardly any incentive to truly report their certaintyequivalents. Even major mistakes cause only negligible costs in terms of forgoneexpected payoffs, while they save much in terms of cognitive effort.

Bostic et al. (1990) put another blame on the BDM elicitation scheme ofcertainty equivalents to be responsible for preference reversals. They addresscertainty equivalents derived from the application of the BDM elicitationscheme21 as judged certainty equivalents, which Bostic et al. (1990, p. 195)juxtapose to choice certainty equivalents. Choice certainty equivalents were firstcomputed by the up-down method for each lottery (or gamble) [Bostic et al., 1990,p. 199]:

... the dollar amount presented at each step of the process, except for theinitial step, depended on the response of the subject to the last choice betweena sum of money and that particular gamble. When the gamble was selected,the dollar amount was increased in the next presentation of the sequence by aconstant of $0.04. If the money was selected, then the subsequent comparisonreduced the money value by $0.04. For the initial comparison, the dollaramount was set at a value well above or below the expected value of thegamble. Half of the subjects began with the high value and half with the lowvalue for each gamble. For each gamble, this sequential procedure wascontinued until the subject’s choice had changed 6 times, to yield an estimateof [the choice certainty equivalent].

For a second experiment, they used a doctored version of the up-down methodcalled PEST [Parameter Estimation by Sequential Testing]. This method workswith variable step sizes instead of fixed step sizes, but does not constitute aprincipal change as compared with the up-down method.22

Table 2 contains the results of the Bostic et al. (1990) experiment. It shows usthat the incidence of preference reversals is somewhat reduced for choice certaintyequivalents, although the phenomenon persists. Moreover, we see that predictedpreference reversals are substantially more reduced than unpredicted or reversepreference reversals. Also, the judged certainty equivalents of $-bets weredistinctly higher than the respective choice certainty equivalents of $-bets. Therewas hardly any difference for the certainty equivalents of P-bets assessed by wayof judged or choice certainty equivalents.23

Although these results are interesting and significant, one must note that themethods to derive judged and choice certainty equivalents are fundamentallydifferent. Whereas a sequential procedure with probable learning effects (which inturn give rise to reflections and more careful pondering on the part of the subjects)is used for choice certainty equivalents, judged certainty equivalents aredetermined as one-shot evaluations without possibilities of further reflection.



This might have contributed to the generation of the elicitation bias of judgedcertainty equivalents.

Thus, experimental work has cast doubt whether the BDM elicitation scheme isreally a major cause for the preference reversal phenomenon. Other work has,however, completely exculpated the BDM elicitation scheme from being the mainoriginator of preference reversals. First, the preference reversal phenomenonpersists if lotteries are evaluated by ratings or by matching. Second, it also persistsunder the ordinal payoff scheme pioneered by Cox and Epstein,24 and widelyapplied by Tversky et al. (1990). The ordinal payoff scheme does not require theincentive-compatible elicitation of certainty equivalents as selling prices of thelotteries. Instead it relies only on the incentive-compatible elicitation of the orderof the certainty equivalents. Subjects are asked to state preferences betweenlotteries as well as their lowest selling prices for the lotteries.25

The subjects are told that one of these [lottery] pairs will be selected atrandom at the end of the session, and that they will play one of these bets. Todetermine which bet they will play, first a random device will be used to selecteither choices or pricing as the criteria for selection. If the choice data areused, then the subject plays the bet chosen. If the pricing data are used, thenthe subject will play whichever gamble was priced higher.

[Under the ordinal payoff scheme,] the prices offered by the subjects are onlyused to order the bets within each pair. Consistency, therefore, requires that theprice orderings and choice orderings should agree, whether or not the subjectsare expected utility maximizers. Thus, if the previously observed reversals werecaused by a failure of expected utility theory, then they should not occur underthe ordinal payoff scheme. This prediction was clearly refuted. The incidence ofreversals was roughly the same (40% to 50%) whether the experimentemployed the BDM scheme, the above ordinal scheme, or even no payoffscheme at all. This finding shows that preference reversal is not caused by theBDM procedure, hence it cannot be explained as a violation of theindependence or reduction axioms of expected utility theory.

Safra et al. (1990) provided another theoretical explanation of preference reversals,namely by fanning out of the indifference curves in a Marschak–Jensen–Machinatriangle. This would constitute another violation of expected utility.26 However,

Table 2. Proportions of Preference Reversals [Source: Bostic et al., 1990, pp. 201, 202,and 206].

Experiment I Experiment IIConditionalPreference Reversals Judged CE Choice CE Judged CE Choice CE

�($) > � (P ), given P u $ 70.0% 35.6% 81.0% 51.0%� (P ) > �($), given $ u P 12.4% 13.1% 3.0% 22.0%Overall rate ofpreference reversals

41.0% 28.0% 46.0% 39.0%

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MacDonald et al. (1992), who investigated the Safra et al. (1990) conjectureexperimentally, did not find any relationship between fanning-out type violations ofexpected utility theory and preference reversals.

Another response mode effect was analyzed by Johnson et al. (1988). Theyobserved a reduction in the rate of preference reversals when the probabilities ofthe lotteries involved are displayed in a format which is more comprehensible forthe subjects.

Preference reversals are also a decreasing function of the number of repetitionsof experiments. This pattern has been observed by several scholars, e.g. Lindman(1971), Pommerehne et al. (1992), Knez and Smith (1987), Wedell and Bockenholt(1990), and Cox and Grether (1991). The reason for this seems to be that‘increasing the number of perceived opportunities to play gambles producesstrong but opposing effects on pricing and choice behaviours. As the number ofplays increases, preference for the P-bet over the $-bet decreases for choice butincreases for pricing. These opposing effects results in a reduction of response-mode induced preference reversals’. [Wedell and Bockenholt, 1990, p. 435.]

