Journal o/ Experimental Psychology1971, Vol. 89, No. 1, 46-55

REVERSALS OF PREFERENCE BETWEEN BIDS ANDCHOICES IN GAMBLING DECISIONS 1

SARAH LICHTENSTEIN AND PAUL SLOVIC 2

Oregon Research Institute, Eugene

The 5s in three experiments chose their preferred bet from pairs of bets andlater bid for each bet separately. In each- pair, one bet had a higher proba-bility of winning (P bet); the other offered more to win ($ bet). Biddingmethod (selling vs. buying) and payoff method (real-play vs. hourly wage) werevaried. Results showed that when the P bet was chosen, the $ bet oftenreceived a higher bid. These inconsistencies violate every risky decisionmodel, but can be understood via information-processing considerations. Inbidding, 5 starts with amount to win and adjusts it downward to account forother attributes of the bet. In choosing, there is no natural starting point.Thus amount to win dominates bids but not choices. One need not call thisbehavior irrational, but it casts doubt on the descriptive validity of expectedutility models of risky decision making.

Utility theory, in one form or another,has provided the guiding principle forprescribing and describing gambling deci-sions since the eighteenth century. Theexpected utility principle asserts that givena choice among gambles, the decision makerwill select the one with the highest expectedutility.

There are a number of ways other thanchoosing by which an individual can ex-press his opinions about the utilities ofvarious gambles. Among these are ratingsof attractiveness, bids to buy, and bids tosell:

1. Ratings of attractiveness: On anarbitrary scale, 5 assigns an attractivenessvalue to each of a set of bets. For anypair, it is assumed that he prefers the oneto which he gives the highest rating.

2. Bids to buy (B bids): E owns a set ofbets. For each bet, 5 indicates the maxi-mum amount of money he would pay inorder to be able to play the bet. (SeeCoombs, Bezembinder, & Goode, 1967;

1 This work was sponsored by the Personnel andTraining Research Programs, Psychological SciencesDivision, Office of Naval Research, under ContractN00014-68-C-0431, Contract Authority NR 153-311,and by Grants MH-15414 and MH12972 from theUnited States Public Health Service. We are grate-ful to Peter Christenson, who was the E in Exp. IIand III , and to Leonard Rorer, Lewis Goldberg,and Robyn Dawes for their helpful suggestions.

2 Requests for reprints should be sent to SarahLichtenstein, Oregon Research Institute, Box 3196,Eugene, Oregon 97403.

Lichtenstein, 1965.) For any pair of bets,it is assumed that he prefers the one forwhich he bids the most money.

3. Bids to sell (S bids): S owns a setof bets. For each bet, S indicates theminimum amount for which he would sellthe right to play the bet (see Becker,DeGroot, & Marschak, 1964; Coombs etal.,1967; Tversky, 1967). For any pair it isassumed that 5 prefers the bet for which hedemands the most money.

Since utility theory assumes that thesedifferent responses are all determined bythe same underlying values, it predictsthat 5s who are asked to choose one of twobets will choose the one for which theywould make the higher bid, or to whichthey would give the higher rating.

In contrast, the view of the presentauthors is that such decisions do not relysolely on expected utilities. Slovic andLichtenstein (1968) have demonstratedthat a gamble is a multidimensional stimu-lus whose various attributes have differ-ential effects on individual decision-makingbehavior. In particular, they presentedevidence that choices and attractivenessratings are determined primarily by agamble's probabilities, while bids are mostinfluenced by the amount to be won or lost.Specifically, when 5s found a bet attractive,their bids correlated predominantly withthe amount to win; when they disliked abet, the amount to lose was the primary

46

REVERSALS OF PREFERENCES IN GAMBLING DECISIONS 47

determiner. It was argued that these dif-ferences between ratings and choices onthe one hand and bids on the other demon-strated the influence of information-pro-cessing considerations upon the method bywhich a gamble is judged. In the biddingtask, 5s had to evaluate a gamble in mone-tary units; this requirement apparentlyled them to attend more to payoffs whenbidding than they did when making choicesor ratings.

