Bayes Rule as a Descriptive Model

Bayes Rule as a Descriptive Model: The Representativeness HeuristicAuthor(s): David M. GretherSource: The Quarterly Journal of Economics, Vol. 95, No. 3 (Nov., 1980), pp. 537-557Published by: Oxford University PressStable URL: http://www.jstor.org/stable/1885092 .

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BAYES RULE AS A DESCRIPTIVE MODEL: THE REPRESENTATIVENESS HEURISTIC*

DAVID M. GRETHER

Results of experiments designed to test the claim of psychologists that expected utility theory does not provide a good descriptive model are reported. The deviation from tested theory is that, in revising beliefs, individuals ignore prior or base-rate information contrary to Bayes rule. Flaws in the evidence in the psychological literature are noted, an experiment avoiding these difficulties is designed and carried out, and the psychologists' predictions are stated in terms of a more general model. The psychologists' predictions are confirmed for inexperienced or financially unmotivated subjects, but for others the evidence is less clear.

There are a number of areas of economic research for which the nature of individual decision processes is important. In addition, there are some substantive areas to which economic theory has been applied in which market forces cannot be relied upon to "discipline" the behavior of economic agents. Recently economists have begun to study the sensitivity of market equilibria to imperfect information. These studies have considered labor markets [Mortensen, 1976, and Wilde, 1977] and consumer products market [Salop and Stiglitz, 1977, and Wilde and Schwartz, 1979]. One of the conclusions which has emerged from this literature is that the properties of market equilibria are sensitive to the search strategies used by individual consumers or workers. For example, consider Salop and Stiglitz [1977] versus Wilde and Schwartz [1979]. Knowledge about individual behavior under uncertainty can also be of importance in several policy areas. Exam- ples are interventions in markets where information is incomplete [Schwartz and Wilde, 1979], the provision of insurance against natural disasters [Slovic et al., 1977b], a variety of "truth in lending" or la- beling-type issues, etc. Also, note the use of economic theory in non- market settings, e.g., the analysis of voting [Riker and Ordeshook, 1973], and legislative behavior [Fiorina, 1974]. In addition, there are situations to which economic theory of individual behavior has been applied in which no trades or arbitrage is possible, e.g., such areas as crime [Becker, 1968, and Erlich, 1975], suicide [Hamermesh and Soss, 1974], marriage [Becker, 1973, 1974], and extramarital affairs [Fair, 1978].

* Financial support by the National Aeronautics and Space Administration is gratefully acknowledged. I would like to thank my colleagues, Forrest D. Nelson, Charles R. Plott, and Louis L. Wilde for their many helpful suggestions and comments, and Steven Matthews, Brian Binger, Elizabeth Hoffman, and Gerhard Befeld for their research assistance.

? 1980 by the President and Fellows of Harvard College. Published by John Wiley & Sons, Inc. The Quarterly Journal of Economics, November 1980 0033-5533/80/0095-0537$02.10



538 QUARTERLY JOURNAL OF ECONOMICS

In a series of papers [Kahneman and Tversky (K and T), 1972, 1973; and Tversky and Kahneman (T and K), 1971, 1973, 19741, D. Kahneman and A. Tversky have suggested that individuals use certain heuristics in making decisions under uncertainty. When adopted, these heuristics (which are called availability, representativeness, and anchoring) lead to certain predictable and consistent biases in individual judgments concerning the likelihood of uncertain events. In their experiments people simply do not behave as predicted by economic theory. K and T present detailed and persuasive experimental evidence which supports their view that, in some circumstances, individuals adopt the simple rules of thumb that K and T have char- acterized. Their experiments, of course, were not designed for an economics audience, and it is not clear from the evidence they present whether or not the behavior they find could be expected to be an important factor in an economic setting. The purpose of this paper is to report the results of a series of experiments designed to test, in a way relevant to economics, one of the heuristics; namely, representativeness.

The plan of the paper is as follows. The heuristic and the nature of the evidence concerning it is discussed in Section 1. The experimental design is presented in Section II, and the results are presented in Section II. The discussion of the results and the conclusions of the paper are given in Section IV.