Prices of lotteries may be elicited in terms of selling prices or in terms of bidprices. Selling prices address subjects’ willingness to accept, whereas bid pricesaddress their willingness to pay. Thus, two different elicitation modes exist forprices of lotteries. Most experiments which have evidenced preference reversalsare cast in terms of selling prices of the lotteries involved. Empirical work hasestablished much higher selling prices (willingness to accept) than bid prices(willingness to pay).27 Therefore, it is interesting to see whether the preferencereversal phenomenon remains manifest also for bid prices. So far, only fewexperiments have actually used bid prices. First, we have to mention ExperimentII of Lichtenstein and Slovic’s (1971) study and part of the experiments carriedout by Knez and Smith (1987). Although Lichtenstein and Slovic’s Experiment IIconfirmed the tendency of relative overpricing of $-bets, the bid ‘technique servesto dampen the tendency towards gross overbidding for $-bets and hence to reducethe rate for predicted reversals. In addition, ... bids for $-bets are closer in rangeto bids for P-bets in Exp. II ... ’ [Lichtenstein and Slovic, 1971, p. 50]. Knez andSmith (1987, p. 151) observed in the first round of their experiments morepreference reversers among bidders than among sellers (63% versus 52%), which,however, dropped in the second round of repetition by 21% (resulting in 42%buyer reversals versus an unchanged 52% of seller reversals). Second, recall thatCasey (1991, p. 236, Table 2, and p. 240, Table 6) observed reverse preferencereversals for bid prices in combination with high payoffs. However, when bidprices were combined with small payoffs, the traditional pattern of preferencereversals remained [Casey, 1991, p. 240, Table 5], which accords well withLichtenstein and Slovic’s (1971) results for their Experiment II.

4.2. Preference reversals caused by intransitive preferences

Loomes and Sugden (1982) established regret theory.28 Regret theory allows alsofor intransitivity of preferences. More precisely, it distinguishes between two types



of preference cycles. One type is consistent with regret theory (predictedpreference cycles), the other type is inconsistent with regret theory (unpredictedpreference cycles). Shortly thereafter, Loomes and Sugden (1983) providedtheoretical reasoning that preference reversals can be explained by intransitivepreferences. In particular, if their incidence is associated with predicted preferencecycles, then regret theory is able to explain the occurrence of preference reversals.Fishburn (1985), too, explained preference reversals as caused by intransitivepreferences. Loomes et al. (1989, 1991) carried out experiments to show thatpreference reversals were indeed associated with intransitive preferences of thepredicted-preference-cycle type, that is, can be explained by regret theory.

More precisely, Loomes et al. (1989) introduced a lottery providing a certainpayoff C such that C lies between the lower and higher payoffs both of the P-betand of the $-bet. According to regret theory, P u $, $ u C and C u P establishes apredicted preference reversal cycle, the pattern $ u P, P u C, and C u $ establishesas unpredicted preference reversal cycle [Loomes et al., 1989, p. 142]. Such cyclesmay result from a choice-only experimental design, whereas the traditionalpreference reversal pattern (involving prices of lotteries) is referred to as thestandard experimental design. However, the relationship between these twodesigns is not immediate:

We cannot make direct comparisons between the frequencies of reversals inthe two kinds of experiment. Even if the regret theory explanation ofpreference reversal is correct, we should expect to observe fewer reversals in achoice-only experiment. For a participant in the standard experiment toreveal a preference reversal, it is sufficient that his valuations of the two betsshould be ordered in the ‘wrong’ way, as compared with his preferencebetween the two; but in a choice-only experiment it is also necessary that onevaluation is greater than the certainty offered by action C while the other isless. [Loomes et al., 1989, p. 143.]

Loomes et al. (1989, p. 143) used the prices from the standard experiment toimpute the {$,C} and {P; C} choices under a hypothetical choice-only experiment.If preference reversal was caused by some information-processing effect, thenLoomes et al. (1989, p. 143) argue that it is not legitimate to impute choices fromvaluations: ‘Thus if information-processing effects are contributing to preferencereversal, we should expect to find a greater tendency for cycles in the imputedchoices than in the actual ones’. For the standard design, Loomes et al. (1989,pp. 148–9) observed 46.2% traditional preference reversals (of these some twothirds in the predicted direction) and only 19.4% actual preference reversal cycles(of these more than three quarters in the predicted direction). There was nostatistically significant difference of cycles in imputed and actual choices. Thediscrepancy of preference reversals in both designs occur because the choice-onlymethod has to rely on some C which lacks the fine tuning of lottery prices. Thus,some standard preference reversals may not be reflected for some feasible C’s.

A rival theory to explain preference reversals by intransitive preferences is itsexplanation by overpricing the $-bets and=or underpricing the P-bet. Explanations

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of this phenomenon are provided in Section 4.3. Tversky et al. (1990) submitted acomprehensive experimental investigation which allows to distinguish betweenintransitive preferences, and overpricing of the $-bet and=or underpricing of the P-bet.

For the experimental design, Tversky et al. (1990) introduce a certain payoff Xwhich should be chosen such that it is likely to lie between the prices of the $-betand the P-bet.29 Subjects are asked for three lottery preferences (or choices), viz. $versus P, $ versus X, P versus X, and for two prices, �($) and � (P ). The priceswere determined according to the ordinal payoff scheme to avoid possibledistortions caused by other elicitation schemes, e.g. the BDM elicitation scheme.

Preference reversal occurs obviously, if

P u $ and � ($) > X > � (P ): (4:6)

Now, the critical data for the type of preference reversal are the observedpreferences (choices) between X and P and between X and $. Ignoring ties, thereare four possible response patterns, which, in addition to (4.6), characterize thetype of preference reversal [Tversky et al., 1990, p. 206].