The notion that the information describ-ing a gamble is processed differently forbids than for choices suggested that itmight be possible to construct a pair ofgambles such that 5 would choose one ofthem but bid more for the other. Forexample, consider the pair consisting ofBet P (.99 to win $4 and .01 to lose $1) andBet $ (.33 to win $16 and .67 to lose $2).Bet P has a much better probability ofwinning but Bet $ offers more to win. Ifchoices tend to be determined by probabili-ties, while bids are most influenced bypayoffs, one might expect that 5s wouldchoose Bet P over Bet $, but bid more forBet $. If such a reversal of orderings wereto occur, it would provide dramatic con-firmation of the notion that bidding andchoice involve two quite different processes,processes that involve more than just theunderlying utilities of the gambles. Thefollowing three experiments tested thishypothesis.

For all three experiments, the generalparadigm was first to present 5 with anumber of pairs of bets. All of the betshad positive expected value and wereviewed by 5s as bets they would like toplay. Every pair was composed of twobets with the same (or nearly the same)expected value: a "P bet," i.e., a bet witha high probability of winning a modestamount and a low probability of losing aneven more modest amount, and a "$ bet,"i.e., a bet with a modest probability ofwinning a large amount and a large proba-bility of losing a modest amount. For eachpair, 5 indicated which bet he would preferto play. After 5 had made all his choices,he then made a bid for each of the bets,

which this time were presented one at atime.

EXPERIMENT I

Experiment I was a group study com-paring choices with S bids. The 5s were173 male undergraduates who were paidfor participating; there was no actualgambling. The stimuli were 13 two-out-come bets, 6 bets with excellent odds (theP bets), and 7 bets with large winningpayoffs (the $ bets). All bets had positiveexpected values, ranging from $1.40 to$4.45. First these bets were combined into12 pairs, each of which had one P bet andone $ bet; no bet occurred more than twice.The 5s were asked to pick, for each pair,the bet they would prefer to play. Aftereach choice, 5s indicated how strongly theypreferred their chosen bet by marking one offour lines on their answer sheet; the firstline was labeled "slight" preference and thefourth was labeled "very strong" prefer-ence. The instructions suggested that thetwo intermediate lines might be labeled"moderate" and "strong." After about1 hr. of intervening work, 5s then madebidding responses to 19 singly presentedbets. The first 6 bets were intended aspractice bets and differed from those usedin the paired comparisons. The responsesto these bets were not analyzed. The next13 bets were the same as those used earlier.In the bidding instructions, 5 was told heowned a ticket to play the bet and wasasked to name a minimum selling price forthe ticket such that he would be indifferentto playing the bet or receiving the sellingprice. For both the bidding and choicetasks, 5s knew their decisions were "justimagine,"

Since the 12 pairs contained severalrepetitions of single bets, only the resultsof a subset of 6 pairs of bets, which con-tained no bets in common, are presentedhere; these bets are shown in Table 1. Theresults for the other 6 pairs were virtuallyidentical to these.

Results.—The first column of Fig. 1 showsthe results of Exp. I. The top histogramindicates that most 5s varied their choicesacross bets, choosing the P bet over the

48 SARAH LICHTENSTEIN AND PAUL SLOVIC

TABLE 1BETS USED IN EXPERIMENT I

Pair

1

2

3

4

5

6

Pbet

.99 Win $4.00

.01 Lose 1.00

.95 Win $2.50

.05 Lose .75

.95 Win $3.00

.05 Lose 2.00

.90 Win $2.00

.10 Lose 2.00

.80 Win $2.00

.20 Lose 1.00

.80 Win $4.00

.20 Lose .50

Expectedvalue

$3.95

$2.34

$2.75

$1.60

$1.40

$3.10

$ bet

.33 Win $16.00

.67 Lose 2.00

.40 Win $ 8.50

.60 Lose 1.50

.50 Win $ 6.50

.50 Lose 1.00

.50 Win $ 5.25

.50 Lose 1.50

.20 Win $ 9.00

.80 Lose .50

.10 Win $40.00

.90 Lose 1.00

Expectedvalue

$3.94

$2.50

$2.75

$1.88

$1.40

$3.10

$ bet about half the time. This does notmean, however, that the 5s felt indifferentabout their choices. The mean strengthof preference, when coded 1 ("slight"), 2,3, or 4 ("very strong"), was 2.94, with astandard deviation of .95. Most of the 5s(65%) never used the "slight" preferencerating.

The second histogram in the first columnof Fig. 1 shows that 5s were far more con-sistent in their bids: 70% of the 5s neverbid more for the P bet than for the $ betwith which it had previously been paired.