The final conclusion reached is that the representativeness heuristic, as interpreted here, is a good descriptive model of behavior under uncertainty for untutored and unmotivated (or at least not financially motivated) individuals. There is some evidence that with increasing experience and financial motivation individuals rely less on the rule of thumb and act more like Bayesians, but the evidence for this latter position is far from conclusive.

I K and T provide evidence concerning the representativeness

heuristic in three papers [T and K, 1971, 1973; K and T, 1972]. "A person who follows this heuristic evaluates the probability of an uncertain event, or a sample, by the degree to which- it is (i) similar in essential properties to its parent population and (ii) reflects the salient features of the process by which it is generated" [K and T, 1972, p. 4311.

An example of the representativeness heuristic is given by the following experiment reported in K and T [19731 and T and K [19741.



BAYES RULE AS A DESCRIPTIVE MODEL 539

Eighty-five subjects were given the following instructions.

A panel of psychologists have interviewed and administered personality tests to 30 engineers and 70 lawyers, all successful in their fields. On the basis of this information, thumbnail descriptions of the 30 engineers and 70 lawyers have been written. You will find on your forms five descriptions chosen at random from the 100 available. For each description, please indicate your probability that the person described is an engineer on a scale of 0 to 100 [K and T, 1973, p. 241].

Another group of 86 subjects was given identical instructions, except that the number of lawyers was changed to thirty and the number of engineers to seventy. Both groups were given the same five descriptions, one of which was the following:

Jack is a forty-five year old man. He is married and has four children. He is generally conservative, careful, and ambitious. He shows no interest in political and social issues and spends most of his free time on his many hobbies which include carpentry, sailing, and mathematical puzzles. The probability that Jack is one of the 30 [70] engineers in the sample of 100 is --% [1973, p. 241].

In addition the two groups were asked the following question:

Suppose now that you are given no information whatsoever about an individual chosen at random from the sample. The probability that this man is one of the 30 [70] engineers in the sample of 100 is --% [1973, p. 241].

Both groups of subjects gave nearly the same posterior probabilities for each of the five descriptions in spite of the substantial change in the priors. Some of the descriptions were intended to sound very much like lawyers, and others very much like engineers, so that a subject using the representativeness heuristic is expected to assign a high probability that a description is of, say, a lawyer, even though the prior probability is small. For one of the five descriptions the median estimate of the probability that the man chosen was an engineer was around 0.05 for both groups of subjects, and another of the descriptions was around 0.95. Both groups made the "correct" response to the question quoted above. Perhaps the most interesting result reported was the response to the following:

Dick is a thirty-year old man. He is married with no children. A man of high ability and high motivation he promises to be quite successful in his field. He is well liked by his colleagues [1973, p. 242].

This description was intended to be neutral, and apparently was judged so by both subjects. For both groups the median estimate was 0.50. Thus, these subjects evaluated useless information and no information quite differently.

The results of K and T certainly do support the hypothesis that individuals judge by something like representativeness and ignore




prior probabilities. The responses to the vacuous description and the fact that the subjects were reminded of the prior odds after each description was given are especially convincing. Nevertheless, this experiment has features that make the applicability of the findings to economic decisions doubtful. First, there is the difficulty of controlling the information given when verbal descriptions or situations are presented. This problem is well illustrated by the results reported by Hammerton [1973], who demonstrated that rewording mathemati- cally equivalent problems could lead subjects to provide different answers. (For a more detailed discussion see Grether [1978].)

A second potential source of difficulty is that, as is often the case in experiments, the subjects are not told the truth about the random process being examined. Clearly, the thumbnail descriptions were not a random sample from the alleged population. The subjects' responses would agree with Bayes rule only if they either "played the game" or believed the experimental instructions and thereby badly misper- ceived what was going on. Also, the use of the word "probability" in the instructions raises questions about the results. After all, the proper definition of the term has been the subject of a major debate among statisticians for years.