INTRANSITIVITY $ u X and X u P; (4.7)

OVERPRICING THE $-BET X u P and X u $; (4.8)

UNDERPRICING THE P-BET P u X and $ u X; (4.9)

OVERPRICING THE $-BET AND UNDERPRICING THE P-BETP u X and X u $. (4.10)

Tversky, et al. (1990, p. 209) observed some 45% predicted and some 4%unpredicted preference reversals. About half of the preference reversal patternsmet the test conditions (4.6); these were 620 individual preference patterns [cf.Tversky et al., 1990, p. 209]. Table 3 gives an account of the results.

Table 3 shows that intransitive preferences can account for only 10% ofobserved preference reversals. The results of Loomes et al. (1989, p. 149, Table 4;1991) show figures for intransitive preferences which are somewhat higher thanthe figures of Tversky et al. (1990). However, both papers follow differentapproaches and cannot easily be compared. Whereas Tversky et al. (1990) test fornon-transitive preferences for subjects which did exhibit preference reversal,Loomes et al. (1989, 1991) used independent samples of subjects and investigatedthe incidence of preference reversals in the choice-only and in the standarddesigns.

Thus, it seems that the explanation of preference reversals by intransitivepreferences is still an open issue. However, the Tversky et al. (1990) explanation ofpreference reversals by overpricing the $-bet and underpricing the P-bet has leftan impressive mark upon the reasoning on preference reversal. The literaturedealing with this explanation is considered in the next section.



4.3. Preference reversals caused by overpricing the $-bet and=or underpricing theP-bet

Table 3 contends that overpricing of the $-bet and=or underpricing of the P-betaccount for some 90% of preference reversals. Although the fulcrum of thisexplanation is the evaluation of the prices for lotteries, we must bear in mind thatits counterpart is the explanation of subjects’ predilection for the P-bet in terms ofchoice between lotteries. Several hypotheses for the occurrence of preferencereversals have been put forward.

Slovic and Lichtenstein (1968, p. 11) were the first to observe that subjects’choices of lotteries (they use the term rating) correlated more highly with theprobability of winning a positive payoff (PW) than with any other component ofthe respective lottery pairs, while the prices of lotteries (they use the term bids)correlated most highly with the amount of the negative payoffs of the lotteries ($L)involved:

Whereas 50% of the [subjects] in the rating group relied predominantly onPW, only 18% of the [subjects] in the bidding group did. Similarly, thepercentage of [subjects] for whom $L was most important was 26% in therating group but 53% in the bidding group.

The differences between rating and bidding seem to illustrate the influenceof information-processing considerations on the method by which a gamble isevaluated. Apparently the requirement that [subjects] evaluate a gamble inmonetary units when bidding forces them to attend more to the payoffdimensions — also expressed in terms of dollars — than they do when ratinga bet.

In their classical paper on preference reversals, Lichtenstein and Slovic (1971,p. 54) surmise that subjects apply different decision processes for choices betweenlotteries and the evaluation of the prices of lotteries, respectively. In pairedcomparisons leading to choices,

each attribute of one bet can be directly compared with the same attribute ofthe other bet. There is no natural starting point ...

In contrast, bidding techniques provide an obvious starting point: theamount to win ... [A subject] who is preparing a bidding response to a bet hesees as favorable starts with the amount to win and adjusts it downwards to

Table 3. Causes of Preference Reversals [Source: Tversky et al. (1990, p. 210, Table 3)].

Causes Percentages

Intransitivity 10.0%Overpricing of $-bet 65.5%Underpricing of P-bet 6.1%Overpricing $ and Underpricing P 18.4%

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take into account the other attributes of the bet. The amount to win translatesdirectly into an amount to bid.

For P-bets, their modest winning amounts are only slightly adjusteddownwards, also due to the high probabilities of winning. However, for $-bets,the high winning amounts are but insufficiently adjusted downwards with respectto the small winning probability [Lichtenstein and Slovic, 1971, p. 54]. This leadsto an overpricing of the $-bet and causes a mismatch between lottery choices andcertainty equivalents.30 Lichtenstein and Slovic consider this effect to be caused bythe influence of a starting point which is a particular kind of an anchor.31

More recently, the idea that different decision processes applied to choicesbetween lotteries and the assessment of lottery prices, respectively, are responsiblefor preference reversals has led to two major lines of explanations: the prominencehypothesis and the compatibility hypothesis. Both view lotteries as two-attributechoice alternatives, one attribute being probability, the other payoff.

The prominence hypothesis developed from the elimination-by-aspects theoryestablished by Restle (1961) and Tversky (1972a, b); later empirical evidence wasprovided by Ranyard (1976). It was succinctly characterized by Slovic (1975,p. 281) as follows: ‘At each stage in the choice process an aspect is selected with aprobability proportional to its importance; alternatives that do not include theselected aspect are eliminated. In choosing a new car, for example, the firstselected aspect might be automatic transmission. If so, all cars without this featurewould be eliminated. Among the remaining alternatives another aspect, perhaps a$4,000 price limit, would be selected and more expensive cars would thus beexcluded. The process would continue until all cars but one were eliminated’.Slovic’s (1975) work, in turn, gave rise to the development of the prominencehypothesis which purports that subjects choose among alternatives according totheir superiority in the more important dimension.

