Stimulus complexity and choice inconsistency among gambles

ORGANIZATIONAL BEHAVIOR AND HUMAN PERFORMANCE

21, 146-159 (1978)

Stimulus Complexity and Choice Inconsistency among Gambles H. R.

L1NDMAN AND JAMES LYONS

Indiana University Tversky (Psychological Review, 1969, 76, 31-48) found that some subjects can be induced to make intransitive choices among gambles. The gambles were so designed that for some pairs the subjects were inclined to choose on the basis of monetary values, while for others they were inclined to choose on the basis of the probabilities. This study replicates and expands Tversky's. The two principal findings are: (1) Tversky's results can be replicated using unselected subjects. (2) If subjects are forced to be transitive, by requiring them to evaluate the bets individually on a monetary scale, most resolve the intransitivity in favor of maximizing expected value.

Laboratory research on decisions under uncertainty (e.g., gambling) has traditionally emphasized what may be called the strictly economic aspects of such decisions (Edwards, 1954, 1961; Edwards, Lindman, & Phillips, 1965). A large body of research exists, for example, testing the subjectively expected utility maximization (SEU) decision model. The SEU model is derived directly from economics (Edwards, 1954) and assumes, essentially, that subjects reach decisions by optimally combining the relevant probabilities and utilities. Apparently unexplained differences in choices are assumed to be due to varying assessments of these probabilities and utilities rather than to other factors. Alternative models of choice (e.g., Coombs and Huang, 1970) have also tended to assume that subjects maximize some economically relevant function of probabilities and outcomes (though it may not be at all similar to the SEU model). More recent research, however, has discovered choice patterns inconsistent with such strictly economic models. Lichtenstein and Slovic (1971, 1973; Slovic and Lichtenstein, 1968) and Lindman (1971) have found that monetary evaluations of gambles are sometimes inconsistent with paircomparison choices; for some pairs of gambles subjects stochastically prefer the gambles to which they have assigned the lower monetary value. Slovic and Lichtenstein explain this by adding an attentional mechanism to choices. Since the evaluation is monetary, subjects focus on the monetary aspects of a gamble when evaluating it. This attentional mechanism is not present when the subject is choosing between gambles. Requests for reprints should be sent to H. R. Lindman, Department of Psychology, Indiana University, Bloomington, Indiana 47401. 146 0030-5073/78/0212- 0146502,00/0 Copyright ~) 1978 by Academic Press, Inc, All rights of reproduction in any form reserved.

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Tversky (1969) has revealed another aspect of the relationship between attentional mechanisms and gambling. When asked to choose between two gambles that differ only slightly in probability and amount to win, subjects stochastically preferred the gamble with the larger amount to win even though that gamble had a lower expected value. When the differences were larger, however, subjects preferred the higher expected value, larger probability gamble. The phenomenon easily lent itself to generating intransitive sets of preferences. Tversky represented the probabilities as pie sections of circles; he conjectured that subjects paid less attention to probabilities that differed only slightly because such probabilities were not very discriminable. (The differences were as small as 1/24.) The monetary outcomes, on the other hand, were always discriminable because they were printed as numerical values. In summary, in the studies of Lichtenstein and Slovic and of Lindman, subjects appeared to be focusing on amounts because of the nature of the responses; in the study of Tversky, subjects appeared to focus on amounts because they were more discriminable. The present study replicates Tversky's results with two differences. First, unselected subjects were used, in contrast to Tversky's deliberate selection of subjects for their intransitive tendencies. Second, each subject made both pair-comparison choices, as in Tversky's study, and singlestimulus evaluations as in the Lichtenstein and Slovic and Lindman studies. Since single-stimulus evaluations cannot be intransitive, the evaluations must be inconsistent at some point with the pair-comparison choices. If subjects are actually focusing primarily on money, they should resolve the problem by consistently placing higher evaluations on gambles with larger amounts to win but smaller expected values. The study differs from those of Lichtenstein and Slovic and Lindman in two ways. First, comparisons are made across gambles with differing expected values. In the earlier studies, comparisons were made only among gambles with the same expected value. Second, the gambles are simpler. There is only one nonzero outcome in each gamble. In the earlier studies the gambles had at least two nonzero outcomes. Thus, in the earlier studies, subjects had to consider at least four quantities--the win, the loss, and both probabilities--in each gamble. In the present study they had to consider only two--the win and the probability of winning. METHOD

Subjects. The subjects were 42 Indiana University undergraduates of both sexes who participated to fulfill partially an experimental requirement in introductory psychology. Stimuli. Twenty-five two-outcome gambles were constructed using all possible combinations of the five probabilities and five amounts shown in Table 1.