Finally, there is the question of incentives; it is not clear that K and T's subjects had a positive incentive to give "correct" answers. The instructions included the following statement:

The same task has been performed by a panel of experts, who are highly accurate in assigning probability to the various descriptions. You will be paid a bonus to the extent that your estimates come close to those of the expert panel [1973, p. 241].

Thus, there was an incentive to behave as the "experts," which may or may not be interpreted as attempting to give the right answer.

K and T present additional evidence in favor of the representativeness hypothesis and some evidence that many truths of mathematical statistics are not intuitive concepts even to individuals trained in them (see also T and K [1971]). In particular, regression effects (e.g., sampling based upon the value of the dependent variable) and sampling variability are often misunderstood. Also, it should be noted that judgment by representativeness has other implications, e.g., insen- sitivity to sample size and certain misconceptions concerning randomness. Consider a sample drawn from a binomial distribution.

The most salient property of a binomial sample is, clearly, the sample proportion. Hence we expect, at least in the aggregate case, that the subjective posterior probability will depend primarily on the proportion of red chips, say, in the sample with little or no regard to the size of the sample or to the difference between the proportion of red chips in the two symmetric populations [K and T, 1972, p. 446].




II

In order to test the representativeness hypothesis, a series of experiments were conducted using students from Pasadena City College, Occidental College, University of Southern California, Cal- ifornia State University Los Angeles, California State University Northridge, and University of California at Los Angeles as subjects. These experiments (not counting a pilot experiment run at California Institute of Technology) involved a total of 341 subjects.

Students were recruited from a variety of classes. They were told that an "economics experiment" was to be held and were given the time and place of the experiment. They were guaranteed that the experiment would last no more than one hour and that they would be paid at least five dollars for participating. They were given no other prior information concerning the nature of the experiment. When the subjects arrived, they were randomly sent to two different rooms. The only difference in practice between the rooms was the method of payment.

Instructions were passed out, and after obtaining the partici- pants' names and social security numbers, the first three paragraphs of the instructions were read aloud to the subjects (see Appendix).

At this time the subjects elected one of themselves to be a monitor. The monitor was allowed to inspect all equipment, and more importantly, to observe all procedures during this experiment. It was stated in the instructions that the monitor "should check the truthfulness of what the experimenter says, but other than that may not communicate any information to you in any way. If the monitor communicates any other information, he or she will be asked to leave without payment." It appeared that this solved the credibility problem, and most monitors were quite conscientious and occasionally even officious.

After the monitor was elected, the remainder of the instructions were read, and subjects' questions were answered. Once it appeared that subjects understood what their tasks were, the procedures were gone through (i.e., a dry run so to speak) to make sure that the subjects fully understood the mechanics of the experiment.

The equipment used consisted of three bingo cages and an opaque screen. One of the cages designated Cage X contained balls numbered one through six. The second cage (Cage A) contained four balls marked with an "N" and two balls marked with a "G," and the third cage (Cage B) also contained six balls, three marked with an "N" and three with a "G." All balls were the same color and it was not possible to distinguish between cages except by close inspection.




The experiment proceeded as follows: Cages A and B were put behind the screen, and the rule used to determine which of them was to be chosen was announced; Cage X served as the "prior." The rules were of the form "if one of the numbers one through k is drawn from Cage X, we shall choose Cage A; otherwise we shall choose Cage B." k was varied between two and four, thereby generating prior odds ratios for Cage A of one-half, one, and two. Next, a ball was drawn from Cage X (the subjects could not see the number on the ball) and, depending upon that number, either Cage A or Cage B was chosen and a sample of size six was drawn (with replacement) from it. After each draw the result ("N" or "G") was announced, written on a blackboard, and also recorded by the subjects on forms provided. After the six draws were completed, subjects were asked to indicate on their forms "the one (cage) you think the balls came from." Subjects were allowed to take as long as they wished, and when all were ready, the procedure was repeated again (with possibly a different prior).