The proportion of times that the bid forthe $ bet exceeded the bid for the P bet,given that the P bet had been chosen fromthe pair, is called the "proportion of condi-tional predicted reversals" in Fig. 1. Ofthe 173 5s, 127 (73%) always made thisreversal: for every pair in which the P betwas chosen, the $ bet later received ahigher bid.

The histogram labeled "conditional un-predicted reversals" shows the proportionof times in which the bid for the P betexceeded the bid for the $ bet, given thatthe $ bet had been previously chosen. Thislatter behavior was not predicted by theauthors and is hard to rationalize underany theory of decision making. It mightbest be thought of as a result of carelessnessor changes in 5's strategy during the experi-ment. Unpredicted reversals were rare inExp. I; 144 5s (83%) never made them.

The mean strength of preference ratingwas as high when 5s made reversals as it

was when 5s were consistent, as shown inTable 2. This finding suggests that re-versals could not be attributed to indiffer-ence in the choice task.

It is clear that when S bids are comparedwith choices, reversals occur as predicted.Would the effect also hold for comparisonsof choices with B bids? There are certainconsiderations suggesting that the reversaleffect might be diminished, since the effectseems to be largely attributable to a ten-dency to overbid for $ bets but not for Pbets. For example, with the P bet, .99 towin $4 and .01 to lose $1, it is hard toimagine a bid much beyond the expectedvalue of $3.95; while with the $ bet, .33 towin $16 and .67 to lose $2, bids greatlyexceeding the expected value of $3.94 arecommon. Since 5s ask to be paid morewhen selling a bet than they pay to playwhen buying, the S bid method leads tohigher bids than the B bid method (Coombset al., 1967; Slovic & Lichtenstein, 1968).Therefore, S bidding should act to enhancethe amount of differential overbidding andthereby lead to more reversals than Bbidding.

EXPERIMENT II

The goals of Exp. II were to test thegenerality of the reversal phenomenon byusing the B bid technique, as well as tostudy the relationships between variousattributes of the bets and the reversalphenomenon by using a larger and morevaried set of stimuli.

Method.—The 5s were 74 college students run infour groups. No bets were actually played; 5swere paid by the hour. The stimuli were 49 pairs ofbets following 11 practice pairs. Contained in

TABLE 2STRENGTH OF PREFERENCE RATING GIVEN

TO CHOICES IN EXPERIMENT I

Bid more for:

Bet chosen

p$

P bet

X

3.062.78

SD

.93

.94

$bet

X

3.102.78

SD

.91

.91

REVERSALS OP PREFERENCES IN GAMBLING DECISIONS 49

EXPERIMENT I

6 Bet Poire173 Subject!

S*l«no MtlhodImaginary Payoff

0 1 2 3 4number of bet pairsCHOSE P-BET

5 6

EXPERIMENT II

49 Btt Pairs

66 Sub/setsBuying Method

Imaginary Payoff

0 7 14 21 28number of bet poirs

EXPERIMENT III

6 Bet Poirs14 Sgb/ecliSelling Method

Real Payoff

0 1 2 3 4 5 6number of bet poirs

0 1 2 3 4 5 6number of bel pairs

BID MORE FOR P-BET

0 7 14 21 28 35 42 49number of bet pairs

0 1 2 3 4 5 6

number of bet pairs

f .2

0 .1 .2 .3 .1 .5 .6 .7 .8 .9 IP 0 .I .2 .3 .4 .5 .6 .7 .8 .9 I.Oproportion of reversals proportion of reversals

CONDITIONAL PREDICTED REVERSALS0 _

0 .I .2 .3 .4 .5 .6 .7 .8 ;9 IDproportion of reversals

0 . 1 . 2 , 3 .4 5 .6 .7 .8 .9 1.0proportion of reversalsCONDITIONAL UNPREDICTED REVERSALS6.

O .1 .2 .3 .4 .5 .6 .7 .8 .9 Ifl

proportion of reversals

0 .1 .2 ,.3 .4 .5 .6 .7 .8 .9 1.0

proportion of reversals

Given that the P-bet was chosen, proportion of times the $-btt bid was larger than the P-bet bid.

^Given that the S-bet was chosen, proportion of times the P-bet bid was larger than the $-bot bid.