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LINDMAN AND LYONS TABLE l PROBABILITIES, AMOUNTS, EXPECTED VALUES(UPPER CELL VALUES), A N D MEAN OBTAINED ASKING PRICES (LowER CELL VALUES) OF GAMBLESa Amount ($) Probability 11/24 10/24 9/24 8/24 7/24

4.00

4.25

4.50

4.75

5.00

1.83 2.18 1.67C,a 2.13 1.50 1.78 1.33 1.56 1.17~ 1.76

1.95 2.62 1.77 2.03 1.590 1.77 1.42 1.67 1.24 1.44

2.06 2.69 1.87 2.12 1.69 2.08 1.50e 1.75 1.31 1.49

2.18 2.56 1.98 2.41 1.78 2.22 1.58 1.79 1.39 1.73

2.290 3.00 2.08 2.49 1.88 2.20 1.67 1.94 1.46 2.02

o Pair comparisons were between gambles on the same diagonal. b These gambles appeared only in the practice asking-price sessions. This pair was presented twice in the pair-comparison task. a This pair was not presented in the pair-comparison task.

In each gamble, the subject either won the specified amount, with the specified probability, or broke even, with the complementary probability. In no case could the subject ever lose. These gambles are identical to those used by Tversky. The gambles were presented to the subjects as pie diagrams, with the winning sector at the top. The amount to be won in each case was printed inside the sector. Consider the seven diagonals in Table 1, from upper left to lower right. Proceeding down any diagonal, the amounts to win increase, but both the probabilities of winning and the expected values decrease. For the paircomparison choices the gambles within each diagonal were paired in all possible ways, for a total of 30 pairs. Through a printing error, however, one pair, the (8/24, $4.50; 10/24, $4.00) pair was not presented. Another, the (9/24, $4.25; 10/24, $4.00) pair was presented twice. Gambles in different diagonals were never paired. The pairs were arranged randomly in booklets, with three pairs on a page. Subjects indicated their preferences by checking the preferred gamble. For the single-stimulus evaluations~ the 23 gambles, excluding those at the lower left- and upper right-hand corners of Table 1 (which did not occur in any pairs) were arranged randomly, three to a page, in booklets. Above the representation of each gamble, a vertical line was drawn and marked in $. 10 steps from zero to the winning amount. Subjects indicated the value of each gamble by marking the vertical line at the appropriate spot and then writing in the value next to the mark.

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At the beginning of the single-stimulus booklet were four practice gambles to acquaint the subjects with the procedure. The first two practice gambles involved amounts and probabilities different from those of any gambles used in the study; the second two were those in the lower leftand upper right-hand corners of Table I. Procedure. The subjects were run in nine groups with from three to six subjects per group. In five of the groups (25 subjects) the evaluations came first, and in the other four (17 subjects) the choices came first. Poker chips were used to represent money, and at the end of each task the total winnings of each subject were noted so that the session had somewhat of a contest atmosphere. No actual money changed hands. In the choice task, the subjects chose among each of the three pairs on a page; then one of the pairs was selected, at random, and each subject played his chosen bet from that pair. Thus only one gamble was played from each set of three pairs. The evaluation task used the selling procedure first described by Becker, DeGroot, and Marschak (1964). The subject was required to give his "asking price," defined as the lowest price for which he would be willing to sell the privilege of playing the bet. The experimenter then gave a "bid." If the "bid" was larger than the asking price, the subject received the "bid" and did not play the bet; if the "bid" was lower than the asking price, the subject played the bet. In the experiment, subjects gave asking prices for all three gambles on each page. Then a randomly chosen bid was made for a randomly chosen gamble, and the subject either played that bet or received the bid. The bets were played by means of a spinner and a disk colored red for the zero outcome and blue for the winning outcome. The disks were of transparent plastic, and the disks and spinner fit on the stage of an overhead projector, so that the spinning of the spinner was projected on a wall where it was easily visible to all of the subjects. RESULTS Choices