The sample size of six and sample proportions were picked so that the probability of picking a sample that "looked like" one of the parent populations (four N's and two G's or three N's and three G's) was large; the idea being that if the representativeness hypothesis is correct, subjects would tend to think that such samples came from the population they "looked like," even though the actual posterior odds favored the other cage, considering the prior odds.

Notice that, although designed to test hypotheses about the forming of subjective probabilities, everything was handled opera- tionally, so that one does not need to speculate about how the words "odds" or "probabilities" are interpreted. In addition, the subjects actually are observing the results of the random process described to them, rather than being shown a series of nonrandom data and asked to believe that (counterfactually) the data were generated by a particular random process. Though the data are generated as described, this does not assure that the subjects believe that they are, and the use of the monitor was intended to provide additional credibility. By asking for judgments about bingo cages and the numbers one through six, the problem of accidentally "importing" information when problems are presented in story form is avoided.

Table I gives the posterior probabilities for each outcome-prior combination, and Table II shows the posterior odds in favor of the most likely alternative. Notice that the experiment has been designed so that several outcome-prior combinations have the same (or nearly the same) posterior odds.

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representing equally difficult problems for the subjects. Thus, all six situations in which the odds favoring the most likely alternatives are 1.4:1 may represent equally difficult choices. Note that in two of these cases the observed data are three N's, and in two cases the data are four N's. As interpreted here, the representativeness hypothesis predicts that when subjects observe four (three) N's, they will tend to believe that the sample came from the population it is represen- tative of-Cage A (Cage B). From Tables I and II, one can see that of the six cases with posterior odds of 1.4:1 (or 1:1.4) a subject using the representativeness heuristic would be led to make the correct response in two cases, the incorrect response in two cases, and be uninfluenced in two cases.

Finally, in order to provide subjects with an incentive to give correct answers, and to measure the effect of the incentive, subjects were paid in two different ways. In one room all subjects were told that they would be paid $7 at the end of the experiment. In the other room the subjects were told that at the end of the experiment one of their decisions would be chosen (again by drawing numbers from a bingo cage), and if the event that the subject had indicated was the most likely occurred, they would receive $15; otherwise they would be paid $5. The only exception was the last experiment that was run using two sections of an introductory logic course at California State University at Northridge. For this experiment both groups were under monetary incentives for accuracy.

III

Table III shows the proportion correct for the several situations for which the posterior odds favoring the most likely alternatives are 1.4 to 1, by school and method of payment. Note that in almost every case the assumption that some of the subjects are using a heuristic similar to representativeness would explain the pattern found. Also, it is apparent that reliance on the heuristic is not markedly affected by the use of monetary incentives.

Analysis of the contingency tables underlying Table III supports the conclusion just mentioned. There are significant differences among the proportions correct across the three types of events shown in the table (X2(2) = 154). The Northridge data are omitted in all the significance tests. On the other hand, there does not appear to be a systematic effect of the monetary incentive. Subjects paid for accuracy were more frequently correct for events of type A (X2(l) = 7), less often correct for events of type B (X2(l) = 14), and more often correct




TABLE III

EFFECTS OF REPRESENTATIVENESS PERCENT CORRECT WITH POSTERIOR ODDS OF 1.4:1

With Without monetary incentives monetary incentives

School Aa Bb cc Aa Bb cc

Pasadena City College 55 86 86 43 76 83 Occidental College 59 72 88 55 76 79 University of Southern

California 64 95 88 58 87 100 California State University,

Los Angeles 67 76 84 51 60 70 University of California,

Los Angeles 60 70 81 62 71 86 California State University,

Northridge 1 56 90 63 California State University,

Northridge 2 69 69 88

a. A = Representativeness favors wrong choice. b. B = No representativeness data. c. C = Representativeness favors right choice.

though not significantly so (X2(l) = 1) for events of type C. Pooling across the three types of events, one finds that those paid for accuracy were correct 73 percent of the time compared with 70 percent for the others. This difference, while in the direction an economist would predict, is rather modest and statistically insignificant (X2(l) = 3.0).