FIG. 1. Summary of results for three experiments.

so SARAH LICHTENSTEIN AND PAUL SLOVIC

these pairs were 88 different single bets. These88 bets, following 10 practice bets, constituted thebidding stimuli. Results from the practice stimuliwere excluded from the analyses. The bets in eachpair were equated in expected value. Each P bethad a winning probability from T% to fj; the proba-bility of winning for the $ bet ranged from rV toV^-. The $ bet always had a larger amount to winthan the P bet. With few exceptions, the winexceeded the loss in a given bet.

The bets were expressed in dollars and cents;the winning amount ranged from 10i to $10; thelosing amount from lOji to $3.70; the expected valueranged from lOfi to $3. A typical bet pair lookedlike this: P bet, H to win $1.00 and A to lose$0.20; $ bet, A to win $9.00 and A to lose $2.00.

The bets were chosen in an attempt to representa variety of P bets and $ bets. Thus, the winningamount in the $ bet always exceeded the winningamount in the P bet, but the ratio of the former tothe latter ranged from 1.3 to 100. The differencebetween the amount to lose in the $ bet and theamount to lose in the P bet varied from —$3.00 to$2.80.

The 5s were first briefly instructed in the choicetask. They were asked to choose, from each pair,which bet they would prefer to play. After choosing,the 5s turned to the bidding task. Instructions forthe buying method of bidding emphasized "thehighest price you would pay to play it. ... Askyourself each time, 'Is that the most money I wouldpay to play that bet?' "

Results.—Comparison of the second col-umn with the first column of Fig. 1 showsthere were fewer predicted reversals(Kruskal-Wallis analysis of variance byranks, X2 = 82.73, p < .01) and more un-predicted reversals (Kruskal-Wallis analy-sis of variance by ranks, X2 = 87.66,p < .01) for B bids (Exp. II) than forS bids (Exp. I). As expected, the B bidsin Exp. II were lower, relative to theexpected values of the bets, than the S bidsin Exp. I. Bids for the P bets averaged 7^below expected value in Exp. I, but 44^below expected value in Exp. II, t = 5.97,p < .01. Bids for the $ bets were $3.56higher than expected value in Exp. I, but4f* below expected value in Exp. II,t = 12.98, p < .01. These results indicatethat the B-bid technique serves to dampenthe tendency towards gross overbidding for$ bets, and hence to reduce the rate ofpredicted reversals. In addition, since bidsfor $ bets are closer in range to bids for Pbets in Exp. II, even fairly small fluctua-tions in bidding could more easily produce

an increase in the occurrence of unpredictedreversals, as observed. Nevertheless, 46 ofthe 66 5s (p < .01) had a higher rate ofconditional predicted reversals than condi-tional unpredicted reversals.

The 49 pairs of bets used in this experi-ment were constrained by the requirementsthat all P bets had high probability ofwinning a modest amount, while all $ betshad low to moderate probability of winninga large amount. Nevertheless, there weredifferences in the degree to which individualpairs of these bets elicited predicted re-versals. Despite the constraints, there wassufficient variability within some of thecharacteristics of the bets to permit analysisof their relationship to 5s' bids and choices.This analysis indicated that the differencebetween the amount to lose in the $ bet andthe amount to lose in the P bet correlated.82 across the 49 bet pairs with the numberof 5s who chose the P bet. Thus, when theamount to lose in the $ bet was larger thanthe amount to lose in the P bet, 5s chosethe P bet 73% of the time. But when thereverse was true, the P bet was chosen only34% of the time. This loss variable had nodifferential effect upon bids.

Variations in amount to win, on the otherhand, affected bids but not choices. Theamount to win in the $ bet was alwayslarger than the amount to win in the P bet,but the ratio of the two winning amountsvaried. This win ratio correlated .55 acrossthe 49 bet pairs with the number of 5swho bid more for the $ bet than for itspreviously paired P bet. The win ratiodid not correlate (r = — .03) with thenumber of 5s who chose the P bet.

These results, that variations in amountto lose affected choices but not bids, whilevariations in amount to win affected bidsbut not choices, are further evidence thatdifferent modes of information processingare used in bidding and choosing.

The probabilities of winning across the49 bet pairs in Exp. II had very narrowranges (•& to ti in the P bets, rV to -fsin the $ bets) and had no differential effectson the frequency of reversals.