The compatibility hypothesis was anticipated by work of Fitts and Seeger(1953). It was rediscovered by Slovic and MacPhillamy (1974) who observed that,in multi-attribute decision tasks, attributes were weighted more heavily in thecomparison of decision alternatives when they were common than when they wereunique. The compatibility hypothesis was later on employed by Tversky et al.(1988), Tversky et al. (1990), Slovic et al. (1990), and Irwin et al. (1993) to explainthe preference reversal phenomenon. We single out a quotation by Tversky et al.(1988, p. 376):

The choice-matching discrepancy, like other violations of procedureinvariance, indicates that the weighting of the attributes is influenced by themethod of elicitation. Alternative procedures appear to highlight differentaspects of the options and thereby induce different weights. To interpret andpredict such effects, we seek explanatory principles that relate taskcharacteristics to the weighting of attributes and the evaluation of options.One such explanation is the compatibility principle. According to thisprinciple, the weight of any input component is enhanced by its compatibilitywith the output. The rationale for this principle is that the characteristics of



the task and the response scale prime the most compatible features of thestimulus. For example, the pricing of gambles is likely to emphasize payoffsmore than probability because both the response and the payoffs areexpressed in dollars.

Tversky et al. (1988, 1990) first attributed the preference reversal phenomenonto the compatibility hypothesis alone. Indeed they disputed any role of theprominence hypothesis: ‘ ... preference reversals are induced primarily by scalecompatibility, not by the differential prominence of attributes that underlies thechoice-matching discrepancy. Indeed, there is no obvious reason to suppose thatprobability is more prominent than money or vice versa.’32 Subsequentexperimental evidence, however, forced Slovic et al. (1990, p. 22) ‘to reconsiderthe hypothesis that probability is more prominent than money, which is furthersupported by the finding that the rating of bets is dominated by probability ... ’ Inthis later work they attribute preference reversals to both hypotheses. Thecompatibility effect causes the difference between probability matching andpayoff matching (see our above portrayal), whereas the prominence effectcontributes to the relative attractiveness of the P-bet in choice [Slovic et al., 1990,p. 22]. — Notice, however, that Fischer and Hawkins (1993) could not evidence amajor impact of the compatibility effect; they observed that the prominence effectwas much stronger than the compatibility effect.

By systematic manipulation of starting points, Schkade and Johnson (1989)corroborated the Lichtenstein-Slovic conjecture, as well as the explanation ofpreference reversals by the compatibility effect. Using the Mouselab softwarepackage for the presentation of lotteries, they started either with high or with lowanchors as starting points for the assessment of lottery prices. This wasaccomplished by setting the pointer on the computer screen on appropriatevalues. The preferences for lotteries were determined not by choices but byratings. Schkade and Johnson (1989) report large effects of starting pointmanipulation. High starting points produced higher lottery prices and higherratings.

If the starting point manipulation is effective, then the greatest frequency ofpredicted preference reversals is to be expected when the starting point in rating ishigher for the P-bet than for the $-bet, and vice versa for the assessment of lotteryprices. Conversely, when the starting point in rating is higher for the $-bet than forthe P-bet, and vice versa for the assessment of lottery prices, then the lowestfrequency of predicted preference reversals is to be expected. For the formerdesign, Schkade and Johnson (1989, p. 226, Table 8, and p. 227) observed a 70%frequency rate of predicted preference reversals, for the latter only a 34%frequency rate of predicted preference reversals. Thus, a starting pointmanipulation was able to reduce predicted preference reversals by more than half.

Several other theories, too, allow for the explanation of preference reversals.They center around overpricing of the $-bet and=or underpricing of the P-bet, andon subjects’ predilection for the P-bet in choice. Moreover, with Lichtenstein andSlovic (1971) and Schkade and Johnson (1989) they share the view that decision

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makers arrive at lottery prices through a process of anchoring and adjustment. Inthe taxonomy of these theories, the explanation of preference reversals by Tverskyand associates by way of the prominence and compatibility effect has been labeledthe contingent weighting theory by Mellers et al. (1992, p. 334).

The next theory to be discussed is the reference-level theory elaborated by Luceet al. (1993). Its intellectual precursor is Hershey and Schoemaker’s (1985)explanation of the biases which they observed from the concatenation ofresponses from certainty and probability equivalence questions.33 For the PEmode bias they offer a reframing explanation. In the PE mode all payoffs are heldconstant, and attention is focused on the variable probability dimension.

Consequently, the gamble’s outcomes may be psychologically coded as ‘gains’or ‘losses’ relative to the fixed sure amount S. If so, the lottery ispsychologically translated upward (for losses) or downward (for gains) bythis sure amount S, which serves as the new reference or zero point. ...If some subjects indeed reframe the PE mode questions as suggested above,

the net effect will be an increase in risk-averse responses ... Such shifts towardmore risk aversion are consistent with a utility (or value) function which isconvex for losses, concave for gains, and steeper for losses than forgains ... [Hershey and Schoemaker, 1985, p. 1224.]

Following a related reframing notion, Luce et al. (1993, p. 117) offer a full-fledgedtheory which assumes that a decision maker engages in a four-stage process whenconfronting a choice set:

1. The subject evaluates each lottery g by its certainty equivalent �1(g).2. In stage 2, the subject determines a reference level � (X ) for the entire choice

set X that is based on the �1’s according to the following rule: If some �1 ispositive, then �(X ) is the smallest of the gains; if all �1’s are negative, then�(X ) is the smallest loss.

3. All lotteries are reframed in terms of gains and losses relative to �(X ), i.e.�(X ) is subtracted from all payoffs.

4. Certainty equivalents �2 are recalculated for all transformed lotteries, andthe lottery with the largest certainty equivalent is chosen.

Using experimental data collected by Mellers, Chang et al. (1992), Luce et al.(1993, p. 131–3) estimated parameter values for their model. Furthermore, theyshow that their reference-level theory (using this data set) provides a gooddescription of preference reversals.