From Tables I and II one can see that there are two remaining cases in which the representativeness heuristic is available (three N's prior 1:2 and four N's prior 2:1), and four other cases with the same or nearly the same posterior probabilities. For the cases for which the representativeness heuristic applies, the subjects were correct 91 percent of the time compared with 88 percent correct in other ones. This difference, while in the direction predicted of the use of the representativeness heuristic, is statistically insignificant.

As discussed in the previous section, one of the implications of the representativeness hypothesis is that individuals judging likelihood of representativeness should tend to ignore prior probabilities. Consider the case when the representativeness heuristic is available. That is, the cases when N = 3 or N = 4 so that the sample "looks like" one of the parent population. Table IV shows that even for these cases




TABLE IV

PERCENT STATING "A" MOST LIKELY (posterior odds 1.4:1 or 1:1.4)

With Without monetary incentives monetary incentives

Prior Odds for A Prior Odds for A School 2:1 1:1 1:2 2:1 1:1 1:2

Data N = 3 Pasadena City College 50 11 16 48 -

Occidental College 72 12 10 55 19 12

University of Southern California 68 5 0 60 0

California State University, Los Angeles 35 15 12 53 15

University of California, Los Angeles 8 64 6

California State University, Northridge 1 58 23 3


Data N = 4 Pasadena City College 76 35 91 83 59 Occidental College 92 88 68 87 77 45

University of Southern California 95 73 55 93 - 43

California State University, Los Angeles 81 82 18 90 70 50

University of California, Los Angeles 94 80 30 96 86 40

California State University, Northridge 1 55 48


the prior odds clearly affected the choices made by the subjects. As before, the effects are statistically significant and hold independently of the use of monetary incentives.

Thus, it appears that something like the representativeness heuristic is being employed by the subjects, but some consideration is also being given to the prior probabilities. Thus, while the representativeness heuristic may partially explain the subjects' behavior, a more general model is required to characterize what is going on even in this simple situation. Let Y* be the subjective log odds in favor of Cage A of subject i at time t, i.e., e Yt is the ith subject's posterior odds




in favor of A. Consider the following model:

Yi ( PA eLRA] PA 1 Uit (1) e< 1-( PA) it= ea[LR(A)I$- [1 PA. t

where (PA/(1 -PA)) is the subjects' posterior odds, LR(A) =

L(XIA)/L(XjB) is the likelihood ratio for A, PA/(1 -PA) is the prior odds in favor of A, and ui is a random variable with mean zero and finite variance. Of course, if subject i correctly follows Bayes rule, then a = 0, /1 = 02 = 1, and uit 0.

In the experiments reported here, direct observations on Y* are not available, but from the subjects' choice behavior we can observe the variable Yit, where

(2) Yit = 1 if Ye > 0 = 0 otherwise.

Taking logarithms of (1) gives

(3) Yit = a + 0 In LR(A)t +02 Inl 1 -P + Uit,

which is suitable for estimating by logit, probit analysis, or other similar methods [McFadden, 1976; Amemiya, 1975; Maddala, 1977, Ch. 9; and Dhrymes, 1978, Ch. 7]. It should be pointed out that the model presented here is different from, and not completely consistent with, the representativeness heuristic if generalized to other situations. This is because opinions formed according to (1) will, of course, be sensitive to sample size through the likelihood ratio. All of the experiments reported here used a single sample size, so this was not of concern.

Since Yt is not observed, the parameters of (3) can be estimated only up to an unknown scale parameter. Nevertheless, one can for- malize the various polar hypotheses to be tested.

Bayes Rule a = 0, /1 = /2 > 0-

Representativeness Hypothesis /13 > 02 > 0-

Thus, the interpretation of K and T's representativeness heuristic suggested here is that individuals place greater weight on the likelihood ratio than on the prior odds and consider both factors when making decisions. An alternative formulation is the following:

(4) Yt = a + /1 In LR(A)t +02 In ( PA) + /3D3t + /4D)4t + Uit,




where D3 = 1 if Nt = 3

= 0 otherwise;

D4t=1 ifNt=4 = 0 otherwise.