The probability of observing a predictedreversal increases both when the probability

REVERSALS OF PREFERENCE'S IN GAMBLING DECISIONS 51

of S choosing the P bet increases and whenthe probability of 5 bidding more for the$ bet increases. This, together with thecorrelational information presented above,implies that the ideal bet pair for observingreversals would have a larger $ bet lossthan a P bet loss (facilitating choice of theP bet), and a large $ bet win relative to theP bet win (facilitating a larger bid for the $bet). In fact, in this experiment, the betpair which had the most predicted reversals(40 of 66 5s reversed) had just these char-acteristics: P bet, •&• to win $1.10 andT3Tr to lose $ .10; $ bet, A to win $9.20 andA to lose $2.00.

EXPERIMENT III

The purpose of Exp. Ill was to testwhether the predicted reversals wouldoccur under conditions designed to maxi-mize motivation and minimize indifferenceand carelessness. Lengthy and carefulinstructions were individually administeredto 14 male undergraduates. The bets wereactually played, and 5s were paid theirwinnings.

The stimuli were the 12 bets, 6 P. betsand 6 $ bets, shown in Table 3. Theprobabilities were expressed in thirty-sixths, so that a roulette wheel could beused to play the bets. The amounts towin and lose were shown as points (rangingfrom SO to 4,000), and a conversion tomoney was established such that the mini-mum win was 80(£ (even for 5s who had anet loss of points), while the maximum winwas $8.00. The concept of convertingpoints to money, and the minimum andmaximum win, were explained to 5s at thebeginning of the experiment. However,the actual conversion curve was not re-vealed until the experiment was over.

Each 5 was run individually. After sixpractice pairs, 5 chose his preferred betfrom each of the six critical pairs threeconsecutive times, with a different order ofpresentation and top-bottom alignmenteach time. The E kept a record of 5'schoices. After 5 had responded to eachpair twice, this fact was pointed out tohim. The E told 5 that he would see these

TABLE 3BETS USED IN EXPERIMENT III

Pair

1

2

3

4

5

6

P bet

35/36 Win 4001/36 Lose 100

34/36 Win 2502/36 Lose 50

34/36 Win 3002/36 Lose 200

33/36 Win 2003/36 Lose 200

29/36 Win 2007/36 Lose 100

32/36 Win 4004/36 Lose 50

Expectedvalue

386

233

272

178

142

350

*bet

ll/J6Win 160025/36 Lose 150

14/36 Win 85022/36 Lose 150

18/36 Win 65018/36 Lose 100

18/36 Win 50018/36 Lose 150

7/36 Win 90029/36 Lose 50

4/36 Win 400032/36 Lose 100

Expectedvalue

385

239

275

175

135

356

same six pairs one last time, that E wouldremind 5 of which bet he had preferred oneach of the first two times and ask 5 forhis final, binding decision. The E empha-sized that his interest was in obtaining 5'scareful and considered judgments and that5 should feel free to change his decision if,by so doing, he would reflect his true feel-ings of preference for one bet over theother. Then, as 5 looked at each pair, Ewould report either "The first two times,you said you would prefer to play Bet A(Bet B); do you still prefer that bet?" or"You chose Bet A once and Bet B once.Which bet do you really want to play?"It was emphasized that this final choicewould determine which bet 5 wouldactually play.

Following this final round of choices, 5was instructed in making S bids. Thisexplanation was far more complex than thatused in Exp. I and included all the persua-sions discussed by Becker et al. (1964).These instructions were designed to con-vince 5 that it was in his best interest tobid for a bet exactly what that bet wasworth to him. The 5 was told that Ewould choose a counteroffer against whichto compare 5's price by spinning theroulette wheel and entering the number soobtained in a conversion table speciallydesigned for each bet. The conversiontable was a list of the 36 roulette numberswith a counteroffer associated with eachnumber. If the counteroffer was equal to,or greater than, 5's previously stated bid,E would buy the bet from 5, paying 5 the

52 SARAH LICHTENSTEIN AND PAUL SLOVIC

TABLE 4PROPORTIONS OF CHOOSING AND

BIDDING RESPONSES

Bet chosen

P

»

Expected under the "null model" : bid more for:

P bet $ bet

pr's' + p'rsa

cprs' + p'r's

ps' + p's

pr's + P'rs1

b

dprs + p'r's'

ps + p's'

pr' + p'r

pr + p'r'

1

Observed

Exp. I

.085a

c.031

.116

.425b

d.459

.884

.510

.490

1

Exp. II

.261a

c.127

.388

.271b

d.341

.612

.532

.468

1

Exp. Ill

.250a

.048

.298

.321b

d.381

.702

.571

.429

1

amount of the counteroffer. If the counter-offer was smaller than 5's price, no salewould be made and 5 would play the bet.The counteroffer tables were constructedon the basis of previous bids for similarbets, with a range chosen to include mostof the anticipated bids. The values of thecounteroffers can influence the expectedvalue of the game as a whole, but they donot affect the optimal response strategy,a fact which was pointed out to 5s.