No systematic differences were found between the two orders in which the tasks were given, so the results presented will be from the combined groups. There were 15 pairs of gambles such that the two amounts of money differed by only $.25, and the two probabilities differed by only 1/24. We will call these "one-step" pairs. Among these 15 pairs, the proportion of subjects choosing the gamble with the higher expected value (lower winning amount) varied from. 12 to .50, with a mean of only .30. Although the variation appears relatively large, note that for not a single one of the 15 pairs was there a stochastic preference for the alternative with the higher

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expected value. Since no consistent relationship between type of pair and strength of preference was found, the data from the 15 one-step pairs were combined for subject-by-subject analyses. In these, only six subjects preferred the higher expected value gambles more often than the lower expected value gambles; one of these preferred the higher expected value 10 times out of 15, and the other five preferred it only eight times. Three subjects never chose the higher expected value gambles. The same analysis was done on the 10 two-step pairs (amounts differ by $.50, and probabilities differ by 2/24), the four three-step pairs and the single four-step pair. The results are summarized in Table 2. The data show a definite increasing preference for the higher expected value gamble as the number of steps between the gambles of a pair increases. TABLE 2 SUMMARY OF CHOICE DATAa

Proportion of subjects choosing larger expected value (N=42) Pair

Number of pairs

Min

Max

Mean

One step Two steps Three steps Four steps

15 10 4 1

.12 .43 .62 (.83)

.50 .74 .76 (.83)

.30 .57 .67 .83

Columns labeled Min and Max contain the minimum and maximum proportions of choices for different pairs of gambles in that set. These group data, from unselected subjects, agree quite well with Tversky's data obtained from selected subjects; subjects cannot prefer the higher expected value gambles on the four-step pairs and simultaneously prefer the lower expected value gambles on the one-step pairs without violating transitivity. A rough Bayesian analysis (Edwards, Lindman, & Savage, 1963) was performed to estimate the number of subjects that were intransitive. (Derivations of this and the Bayesian test of asking prices are in the Appendix.) Five hypotheses were analyzed: (a) The Subject was transitive and stochastically preferred the higher expected value; (b) the subject was transitive and stochastically preferred the lower expected value, (c) the subject was intransitive, preferring the lower expected value on one-step pairs and the higher expected value on four-step pairs, with the change in preference occurring somewhere in between; (d) the subject was intransitive in the direction opposite to that of hypothesis c, (e) the subject responded in a pure chance manner with probability of 1/2. The analysis was performed under the following assumptions: (1) All five

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hypotheses were equally likely a priori; (2) if the subject was intransitive his change of preference was equally likely to occur (a priori) at any of the three possible transition points; (3) the subject's choices were independent; (4) the subject's probability of choosing his preferred gamble had a prior distribution with a mean of 3/4. This is consistent, for example, with a uniform prior over the range .50 to 1.00. While these assumptions are arbitrary, they are not unreasonable, and the Bayesian analysis was relatively insensitive to moderate changes in the assumptions. Subjects were classified as definitely supporting a hypothesis if the posterior probability of that hypothesis was at least twice as large as that of the second most likely hypothesis. Thirty three subjects were unambiguously classified in this way, and Table 3 gives the number satisfying each hypothesis. In addition, for eight of the nine subjects that could not be unambiguously classified, hypothesis c was one of the two most likely, and it was the most likely hypothesis for seven of these. It appears that two-thirds or more of the subjects in this study showed intransitive preferences in the predicted direction. TABLE 3 NUMBERS OF SUBJECTS SATISFYING EACH CHOICE HYPOTHESIS

Hypothesis a. b. c. d. e.

Number definitely satisfying

Subject transitive, prefers higher EV Subject transitive, prefers lower EV Subject intransitive, predicted direction Subject intransitive, opposite direction Subject random, p = 1/2

2 4 22 0 5

Evaluations

The geometric mean value assigned to each gamble is shown in Table 1 and Fig. 1, along with a line of best fit to be explained below. Gambles that lie on the same diagonal are connected with solid lines. The values were analyzed according to the following model: xi~ = e ~ p~l+~r mj~+~e~J+t3J+~o