Here the binary variables single out those observations that "look like" one of the parent populations.

In the model, Tversky and Kahneman's hypothesis could be interpreted as

/33<0, /34>0.

In this model, unless /1 < /2 individuals will be giving "too much" weight to the evidence and too little to the prior. Note that if individuals cue only on the observations that "look like" one of the two parent populations, /1 and /2 could be equal, while in the first model /1 > /2. To the extent that priors are consistently underweighted, one would expect /1 to be greater than /2. In any event both parameters should be positive.

The results of estimating the parameters of (3) are shown in Table V. All estimates reported are based on maximum likelihood logit analysis. For the purposes of the analysis the subjects were di- vided into four groups, depending upon whether or not there was a monetary incentive to be accurate and whether the subjects were "experienced." A subject was classified as experienced on the jth trial if the subject had previously been exposed to (i.e., had observed) the same prior-outcome combination as occurred on the jth trial. This separation of the observations was done because inspection of the data suggested that subjects were more likely to give the correct response if they had seen the same problem before.

Note that in every case the coefficient on the likelihood ratio exceeds the coefficient of the log of the prior odds, and that the differences are statistically significant. This result is as predicted by our generalization of the representativeness heuristic. Thus, it does appear, as T and K have suggested, that individuals tend to give too little weight to prior information. Note, however, that the prior odds do appear to enter significantly in the equations.

The hypothesis that a single function holds for the several data sets can be tested with a likelihood ratio test. Using the result that asymptotically minus twice the logarithm of the likelihood ratio is distributed as chi squared with degrees of freedom, given by the number of restrictions tested [Kendall and Stuart, 1967, pp. 230-31],




TABLE V

LOGIT RESULTS

Y* =a+I31nLR(A) +321nPA/(l-PA) +u (numbers in parentheses are t-ratios)

Groupa a 01 /2 011-132 -2 In L n

PF -0.09 2.01 1.43 0.58 1,589.0 1,747 (1.4) (19.9) (14.2) (4.9)

PS -0.25 2.86 2.34 0.52 996.0 1,170 (2.8) (14.0) (15.9) (2.8)

NF 0.17 1.78 1.21 0.57 1,485.1 1,630 (2.5) (19.5) (10.5) (4.6)

NS 0.24 2.65 2.17 0.48 874.9 1,061 (2.8) (15.3) (13.6) (2.9)

P. -0.11 2.25 1.82 0.43 2,612.8 2,917 (2.2) (24.4) (20.0) (4.4)

N- 0.18 2.02 1.58 0.44 2,393.9 2,691 (3.5) (24.7) (17.3) (4.5)

*F 0.04 1.86 1.33 0.53 3,085.8 3,377 (0.9) (27.9) (16.3) (6.4)

.S 0.00 2.62 2.20 0.42 1,887.9 2,231 (0.0) (20.5) (20.9) (3.4) 0.04 2.08 1.69 0.39 5,028.0 5,608

(1.1) (34.8) (26.4) (5.8)

a. Key to groups: * = summation;*. = all subjects; P = monetary incentives were available; N = no monetary incentives were available; F = inexperienced subjects, i.e., first time have seen a particular prior-outcome combination; S = experienced subjects.

the tests can be performed using the figures in the fifth column of Table V. The results show that the same equation can be used to describe the behavior of inexperienced subjects whether or not they are financially motivated. On the other hand, the hypothesis of homogeneity is rejected when one pools experienced and inexperienced subjects or the two groups of experienced subjects.

The results from logit analysis using the extended model (4) are shown in Table VI. The coefficient of the log likelihood ratio exceeds the coefficient of the log prior odds for three of the four groups of subjects, but the differences are not significant at the 0.01 level. For all groups the sign of f3 is negative as predicted from the representativeness hypothesis, and in three out of four cases the sign of /4 is

positive. The individual coefficients /3 and 34 are frequently not significantly different from zero and never significantly differ from each other in absolute value. Comparing the figures in column 8 in Table VI with those in column 5 of Table V, one sees that the pair of




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variables (D3,D3) is jointly significant for inexperienced subjects, but not significant or less significant for experienced subjects. Thus, for these data the qualitative predictions of the representativeness hypothesis appear to be borne out.