Further discussion of the technique em-phasized two points: (a) The strategy thatS should follow in order to maximize hisgain from the game was always to name ashis price exactly what he thought the betwas worth to him. (b) A good test ofwhether 5's price was right for him was toask himself whether he would rather playthe bet or get that price by selling it. Theprice was right when S was indifferent be-tween these two events. The E thenpresented several (up to 12) practice trials.These practice trials included the completeroutine: S stated his selling price, R ob-tained a counteroffer, and the bet waseither sold or played.

The 12 critical bets were then presentedthree times successively. However, theplaying of the critical bets (including selec-tion of a counteroffer) was deferred until 5had stated his price for all bets. On the

third presentation, while S was studying aparticular bet, but before 5 had stated hisprice, E told 5 what his price had beenfor the first two times and urged him nowto reconsider and name the final, bindingprice to be used later in playing the game.

After these decisions, S played the game.First, he played his preferred bet for eachof the six pairs of bets. Then the bidswere played, starting with the selection by£ of a counteroffer and ending with eithersale or play of the bet for all 12 bets. The5 kept track of his own winnings or lossesof points, which were, at the end, convertedto money.

Results.—All data analyses were basedonly on the third, final choices and bids5s made. Results from these carefullytrained and financially motivated 5s givefurther credence to the reversal phenome-non. As shown in Column 3 of Fig. 1,six 5s always made conditional predictedreversals and five 5s sometimes made them.Unpredicted reversals were rare.

Is it possible that the reversals in allthree experiments resulted solely from theunreliability of 5s' responses? This hy-pothesis can be examined by assuming thatthe probability that 5 truly prefers the Pbet in the choice task is equal to the proba-bility that he truly values the P bet morein the bidding task. Call this single param-eter p, and let p' = 1 — p. Suppose fur-ther that because of unreliability of re-sponse, 5 will report the opposite of histrue preference with a probability of r(r' = 1 — r) in the choice task and willreverse the true order of his bids with aprobability of 5 (sr = 1 — s) in the biddingtask.

Under this "null model," 5 will choosethe P bet while bidding more for the $ betif he truly prefers the P bet and respondswith his true preference in the choice taskbut responds with error in the biddingtask, or if he truly prefers the $ bet andresponds with error in the choice task buttruly in the bidding task. The probabilityof observing this response is thus: pr's+ p'rs'. The probabilities of all possibleresponses, constructed in similar fashion,are shown in Table 4.

REVERSALS OF PREFERENCES IN GAMBLING DECISIONS 53

These expected probabilities can be com-pared with the proportions actually ob-tained. For each experiment, three of theseproportions are independent and yield threeequations in the three unknowns, p, r, ands; these equations may be solved for p.In general, if the actual cell proportionsare a, b, c, and d as shown in Table 4,solving for p yields the equation:

PP' =ad — be

(b + c) - (a + d)'

For the three experiments here reported,the obtained values for pp' were .295, .315,and .270 respectively. All of these yieldonly imaginary solutions for p. However,they are all close to the maximum possiblereal value for pp', .25, which would implythat p = .5. When p = .5 is substitutedinto the expressions in Table 4, then regard-less of the rates of unreliability, r and 5,all the following conditions must hold: (a)All marginals must be equal to .5; (b) Cella must equal Cell d ; (c) Cell b must equalCell c.

These three conditions are not indepen-dent; only the last, that Cell b must equalCell c, was subjected to statistical test.For the data shown in Table 4, McNemar'stest for correlated proportions yieldedX2 = 338.27 for Exp. I, X2 = 167.29 forExp. II, and X2 = 15.61 for Exp. I l l ; forall, p < .01. Since the "null model" canin no way account for the data, it is reason-able to reject the notion that unreliabilityof response is the sole determiner of thereversal effects.