,

where x~j is the value assigned to the gamble with probability p~ of winning ms; K is an " o p t i m i s m " parameter, representing an amount by which the subject increases his asking price for all gambles; 7r and ~ represent the relative attention paid to the probability and amount, respectively, and o~,/3j, and y~j represent the extent to which the power model in the first part of the function cannot fit the data. In addition, the complete model had similar terms representing differences in these parameters for different subjects. The model looks complicated, but if logarithms are taken on both sides, it becomes a straightforward, linear model, analyzable by

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3.0

l

2.8

2.6

2,4

2.2

2.0

1.8

1.6

x /

/,

,_

~.2

14

~'~E'V G

2:0

2:2

Flc. 1. Mean asking prices as a function of expected value.

standard analysis of variance techniques (Scheffe, 1959). The logarithmic model can be written: log(x~j) - log(p0 - log(mj) = log(K) + log(p0 + tz log(m r) + o~ + flj + Yu , where log(K) = the grand mean; 7r = linear trend term in log (P0;/x = linear trend term in log (mr); o~,/3~, yo = other main effects and interactions. The results of the analysis are shown in Table 4. (Tests of individual subject effects could not be made for lack of suitable error terms.) Although the o~ and Yu are significant, they represent only a tiny percentage of the total variance. TABLE4 ANALYSIS OF VARIANCE ON LOG ASKING PRICES

Effect

Estimate

F

df

a

&2

K

.1836 .0540 .1111

12.82 0.12 1,05 3.89 0.24 1.78

I/8 1/8

,01 --

.159 .000

l/8

--

.000

3/24 3/24 14/128

.05 -.05

.001 .000 .002

Yo

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The only general term accounting for a reasonable percentage of the variance is ~, the general optimism parameter. This suggests that an appropriate model might be x~ = k E V ~ .

That is, the asking price can be characterized as a constant multiplied by the expected value of the gamble. The value of the constant varied somewhat from subject to subject and from gamble to gamble within subjects. A good general model would be xs~j = Ks C ~ E V .

,

where x,o is the asking price for subject s on gamble Go ; Ks is a constant for the given subject; and C~o represents individual variations in response to each gamble. Ninety-five percent confidence limits for K (the average K), K,, and Cso are 1.09 <~ K ~ 1.32; .75 ~< Ks ~< 1.93; and .53 <~ Cs~ ~< 1.88. The Cso do not appear to represent any determinable systematic variation. They appear to be composed at least partly of differences between groups. (Different groups experienced different patterns of wins and losses.) These values indicate that subjects are generally optimistic in their asking prices, asking more than the expected value of the gamble. The best estimate of/~ is 1.20; the best-fitting line, X~o = 1.20 EVo, for the group data, is plotted as the dotted line in Fig. 1. A rough subject-by-subject Bayesian analysis was performed on the asking prices (see Appendix), testing three hypotheses: (a) The subject's asking prices on each diagonal are basically monotonic with the probabilities of winning (i.e., with the expected values); (b) the subject's asking prices are basically monotonic with the amounts (i.e., the inverses of the expected values); (c) the subject's asking prices are not monotonically related to either but are more or less random. The statistic was the number of pairs, T, out of the entire 30 possible pairs in the table, for which hypothesis a was confirmed, minus the number for which hypothesis b was confirmed, If we calculated Kendall' s ~-on the asking prices on each diagonal of the matrix, T would be the sum of the numerator terms in the separate ~-values. The range of T is from - 3 0 to +30. The standard deviation of T under hypothesis c can be calculated to be 6.58 (Kendall, 1970). From this and the fact that T has a practically normal distribution the standard deviation of which does not vary greatly from 6.58 unless its expected value differs considerably from zero, we can obtain likelihood ratios, using the statistical procedures in Edwards et al. (1963), comparing the hypotheses. For hypothesis a a uniform prior over the possible positive values orE(T) was used. For hypothesis b, a uniform prior over the possible negative values of E(T) was used. If the likelihood of the most likely hypothesis was at least twice as large as that of the next

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most likely, the hypothesis was assumed to be clearly supported. If the ratio was less than 2, the most likely hypothesis was assumed to be ambiguously supported. The results of this analysis are in Table 5. A number of subjects gave ambiguous or random responses. Nearly all of the remainder, however, gave asking prices that were a monotonic function of the expected values of the gambles. It must also be pointed out that those subjects who gave "random" responses may have given fairly systematic responses over the entire set of 23 gambles. The test used assessed the systematicity of responses only within the critical diagonals in which probability and outcomes varied inversely. TABLE 5 SUBjECT-BY-SuBJECT ANALYSIS OF ASKING PRICES Number of subjects satisfying hypothesis Hypothesis