As before, monetary incentives do not appear to affect the behavior of inexperienced subjects, in that the hypothesis that the same functions describe both sets of subjects cannot be rejected at any conventional level of significance. This is not true, however, for the experienced subjects. Overall, the specification (4) seems to be su- perior to (3) in that it generally fits the data better and also eliminates the embarrassingly high t -ratios for the intercepts reported in Table V. In these models, nonzero constant terms do not make any sense, so leaving them in is a kind of test of the specification of the model. Overall, the conclusion seems to be that the representativeness hypothesis as developed here holds. That is, in forming subjective log- odds subjects give relatively too much weight to the evidence and too little weight to the prior odds. This conclusion follows for both forms of the model considered here. When the simpler version (3) is used, the coefficient of the log likelihood ratio is significantly greater than the coefficients of the log prior odds. For those cases for which (4) is the better model, the coefficient of the log likelihood ratio is either greater than or not significantly different from the other coefficient, and the "special event" factors are significant.

The estimates in Tables V and VI all assume that the subjects from several educational institutions make up a homogeneous population. To test this hypothesis, (3) and (4) were reestimated for each institution with the data broken into groups as in Tables V and VI. The results obtained were consistent with those previously reported, that is, generally subjects gave too much weight to the data as described in the previous paragraph. For each of the categories shown in Tables V and VI, the hypothesis of homogeneity across institutions was usually rejected at conventional levels of significance. Since qualitatively the results are similar to those in Tables V and VI, to save space only the aggregate results are reported.

The final set of results is a generalization of the previous models, designed to explore the definition of experience used in the previous estimations. The model used is basically a heteroskedasticity model; the assumption being that the logistic coefficients are the same for both experienced and inexperienced subjects, but that the disturbance distributions differ by a scale factor. The basic idea was that with experience the amount of randomness in subjects' decisions might be reduced, and that this would show up as a decrease in the disper-




sion of the distributions of the disturbances in (3) and (4). It should be pointed out that for limited dependent variable

models heteroskedasticity leads to inconsistent parameter estimates, though, of course, estimating the equation for each population sepa- rately provides consistent but inefficient estimates. The estimates in Table VII are maximum likelihood estimates for a logistic model with heteroskedasticity. The form of the likelihood function is such that no serious computational burdens are introduced by this generalization of the standard model. The primary simplification comes from the cross partials of the likelihood function with respect to the various population scale parameters being zero. The full details of this are beyond the scope of this paper.

Table VII presents the results of refitting the model (4) under the assumption that both "experienced" and "inexperienced" subjects use the same functions (i.e., the same parameter values), but differ only with respect to the dispersion of the distributions of the disturbances. The model (3) was also reestimated, but since the coefficients of the binary variables were jointly significant, only the results from (4) are reported. Note that there was no feedback during the course of the experiment, so any "learning" that might have taken place results from familiarity with a problem or from further reflections on it, etc.

The results of estimating the heteroskedastic model are rather striking. In all but one case (one of the Northridge groups), the experienced subjects showed less random or "more precise" behavior than the inexperienced subjects, though the result is statistically significant for only two cases. Thus, in some sense opinion becomes "firm" or a "consensus" emerges with repetition of a problem.

IV

In this paper some experimental work of Kahneman and Tver- sky, presenting evidence that individual reasoning processes do not follow the law of mathematical statistics, has been discussed. The results of experiments designed to test the representativeness heuristic in a manner relevant to economics were presented.