Postexperimental interviews.—The 5s inExp. Ill who gave predicted reversals wereinterviewed at the end of the experiment inan effort to persuade them to change theirresponses. The inconsistency of the re-sponses was explained to S. If 5 was ini-tially unwilling to change any responses, E'sinterview comments became more and moredirective. If pointing out that 5's patternof responses could be called inconsistentand irrational did not persuade S, E ex-plained to S a money-pump game by meansof which E could systematically and con-

tinuously get 5 to give E points withoutever playing the bets.3 This was intendedto illustrate to 5 the consequences of hisresponses. After S understood the natureof the money-pump game, he was againurged to resolve the reversal.

Eleven of the 14 5s were interviewed.Of these, 6 5s changed one or more re-sponses after only a little persuasion, 3changed only after the money-pump gamewas presented to them, and 2 5s refused tochange their responses at all, insistingthroughout the interview that their originalresponses did reflect their true feelingsabout the bets.4

Comments by 5s supported the authors'contention that 5s process the bet informa-tion differently in the two tasks. Some 5sshowed this indirectly, by mentioning onlythe probabilities when discussing choiceswhile focusing on the amount to win andentirely disregarding the amount to losewhen discussing bids.

Other 5s explicitly stated they were usingdifferent processing methods in choosing andbidding. For example, one 5, L. H., triedto justify his bid for the $ bet of Pair 5 bynoting that his bid was less than half of theamount to win. When E pointed out tohim that his chance of winning was muchless than one-half, he replied,

7 against 29 . . . I 'm afraid I wasn't thinking of the7 to 29 [the winning odds], I was thinking of it interms of relative point value . . . In other words,I looked at the 29 and the 7 [when I was choosing]but as for bidding, I was looking at the relativepoint value, which gave me two different perspec-tives on the problem.

3 Paraphrased:Suppose I own both bets and you own some

points. You have said that the $ bet is worth Xpoints to you. Are you willing then to buy the$ bet from me for X points? OK, now you ownthe $ bet and I have X of your points. But yousaid you really would rather play the P bet thanthe $ bet. So would you like to trade me the $bet for the P bet which you like better? OK, noware you willing to sell me the P bet for Y points,which is what you told me it is worth? OK, nowI have both bets back again and also I now have(X — Y) of your points. We are ready to repeatthe game by my selling you the $ bet again forX points.4 An edited transcript of one of these 5s is avail-

able from the authors.

54 SARAH LICHTENSTEIN AND PAUL SLOVIC

Subject L. F. said,I don't arrive at the evaluation . . . quite the sameway that I arrive at the preferable bet. And there'ssome inconsistency there. Now, whether in aparticular case this leads to an inconsistency maybe'snot too important. It's just that they're two dif-ferent models . . . I imagine that shows that themodels aren't really accurate, but in terms of justbetting, they're sound enough. I wouldn't changethem.

Subject M. K. said,

You see, the difference was, that when I had to pickbetween [the P bet] and [the $ bet] I picked thebet which was more sure to produce some pointsfor me. When I was faced with [the $ bet alone],my assessment of what it was worth to me, youknow, changed because it wasn't being comparedwith something.

DISCUSSION

In three experiments, 5s frequently choseone bet from a pair of bets and subsequentlybid more for the bet they did not choose. Thefrequency of such reversals varied somewhatas experimental conditions changed, but wasalways far greater than could be explained byunreliability alone. Similar results have re-cently been found by Lindman (1971).

These reversals clearly constitute incon-sistent behavior and violate every existingtheory of decision making. For example,subjectively expected utility theory (Edwards,1955) postulates both a subjective probabilityfunction and a subjective worth function, butdoes not allow either function to change itsshape as the response mode changes. Bidsand choices should both be predictable fromthe same functions; reversals are thereforeimpossible under the model.6

The present results imply that attempts toinfer subjective probabilities and utility func-

6 This statement is not strictly true for B bids.When S offers a B bid for the bet, the utility of thisbid cannot be directly equated to the expectedutility of the bet. Rather, when a bid of b is givento a bet with outcomes X and Y, the utilities ofthe quantities (X — b) and (Y — b) are relevant.In a choice situation, however, the utilities of X andY|are relevant. Thus reversals could occur withsuitably chosen utility curves (Raiffa, 1968, pp.89-91). Utility theory does not, however, help oneunderstandjwhy there were more predicted reversalsin the present study with S bids (where reversalsare normatively impossible) than with B bids (wherereversals are normatively permitted).

tions from bids or choices should be regardedwith suspicion. Since these subjective factorsare not the sole determiners of decisions, infer-ences about them are likely to vary from oneresponse mode to the next, because of changesin the decision processes used by 5s.