Clearly

Ambiguously

Monotonic with probabilities Monotonic with amounts Random

19 3 13

3 0 4

Finally, the consistency of subjects' responses from pairs to asking prices was tested by correlating, across subjects, the values of T and the posterior probabilities of each of the five pair-comparison hypotheses. A separate Kendall's z was calculated for each of the five hypotheses. Two correlations were significant--both beyond the .01 level. The correlation with hypothesis a (choice of higher expected value) was .48, and the correlation with hypothesis b (choice of lower expected value) was -.37. Subjects who most strongly choose according to expected value also tend to set asking prices according to expected value, and the same kind of consistency occurs with subjects who choose the opposite way. The correlations are not extremely high, but as Table 3 shows, very few subjects gave high probabilities for either of these hypotheses. The three subjects that supported hypothesis a most strongly in the pairs had T values ranging from 12 to 23. The other four subjects who had probabilities higher than .20 for hypothesis a had T values in the same range. Three of the four subjects who most strongly supported hypothesis b in the pairs had T values between - 2 and -12. There are therefore consistent differences-not revealed in the analysis of variance--in the tendency of subjects to behave according to Eq. (1). In addition to differences between subjects, trial by trial differences also appeared to play a role. These were evident from comments by subjects, although no such differences are clearly revealed in the data. Such differences probably involve complex interactions between the stimulus,

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the previous outcomes, and the subject; therefore they would be difficult to analyze. However, they probably are a major factor in the nonlinearities in Fig. I. DISCUSSION These results limit the generalizability of, but do not invalidate, the theory of Lichtenstein and Slovic, that subjects concentrate on amounts of money when selecting asking prices. The difference may well be in the complexity of the task with which the subject is faced. Previous studies have used mostly gambles with two or more distinct outcomes. There were thus four quantities involved, two amounts and two probabilities. In the present study, the subject needed to concentrate on only two values, the winning amount and the winning probability. This assumes, of course, that the subject felt he could ignore probabilities associated with zero outcomes. The assumption is reasonable however; even though the subject was told nothing about how to set his asking prices, he may have guessed that if one has a 2/3 chance of winning $X, the value of the gamble is at least approximately $(2/3)X. In the choices, however, the problem was more complicated. Here there were again four quantities--an amount and a probability for each gamble--to be evaluated. Subjects simplified the problem by focusing on the dimension that seemed most relevant. When the differences were small, that dimension was money; when the differences were more obvious the dimension was probability. In short, subjects tend to simplify a decision problem before attempting to solve it. In this case, the simplification took the form of a reduction in the number of dimensions. A similar effect has probably been responsible for a number of other results in human judgment. A classic example is the combining of different kinds of information, such as grade point averages, letters of recommendation, test scores, etc., to reach decisions on graduate school admissions. On these and similar tasks, even "expert" human evaluators do poorly when compared with a simple linear model (for a review, see Slovic and Lichtenstein, 1971). Moreover, evaluators do poorly when compared with a random linear model (Dawes, 1975; Dawes and Corrigan, 1974). A likely explanation is that evaluators base their evaluations primarily on a small number of dimensions only, tending to ignore other dimensions. In most such studies only three cues usually predict more than 80% of the variance. A number of linear prediction studies have suggested that subjects discount all but a few cues. Hoffman and Blanchard (1961) and Einhorn (1971) found decreased linear consistency with increased numbers of cues. Dawes (1975) and Einhorn (1971) found that conjunctive and disjunctive models sometimes fit responses better than do linear models.