The change-of-process theory ensued from the conjecture of Lichtenstein andSlovic (1971), Payne (1982), and Schkade and Johnson (1989), that subjectiveprobabilities and utilities remain constant over response modes, but the processesused to combine information vary with respect to the task performed by thesubjects. Mellers et al. (1992, pp. 346 and 352) and Mellers, Chang et al. (1992,pp. 347–8) propose that lottery ratings are additively composed of probabilitiesand utilities, whereas the assessment of buying prices of lotteries is combined in amultiplicative way. More precisely, let uG denote the utility of a gain, uL the utility



of a loss, and p the subjective probability of gaining in the respective lottery, then[Mellers et al., 1992, p. 352]

R¼ J(kpþ uG þ uL); (4:11)

� ¼ Z(puG þ (1� p)uL); (4:12)

where R denotes rating, � denotes the bid price of a lottery, J(�) denotes a strictlyincreasing judgment function, Z(�) a strictly increasing evaluation function, and kdenotes a scaling constant. Mellers et al. (1992) and Mellers, Chang et al. (1992)and associates test change-of-process theory, contingent weighting theory, andexpression theory for the explanation of preference reversals. They found thatchange-of-process theory outperforms its competitors.

Expression theory as an explanation of preference reversals was proposed byGoldstein and Einhorn (1987). They assume that the decision maker undergoes athree-stage process. First, he or she encodes lotteries in the form of subjectiveprobabilities and utilities. Second, the subject evaluates the encoded elements oflotteries into a worth of the lottery, u(G). Suppose W denotes a gain and L a loss,then Goldstein and Einhorn (1987, p. 240) suppose the following relationship

u(G)¼ u(W )��[u(W)� u(L)]; (4:13)

where �, 0 ˘ � ˘ 1, denotes a decision weight. In this formulation, Wrepresents an anchor for the evaluation of lotteries. The third stage of thedecision process encompasses the expression of the encoded lotteries on aresponse scale of some type. Now Goldstein and Einhorn (1987) assume thatsubjects use different expressions for different tasks. For instance, to answerthe choice=attractiveness question, subjects prefer lottery G1 to lottery G2, ifu(G1) > u(G2). For other tasks, such as the assessment of different certaintyequivalents, they assume that the subject employs different expressions.Preference reversals are explained as a consequence of the use of differentexpressions for different tasks.34 They succeed in satisfactory explanations oftheir six types of preference reversals.

Casey (1991) blames also anchoring and adjustment heuristics to be responsiblefor the reverse preference reversal which he discovered. If lotteries have nonegative outcomes, then losses cannot occur in the choice task. Yet buying carrieswith it the possibility of a loss. Moreover, the buyer incurs an out-of-pocket loss ifthe lottery’s payoff undercuts the bid price. This reframes lotteries in terms ofgains and losses, which increases subjects’ risk aversion. Casey (1991, pp. 244–7)presents an aspiration level explanation of preference reversals relying on twocomponents, satisfying (subjects accept only a small probability of incurring a lossin bidding) and loss aversion.

4.4. Preference reversals caused by nonlinear probabilities and discount reversal

Consider a lottery (x; p; 0). In analogy to hyperbolic discount functions fordelayed payoffs, Rachlin et al. (1991, p. 235) applied such functions also to the

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assessment of lottery prices [cf. also Rachlin and Raineri, 1992, p. 100]:

� (x; p; 0)¼px

pþ h(1� p)¼

x

1þ h�; (4:14)

where �¼ 1=p� 1 are the odds against x. Rachlin et al. (1991, p. 238) observed arather good fit of their experimental data for this function yielding an estimateh¼ 1:6.

Substituting h¼ 1:6 into (4.14) shows [Rachlin et al., 1991, p. 238]

� (1000; 1; 0)¼ 1000 > 962¼ � (2500; 0:5; 0): (4:15)

Multiplying both winning probabilities by 0.1 yields, however,

� (1000; 0:1; 0)¼ 65 < 80¼ � (2500; 0:05; 0); (4:16)

which would be impossible if the formula for lottery prices were linear in theprobabilities. If the subject asserts, at the same time,

(1000; 0:1; 0) u (2500; 0:05; 0); (4:17)

then we have another instance of preference reversals.If, on the other hand, the subject asserts, consistently with the lottery prices,

that

(1000; 1; 0) u (2500; 0:5; 0)

and (1000; 0:1; 0) e (2500; 0:05; 0); (4:18)

then we encounter an instance of the common ratio effect of Allais’ paradox.35

Nonlinearity in probabilities can thus explain both the occurrence of preferencereversals and of Allais’ paradox.

Notice that (4.14) is akin to the hyperbolic discounting formula. In a seminalpaper, Strotz (1955=56) pioneered in discounting with nonlinear discount rates.Suppose x denotes a payment to be received at some future time t1, and y, y > x,denotes a payment to be received at some future time t2, where t2 > t1, and

(x; t1) u (y; t2): (4:19)

Then Strotz (1955=56) analyzed under which conditions this preference wouldhold for all shifts of the date by T >�t1, i.e.