What are the different decision processesthat underly bids and choices? The mechan-isms determining choices are not clear, but itmust be kept in mind that in paired compari-sons each attribute of one bet can be directlycompared with the same attribute of the otherbet. There is no natural starting point, andS may be using any number of strategies todetermine his choice. Tversky (1969) hasgiven several examples of such strategies.

In contrast, bidding techniques provide anobvious starting point: the amount to win.Evidence from the present experiments, aswell as from the previous study by Slovic andLichtenstein (1968), indicates that the S whois preparing a bidding response to a bet hesees as favorable starts with the amount towin and adjusts it downwards to take intoaccount the other attributes of the bet. Theamount to win translates directly into anamount to bid. However, the probabilities ofwinning and losing, presented in probabilityunits, are more difficult to translate intomonetary units. In trying to adjust his bidto take into account this more complex infor-mation, 5 is not very precise. Some 5s simplysubtract a constant from the amount to win.Others multiply-the amount to win by theprobability of winning, entirely disregardingthe amount to lose. All such schemes producebids that are highly correlated with amount towin but poorly reflect the variations in proba-bilities and amount to lose.

The P bets of the present study offered suchhigh probabilities of winning that their modestwinning amounts needed to be adjusted down-ward only slightly to take the other factorsinto account when bidding. In contrast, the$ bets offered only a modest chance to win alarge amount of money. Thus 5s should havemade a sizable downward adjustment of theamount to win when bidding for these bets.Their failure to do so led them to bid more forthe $> bet than for the corresponding P bet.This, in turn, led to bids which often did notmatch their choices. Predicted reversals arethus seen as a consequence of the differingevaluation techniques used in the two tasks.

REVERSALS OF PREFERENCES IN GAMBLING DECISIONS 55

Is the behavior of 5s who exhibit reversalstruly irrational? Tversky (1969) posed asimilar question about 5s in whom he hadinduced systematic intransitivities, and forthe following reasons answered it in the nega-tive. He noted that it is impossible to reachany definite conclusion concerning humanrationality in the absence of a detailed analysisof the cost of the errors induced by the strate-gies 5 follows as compared with the cost to 5of evaluating alternative strategies. The ap-proximations 5s follow in order to simplifythe difficult task of bidding might prove to berather efficient, in the sense that they reducecognitive effort and lead to outcomes not toodifferent from the results of optimal strategies.In using such approximations, the decisionmaker assumes that the world, unlike thepresent experiments, is not designed to takeadvantage of his approximation methods.

In sum, this study speaks to the importanceof information-processing considerations toooften neglected by decision theorists. Thereversal phenomenon is of interest, not be-cause it is irrational, but because of the in-sights it reveals about the nature of humanjudgment and decision making.

REFERENCES

BECKER, G. M., DEGROOT, M. H., & MARSCHAK, J.Measuring utility by a single-response sequentialmethod. Behavioral Science, 1964, 9, 226-232.

COOMBS, C. H., BEZEMBINDER, T. G., & GOODE,F. M. Testing expectation theories of decisionmaking without measuring utility or subjectiveprobability. Journal of Mathematical Psychology,1967, 4, 72-103.

EDWARDS, W. The prediction of decisions amongbets. Journal of Experimental Psychology, 1955,51, 201-214.

LICHTENSTEIN, S. Bases for preferences amongthree-outcome bets. Journal of ExperimentalPsychology, 1965, 69, 162-169.

LINDMAN, H. R. Inconsistent preferences amonggambles. Journal of Experimental Psychology,1971, 89, in press.

RAIFFA, H. Decision analysis: Introductory lectureson choices under uncertainty, Reading, Mass.:Addison-Wesley, 1968

SLOVIC, P., & LICHTENSTEIN, S. Relative import-ance of probabilities and payoffs in risk taking.Journal of Experimental Psychology, 1968, 78,(3, Pt. 2).

TVERSKY, A. Utility theory, additivity analysis,and decision making under risk. Journal ofExperimental Psychology, 1967, 75, 27-36.

TVERSKY, A. Intransitivity of preferences. Psy-chological Review, 1969, 76, 31-48.

(Received October 1, 1970)