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Slovic (1966) and Wyer (1970) found that, when important cues disagreed, subjects tended to disregard one or both. Miller (1956) argued that the maximum number of items that could be simultaneously stored in short-term memory was approximately seven. Assuming that some of that storage must ordinarily be used for intermediate "calculational" results, etc., the number of stimuli that a subject can combine easily is probably much less than seven. The present study and those cited above suggest that it is closer to three. In the Lichtenstein and Slovic (1971, 1973) and Lindman (1971) studies, each gamble had four quantities--two probabilities and two outcomes. In these studies, subjects appear to have concentrated on only a few of these dimensions when making decisions. In the present study, each gamble had only two relevant cues; consequently, subjects did relatively well when responding to the gambles one at a time. When required to choose between two gambles, however, they were again faced with four cuesm two from each gamble--and most subjects responded by cuing primarily on either the probabilities or the amounts, depending on the particular pair presented. I.t is not clear whether people are unable to combine more than three cues or whether they simply don't. However, one implication is that psychologists must look to different sorts of mathematical models for cognitive tasks. The models we tend to use involve simple rules of combination with large numbers of variables. We are likely to be more sucessful with models that first systematically discard or at least discount all but a few variables and then combine the remainder. Moreover the rules for discounting and subsequent combination may not be at all simple mathematically. For example, stepwise regression procedures may appear to be appropriate for developing a behavioral model. These procedures retain only those few variables that predict the most variance. They then use these same variables in the same linear model for all predictions for which the model is relevant. The research cited above, as well as other studies, indicates that subjects are less consistent in their use of cues. For one pair of gambles, the relevant cue might be differences between probabilities. Similarly, Slovic's (1966) and Wyer's (1970) subjects changed their use of cues from one prediction to another. In short, mathematical models have tended to concentrate on processes that are simple for the modeler but difficult for the subject. Conversely, processes that are simple for the subject may turn out to be difficult to model (Einhorn, 1971). We wish to add a final note on the subject-by-subject analyses. In most psychological research, such analyses are ignored, largely because reasonably usable statistical techniques have not been available. Such analyses,

STIMULUS COMPLEXITY AND CHOICE INCONSISTENCY

]5"7

however, clearly are of value in estimating the actual number of subjects that can be characterized by the overall group results. Such analyses can be useful, even if they must be only approximate. Bayesian methods are promising for such analyses, since they allow the calculation of likelihood ratios which are comparable across hypotheses other than simple null hypotheses. The methods used in this study were not presented in detail because they are specific to these data. However, there is great potential value in the development of general Bayesian methods for subject-bysubject analyses. APPENDIX In this appendix we derive the two Bayesian tests referred to previously. The first result, on choices, is derived as follows. Let Hi be one of the five hypotheses. Then,

P(H, [D) oc P(D IHOP(Hi) ~ P(D IH,). The last proportionality results from assumption l, all five hypotheses were equally likely a priori. The constant of proportionality is easily found when the five P(D IHi) are known. For hypothesis e, pure chance,

P(ntHe) = (1/2)v , where N is the total number of choices. For the remaining hypotheses, let A5 be the hypothesis that the subject prefers the higher E V on pairs of gambles of fewer thanj steps, and that he prefers the lower E V on pairs of j or more steps. The range o f j is from 1 to 4. F o r j = 1, the subject always prefers the lower EV. Also, let P~k be the probability of picking the gamble actually chosen from the kth/-step pair. Then,

P(D IA~) = f zr P~k f(s)dS , si,k where S is the N-dimensional space of the P~k andf(s) is the prior density over S. Under assumption 2 that choices are independent,

P(D IAj) = IIf Pik f(P~k) d p~k = II p~k, ik ik the product of the expected values of the P~k. Under assumption 4/~k --- 3/4 if the subject chooses the preferred value, and/~k = 1/4 if he chooses the nonpreferred. Therefore, if vi is the number of preferred gambles, and w~, the number of nonpreferred gambles, chosen from the/-step pairs,

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j-1 Z Vi

P(DIA ~) = (3/4)i=~

4 ZWi

(1/41 i=j

Similarly, let Bj be the h y p o t h e s i s that the subject prefers the l o w e r EV g a m b l e f r o m pairs with f e w e r t h a n j steps, and the higher EV g a m b l e f r o m pairs w i t h j or m o r e steps. T h e n , by the s a m e r e a s o n i n g , j=l 2 wi

P(D [Bj) = (3/4t '='

4

~ vi

(1/4) i=j

Then 4

P(DIHa) = P(D IBm) = (3/4) i ' '