(x; t1 þ T ) u (y; t2 þ T ) 8 T >�t1: (4:20)

Strotz showed that the only evaluation function (�) of future payments, whichsatisfies both (4.19) and (4.20), is discounting with a constant discount rate, that is:

(x; t1)¼x

(1þ r) t1; r const:; or

xe��t1 ; � const:

8>>><

>>>:

(4:21)



However, it has often been observed that for some T 0, subjects exhibit thepreferences

(x; t1) u (y; t2) and (x; t1 þ T 0) e (y; t2 þ T 0): (4:22)

In order to accommodate for such a preferential pattern, scholars such as Ainslie(1975, 1991), Ainslie and Haslam (1992), Benzion et al. (1989), Loewenstein andPrelec (1992), Mazur (1987), and Roelofsma and Keren (1995), suggestedevaluation functions with nonconstant discount rates, mostly hyperbolic discountfunctions. For instance, Loewenstein and Prelec (1992, p. 580) proposed theevaluation function for a payment x due at time t, where � denotes the presenttime, as

� (x; t� �)¼x

[1þ �(t� �)]=�; � � 0; > 0: (4:23)

The parameter � determines the departure of the evaluation function � (�) fromexponential discounting. In particular, we have

lim�2 0 � (x; t� �)¼ e� (t� �): (4:24)

By looking for evaluation functions like (4.23), the description of individualbehaviour is established by a mathematical contrivance. Starting from anoberservation as shown in (4.22), the analyst develops an evaluation function like,e.g. (4.23), which assigns present prices to payments effectuated at different dates.Such an approach allows for preference switching, as shown in (4.22), but, at thesame time, excludes preference reversal by definition.

However, another approach asks subjects for their preferences for (x; t1) vis-a-vis (y; t2) on the one hand, and for their immediate selling prices of the (certain)future claims, � (x; t1) vis-a-vis � (y; t2), on the other. The case

(x; t1) u (y; t2) and � (x; t1) < � (y; t2) (4:25)

establishes then a preference reversal, which has been called discount reversal.Tversky et al. (1990, pp. 212–4) investigated discount reversals. Administering

four triples of prospects of hypothetical payments to 169 subjects, they registered52% of discount reversals of the type described in (4.25). Using also preferencesexpressed for given fixed immediate payments, Tversky et al. (1990, p. 214,Table 5) were able to explain 15.4% of discount reversals by intransitivity, 54.8%by overpricing of (x; t1), 11.5% by underpricing (y; t2), and 18.3% bysimultaneous overpricing (x; t1) and underpricing (y; t2). Thus, they draw theconclusion that ‘payoffs are weighted more heavily in pricing than in choice’.

Bohm (1994a) carried out similar experiments as Tversky et al. (1990). Forhypothetical payoffs and selling prices, Bohm (1994a, p. 1375; own calculationsfrom Table 2) observed 33.9% predicted and 1.6% unpredicted discountreversals. For hypothetical payoffs and bid prices, Bohm (1994a, p. 1374; owncalculations from Table 1) observed 27.4% predicted and 5.3% unpredicteddiscount reversals. For real payoffs and bid prices, Bohm (1994a, p. 1374, own

644 CHRISTIAN SEIDL


calculations from Table 1) found 9.4% predicted and 6.3% unpredicted discountreversals. Thus, financial incentives and bid prices seem to reduce the incidence ofpredicted discount reversals and to increase the incidence of unpredicted discountreversals.

Interestingly enough, the same hyperbolic discount functions, which canestablish preference switching [as shown in (4.22)] and respective presentprices of options, may also be used to explain preference reversals and showtheir close relationship with probability transformation functions. Suppose(4.19) holds. Then an evaluation function akin to (4.23) may be used todemonstrate

(x; t1) u (y; t2) and � (x; t1 � �) < � (y; t2 � �): (4:26)

This has let Tversky et al. (1990, pp. 212–4), Rachlin et al. (1991), Prelec andLoewenstein (1991), Rachlin and Raineri (1992), and Bohm (1994a) to stress theclose relationship between preference reversals and discount reversals. Assume ¼ � in (4.23), equate the denominators in the right hand sides of (4.14) and(4.23), and take the logarithms, then we engender

ln(t� �)¼ ln �þ ln(h=�): (4:27)

In a doubly logarithmic diagram of the delay of a (certain) payment and �[the odds against x in a lottery (x; p; 0)], (4.27) gives, therefore, a straight linewith a slope of 1, a result which was remarkably well confirmed [Rachlin et al.,1991, p. 242, Figure 10].

5. Conclusion and outlook

This article provides a literature survey on the preference reversal phenomenon.When comparing a so-called P-bet (a bet with a high probability of winning amodest payoff) and a so-called $-bet (a bet with a modest probability of winning ahigh payoff), psychologists Lindman, Lichtenstein and Slovic observed in the latesixties and early seventies that subjects tend to prefer the P-bet, but, at the sametime, associate a higher lottery price with the $-bet.

This stunning phenomenon provoked much follow-up studies, first bypsychologists and, somewhat later, by the sceptical economics profession whicheventually had to acknowledge the robust manifestation of what it perceived asirrational behaviour of individuals. This work led to the discovery of sundryvarieties of preference reversals, which disclose a general pattern of humanbehaviour rather than a peculiar characteristic of choice between bets.Moreover, certain experimental designs beget different incidences of preferencereversals.

Several explanations of the preference reversal phenomenon have been broughtforward. First, it has been explained as a consequence of using the BDMelicitation scheme of certainty equivalents. This elicitation scheme constitutes infact a two-stage lottery with different payoffs, which distorts subjects’ true



revelation of selling prices if the reduction axiom and=or the independence axiomare violated. Yet experimental work showed that the isolation effect invalidatesthe perception of the BDM elicitation scheme as two-stage lotteries.

Intransitivity of preferences has been blamed as another cause of preferencereversals. Regret theory in particular lends itself to the rationalization ofintransitive preferences and thus the occurrence of preference reversals.

An appealing behavioural explanation of preference reversals is overpricing ofthe $-bet and=or underpricing the P-bet, where overpricing of the $-bet is theprevailing phenomenon. This was reasoned by the prominence and compatibilityhypotheses, respectively, by the reference-level theory, and by expression theory.

Nonlinear probabilities, somewhat akin to hyperbolic discounting and discountreversals, which are also given thorough attention, provide a fourth cause ofexplaining preference reversals.