In addition to the basic model (1), two generalizations were estimated. The same qualitative results were obtained: individuals tend to give too much weight to the "evidence" and thus too little weight to their prior beliefs, though priors are not ignored. In addition, there is some evidence that repetition of a problem or choice task even without feedback leads to more firmly held opinions or less random-




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ness in judgment, in the sense that the distribution of the ui becomes more concentrated. Finally, and to the author the most surprising, the evidence that financial incentives affect the behavior is far from compelling. Responses to unfamiliar problems (that is, problems that had not previously been seen) showed no sensitivity to the presence or absence of monetary incentives for accuracy, though as experience is gained, the picture is not so clear.

Some may ask why we should care about the behavior of individual students in the sorts of environments used in this paper. That is, economic theory is really a theory of market behavior, i.e. of ag- gregates, and that the repetitive nature of market interactions, to- gether with the reinforcement provided by the profit system (economic agents either learn to process information efficiently or are forced to act as if they have), render single observations on individual behavior irrelevant. Also, a relatively few individuals sensitive to arbitrage possibilities may make markets work as predicted by the theories. Thus, we really do not care much about individual behavior or information processing capacity. At the beginning of the paper it was argued that there are areas of economic research that require economists to be concerned about the nature of individual decision processes. It may be that the behavior of members of dissimilar groups differs systematically, and no claim is made that the results reported here would replicate with qualitatively different subject pools (though this is a testable proposition). However, conventional economic theories as exposited do not discriminate among types of individuals; i.e., they do not contain parameters for students, stock market specialists, housewives, business executives, etc. Thus, as long as individual decision processes are a matter of legitimate concern for economics, the results of the sort presented here are relevant to economics.

APPENDIX

INSTRUCTIONS

Name Social Security Number Address

The experimenters are trying to determine how people make decisions. We have designed a simple choice experiment, and we shall ask you to make decisions at various times. The amount of money you make will depend on how good your decisions are. During the exper-




iment you will be asked to make a number of decisions. At the end of the experiment we shall randomly choose one of your decisions and if it is correct, you will receive $15 and if it is incorrect, you will receive $5.

If you look at the person in the front of the room, you will see that he or she has three randomizing devices, otherwise known as bingo cages, which are designated as Cage A, Cage B, and Cage X. Inside both Cage A and Cage B are six balls, some of which are marked with an N and some with a G. Cage A has four N's and two G's and Cage B has three N's and three G's. Inside Cage X there are six balls numbered one, two, three, four, five, and six.

The experiment will proceed as follows. First, we shall ask you to select one individual as a monitor to watch the procedures, to ex- amine the equipment, and to make sure that the experimenters really are doing what they say they are doing. The monitor should check the truthfulness of what the experimenter says, but other than that may not communicate any information to you in any way. If the monitor communicates any other information, he or she will be asked to leave without payment. The monitor will receive $10 for his efforts.

(pick volunteer) Now Cages A and B are put behind this screen, and we spin Cage

X. Before each run we will tell you that if certain numbers come up, we will choose Cage A, and otherwise we will choose Cage B. For example, if 1, 2, 3, or 4 are drawn, we will pick Cage A: if a 5 or 6 is drawn, we will pick Cage B. After drawing from Cage X, the monitor will be asked to choose the appropriate cage (A or B) from behind the screen. Now the experimenter will make six draws from the cage, replacing the drawn ball each time. He will write the results of the draws on the board. You will be asked to indicate on your answer sheet whether you think the draws come from Cage A or Cage B....

The only talking allowed during the course of the experiment will be to clarify questions you may have about the procedure, and these questions should be directed to the experimenter. We will now walk through a complete run of the experiment to make sure everyone understands the procedure. Cage A has 4 N's and 2 G's Cage B has 3 N's and 3 G's Run No. 0. In this run there are -- chances out of 6 for choosing Cage A and -- chances for choosing Cage B. Record the results for this run: Circle the one you think the balls come from. Cage A Cage B

Run No. 1. In this run there are -- chances out of 6 for choosing A and -- chances for choosing Cage B. Record the results for this run: Circle the one you think the balls come from. Cage A Cage B

CALIFORNIA INSTITUTE OF TECHNOLOGY




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Bayes Rule as a Descriptive Model

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