,

4

2w, P(D tHu) = P(D IA,) = (3/4) '=~ U n d e r a s s u m p t i o n 2, (A2, Aa,

A4) and (B~, B> B4) are

equally likely a priori,

SO

P(D JHc) = 1/3[P(D [B2) + P(D IBa) + P(D IB4)] , e(D IHa) = 1/3[P(D Ins) + e(D Ida) + e(D IA41]. F o r the s e c o n d analysis, on the bet values, T is n o r m a l l y distributed with m e a n O and s t a n d a r d deviation a b o u t 6.58. T h e n ,

P(H,:T) cc f(TIH~) = If(T[ O ) f ' ( O ) d O .

jf(rl

U n d e r H , : O > O,f'(O) = 1/30, 0 < O < 30, andf(]C IHa) = o)f'(O)dO, w h i c h is a b o u t 1/30 of the a r e a a b o v e zero of a n o r m a l c u r v e with m e a n T and s t a n d a r d deviation 6.58. T h e s a m e d e r i v a t i o n applies to Hb: O < 0, e x c e p t that w e find the a r e a b e l o w zero. U n d e r h y p o t h e s i s Hc: O - O,f(TIHc) is the height, at T, of a n o r m a l c u r v e with m e a n 0 and s t a n d a r d deviation 6.58.

REFERENCES Becker, G. M., DeGroot, M. H., & Marschak, J. Measuring utility by a single response sequential method. Behavioral Science, 1964, 9, 226-232. Coombs, C. H., & Huang, L. C, Polynomial psychophysics of risk. Journal of Mathematlcal Psychology, 1970, 7, 317-338. Dawes, R. M. The mind, the model, and the task, In F. Restle, R. M. Shiffrin, N. J. Castellan, Jr., H. R. Lindman, & D. B. Pisoni (Eds,), Cognitive theory, Vol. 1. Hillsdale, NJ: L. Erlbaum, 1975. Vol. I, pp. 119-129,

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Dawes, R. M., & Corrigan, B. Linear models in decision making. Psychological Bulletin, 1974, 81, 95-106. Edwards, W. The theory of decision making. Psychological Bulletin, 1954, 51, 380-418. Edwards, W. Behavioral decision theory. Annual Review of Psychology, 1961, 12, 473-498. Edwards, W., Lindman, H., & Phillips, L. D. Emerging technologies for making decisions. In F. Barron (Ed.), New directions in psychology, Vol. 2. New York: Holt, Rinehart, and Winston, 1965. Edwards, W., Lindman, H., & Savage, L. J. Bayesian statistical inference for psychological research. Psychological Review, 1963, 70, 193-242. Einhorn, H. J. Use of nonlinear, noncompensatory models as a function of task and amount of information. Organizational Behavior and Human Performance, 1971, 6, 1-27. Goldberg, L. R. Simple models or simple processes? Some research on clinical judgments. American Psychologist, 1968, 23, 483-496. Hoffman, P. J., & Blanchard, W. A. A study of the effects of varying amounts of predictor information on judgment. Oregon Research Institute Research Bulletin, 1961. Kendall, M. G. Rank correlation methods, 4th ed. London: Griffin, 1970. Lee, W. Decision theory and human behavior. New York: Wiley, 1971. Lichtenstein, S., & Slovic, P. Reversals of preference between bids and choices in gambling decisions. Journal of Experimental Psychology, 1971, 89, 46-55. Lichtenstein, S., & Slovic, P. Response-induced reversals of preference in gambling: An extended replication in Las Vegas. Journal of Experimental Psychology, 1973, 101, 16-20. Lindman, H. R. Inconsistent preference among gambles. Journal of Experimental Psychology, 1971, 89, 390-397. Miller, G. A. The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 1956, 63, 81-97. Scheff6, H. The analysis of variance. New York: Wiley, 1959. Slovic, P. Cue consistency and cue utilization in judgment. American Journal of Psychology, 1966, 79, 427-434. Slovic, P., & Lichtenstein, S. C. The relative importance of probabilities and payoffs in risk-taking. Journal of Experimental Psychology Monograph Supplement, 1968, 78, (No. 3, Part 2). Slovic, P., & Lichtenstein, S. Comparison of Bayesian and regression approaches to the study of information processing in judgment. Organizational Behavior and Human Performance, 1971, 6, 649-744. Tversky, A. Intransitivity of preferences. Psychological Review, 1969, 76, 31-48. Wyer, R. S., Jr. Information redundancy, inconsistency and novelty and their role in impression formation Journal of Experimental Psychology, 1970, 6, 111-127.