Finally, let us turn to the significance of the preference reversal phenomenon forthe economics profession. The preference reversal phenomenon has played adecisive role to arouse economists’ interest in experimental economics which hasvirtually ushered the end of armchair economics [Simon, 1986]. In particular thepreference reversal phenomenon has contributed to acknowledge

that decision-making is a constructive process. In contrast to the classicaltheory that assumes consistent preferences, it appears that people often do nothave well-defined values, and that their choices are commonly constructed,not merely revealed, in the elicitation process. Furthermore, differentconstruction can give rise to systematically different choices, contrary tothe basic principles that underlie classical decision theory. [Tversky, 1996,p. 185.]

This means that economics is at the verge of a revolution which replacesneoclassical armchair economics with behavioural economics. In particular, theclassical paradigm of value discovery is now being replaced with the new paradigmof value construction:

The idea of constructive preferences goes beyond a mere denial that observedpreferences result from reference to a master list in memory. The notion ofconstructive preferences means as well that preferences are not necessarilygenerated by some consistent and invariant algorithm such as expected valuecalculation ...[Payne et al., 1992, p. 89].

Behavioural economics centres on problems arising from the violation ofprocedure invariance and description invariance, which are the two main pillarsthat have propped up classical economics. This means that both the elicitationmode and the framing of the underlying problem matter for the pattern ofindividual preferences. This means, e.g. that marketing, which is completely aliento neoclassical economics, becomes an integral part of behavioural economics.

A plethora of empirical phenomena, so far hardly ever noticed by theeconomics profession, will become centerpieces of applied economic research:Anchoring moves individuals’ values and preferences in the direction marked by

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the anchor [cf. e. g., Slovic, 1972; Tversky and Kahnemann, 1974; Slovic et al.,1977, p. 16; Edwards and von Winterfeldt, 1986, p. 247; Northcraft and Neale,1987; Kahnemann, 1992]. The background contrast effect purports that analternative appears attractive on the background of less attractive alternatives andunattractive on the background of more attractive alternatives [Simonson andTversky, 1992; Tversky and Simonson, 1993]. The tradeoff contrast effect meansthat the relative scarcity of attributes of choice alternatives influences theweighting of an option’s attributes for subsequently presented alternatives[Tversky and Simonson, 1993, p. 1181]. The asymmetric dominance effect notesthat the presentation of a choice alternative Z, which is dominated by X, but notby Y, shifts preferences in favor of X [Huber et al., 1982; Huber and Puto, 1983;Tyszka, 1983; Ratneshwar et al., 1987; Wedell, 1991]. The endowment effectobserves that people demand more to give up an object than they are willing topay to acquire it, which causes differences in willingness-to-accept and will-ingness-to-pay on the one hand, and nonreversibility of indifference curves on theother [Thaler, 1980; Knetsch, 1989, 1992; Kahnemann et al. 1990, 1991]. Theavailability bias means that subjects judge the probability of events by the ease ofgetting information [Tversky and Kahnemann, 1973; Lichtenstein et al., 1978].

Phenomena like these cannot be longer ignored by the economics profession.They will pose the main future challenges for the science of economics. Many ofthem can serve to explain preference reversal phenomena which extend wellbeyond the mere comparison of lottery preferences and lottery prices. However,work on these simple roots of preference reversal is so exuberant that a thoroughliterature survey had to concentrate on the pure preference reversal phenomenonunless it would have to fill a whole textbook.

Acknowledgements

I am indebted to two anonymous referees for helpful comments. The usualdisclaimer applies.

Notes

1. Cf. Lindman (1965, 1971); cf. also Slovic and Lichtenstein (1968, p. 16). It seems thatEdwards (1954a, p. 396), and Edwards (1962, p. 127), had a presentiment of preferencereversals, which he, however, attributed to probability preferences. See also Dale (1959).

2. Cf. Lichtenstein and Slovic (1971, pp. 51–2). For a concise demonstration of theBecker-DeGroot-Marschak elicitation scheme cf. Section 4.1.

3. Lichtenstein and Slovic (1971, p. 48), report that 73% of all subjects of Experiment I

exhibited predicted preference reversals: Whenever they chose lottery P, lottery $received a higher price than P.

4. For earlier surveys of the literature cf. Slovic and Lichtenstein (1983, pp. 596–9); Berg etal. (1985, pp. 32–4); Goldstein and Einhorn (1987, pp. 236–8); Tversky and Thaler

(1990); Hausman (1992, pp. 227–44). I was told that there exists also a PhD dissertationby Timo Tammi, Essays on the Rationality of Experimentation in Economics: The Case ofPreference Reversal. University of Joensuu, Publications in Social Science, No 27.



5. Cf. Zagorski (1975, p. 490). [There seems to be an error, because the messages on

p. 490 are contradictory.]6. The effect of repetitions of experiments [Pommerehne et al., 1982, p. 573, Conclusion 3]

had obviously been anticipated by Lindman (1971). Slovic and Lichtenstein (1983,

p. 597), note: ‘Readers of the papers by Pommerehne et al. and Reilly would hardlyknow there was considerable scrutiny of preference reversals prior to the publication byGrether and Plott’.

7. That is, u 00(y)=u 0(y) is constant for all feasible y.

8. Cf. Berg et al. (1985, pp. 34–5).–Indeed, Kachelmeier and Shehata (1992, pp. 1132–3),report minimum selling prices which were almost twice as high as the respectivemaximum bid prices.

9. All figures are computed from Berg et al. (1985, p. 43, Table 3).10. Matching was also used by Hershey and Schoemaker (1985); Johnson and Schkade

(1989); Schoemaker and Hershey (1992); Delquie and de Neufville (1991); Delquie (1993).

11. Cf. Schkade and Johnson (1989, p. 212, Table 2, and p. 219, Table 5); Goldstein andEinhorn (1987, p. 239, Figure 2 a, c, and d); Tversky et al. (1988, pp. 381–82).

12. Bohm (1994b, p. 187), too, has criticized that the lotteries involved in testing preferencereversals do not represent real-world situations. In particular, he blamed the traditional

experimental design of preference reversals for having used unrealistic settings, inparticular, that prices for lottery tickets are zero, that lottery payoffs are not multi-valued, that payoffs are trivially low, and that the experiments include also subjects

who do not wish to buy lottery tickets.13. Cf. Tversky et al. (1990), pp. 212–14; Rachlin et al. (1991); Rachlin and Raineri (1992).14. We must not forget the logical requirement that the subject’s selling price is greater or

equal than the subject’s own bid price.15. For a detailed presentation see Davis and Holt (1993, pp. 487–93, and pp. 498–9);

Seidl and Traub (1999, pp. 218–9). Interestingly enough, Vickrey (1961, p. 26), has

produced an intellectual counterpart of this procedure for the bid side.16. In the specification of the compound independence axiom. For more details see Segal

(1990, p. 350 and p. 355).17. For details see Quiggin (1982); Chew (1983); Chew et al. (1991).

18. For group B, (4.3) was posed before (4.4), and for group C vice versa. However, thebehavioural pattern of both groups did not differ significantly.

19. For details see Allais (1979, p. 89).–In case of (4.3) and (4.4) the component (£10; 0:2)of R 0 and R 00 is the common consequence.

20. The isolation effect means that subjects disregard the first stage of a lottery andcompare the options only according to the second stage of the lotteries. Cf. Tversky

(1975, p. 167); Kahneman and Tversky (1979, pp. 271–3). Tversky and Kahneman(1986, p. S252 and p. S268) have shown that the violation of cancellation–and thus thecommon violation of the independence and reduction axioms — disappears when thedecision problem is presented in a sequential form.

21. For some reason they attribute it to Grether and Plott (1979); cf. Bostic et al. (1990,p. 195).

22. Therefore, we need not go into details; Cf. Bostic et al. (1990, pp. 205–6). — The

Quickas [Quick Assessment] method mentioned by Kimbrough and Weber (1994,p. 623), is another species of the up-down method.

23. Cf. Bostic et al. (1990, p. 209). Recall that using the O’Brien elicitation scheme instead

of the BDM elicitation scheme had much smaller impact on the incidence of preferencereversals.

648 CHRISTIAN SEIDL


24. Cf. Cox and Epstein (1989, p. 412). Note that they use the term choice reversals (p. 409)

instead of preference reversals. Recall that, in contrast to Tversky et al. (1990), Cox andEpstein observed roughly the same percentages of ‘predicted’ and ‘unpredicted’preference reversals.

25. The following quotation is taken from Tversky and Thaler (1990, p. 206). For theexperimental results cf. Tversky et al. (1990, pp. 208–11).

26. The indifference curves in a Marschak–Jensen–Machina triangle are parallel straightlines if expected utility holds. Fanning out means that the slope of indifference curves

increases as the utility level rises.27. Cf., e.g. Bishop and Heberlein (1979); Knetsch and Sinden (1984, 1987); Marshall et al.

(1986); Coursey et al. (1987); Birnbaum and Sutton (1992); Mellers et al. (1992); Davis

and Holt (1993, pp. 457–60). For contrary results cf. Harless (1989).28. Regret theory assumes that subjects make comparisons between lotteries. In particular

they compare ‘what is’ under the established state of nature of the chosen lottery and

‘what might have been’ under the same state of nature if some other lottery had beenchosen. If the current outcome is worse, the subject feels regret for not having chosenthe alternative lottery, otherwise, he or she feels elation. Regret or elation is seen as afunction of the possible payoff pairs.

29. Notice the difference from the Loomes et al. (1989) design. X should lie between theprices of the $-bet and the P-bet, C has to lie between the payoffs of the $-bet and theP-bet.

30. Recall the results obtained by Zagorski (1975) that the sum of the selling prices forexchanging lotteries was greater if the first pair of the sublotteries involved only adifference in payoffs. This confirms Lichtenstein and Slovic’s conjecture.

31. Anchoring means that subjects’ judgments depend on some reference point, whichcould be the initial value presented to them, an aspiration level, or a subject’sendowment. Anchoring has been studied by Hunt and Volkmann (1937); Rogers

(1941); McGarvey (1942=43); Helson (1947). Its importance for decision science wasstressed by Tversky (1974, p. 154), and Tversky and Kahneman (1974, p. 1128).

32. Cf. Tversky et al. (1988, p. 382); Tversky et al. (1990, p. 211) contend that ‘the weightof any aspect (for example, probability, payoff) of an object of evaluation is enhanced

by compatibility with the response (for example, choice, pricing). ... Because theselling price of a bet is expressed in dollars, compatibility entails that the payoffs,which are expressed in the same units, will be weighted more heavily in pricing than

in choice’.33. Hershey and Schoemaker (1985) concatenated the responses from certainty and

probability equivalence questions. Their subjects had to answer assessment questions in

both the certainty and probability equivalence response modes, with the first replybecoming a lottery element in the second question. This provided two patterns ofresponse modes, viz. CE� PE and PE� CE. Each of them was used both for gainsand for losses. For gains they observed second-stage responses which were higher than

the corresponding first-stage parameters (i.e. a higher probability in the CE� PE case,and a higher certainty equivalent in the PE� CE case). For losses they observedsecond-stage responses which were lower than the corresponding first-stage para-

meters.34. We cannot go into details here. The reader is referred to Goldstein and Einhorn (1987,

pp. 241–4).

35. Cf. Allais (1979, p. 89); note that the win probabilities of the right hand lotteries in(4.18) are scaled down by one tenth.



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