The effects of biases in probability judgments on market prices

~ )

Pergamon

Accounting Organizations and Soctetv, Vol. 19, No. 8, pp. 675-700, 1994 Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0361-3682/94 $7.00+0.00

0361-3682(93)E0002-X

THE EFFECTS OF BIASES IN PROBABILITY JUDGMENTS ON MARKET PRICES* ANANDA R. GANGULY University of Pittsburgh and

JOHN H. KAGEL University of Pittsburgh and

DONALD V. MOSER University of Pittsburgh Abstract Experimental markets were used to examine w h e t h e r individual probability judgment biases affect market prices. This issue is important to accountants because users of accounting information (especially investors) face competitive market environments. The expectation was that it would be more diffaoflt for prices to be unbiased in markets w h e r e biased traders had the highest expected payoffs than in markets where unbiased traders had the highest expected payoffs. This ~ t i o n arose from the observation that competitive forces would produce biased prices w h e n biased traders had the highest expected payoffs unless either ( 1 ) biased traders learned to be unbiased as a result of market experience, or ( 2 ) biased traders were inactive, thus allowing unbiased traders to set prices. Consistent with expectations, prices were biased in a market w h e r e biased traders had the highest expected payoffs. That is, individual judgment biases persisted, biased traders remained active, and prices were biased accordingly. Results were less clear in a market w h e r e unbiased traders had the highest expected payoffs, with prices moving toward unbiased prices but remaining more biased than unbiased overalL The results of this study suggest that individual judgment biases can have a substantial effect on market prices, and, consequently, demonstrations of individual investor judgment biases should be of concern to accountants.

T h e r e is considerable evidence indicating that the probability judgments of individuals are often systematically biased. This evidence comes from studies in experimental psychology as well as a variety of applied fields (including

accounting), and poses a challenge to theories that assume rational individual behavior (e.g. principle-agent theory, standard asset pricing models, and standard market theories). 1 Such theories form the basis for a large part of the

* This research was partially supported by grants from t h e Economics Division of t h e National Science Foundation and the Joseph M. Katz Graduate School of Business. W e thank the reviewers, Jake Birnberg. Colin Camerer, Harry Evans, Vicky Heiman-Hoffman, Yuhchang Hwang, Jim Patton and the participants of a University of Minnesota accounting workshop for helpful c o m m e n t s on an earlier draft of this paper. W e alone are responsible for any errors or omissions. 1 In this paper, Camerer's (1992, p. 239) definition of "rational behavior" is adopted. That is, rational behavior is taken "to be j u d g m e n t consistent with laws of statistics and probability (including Bayes' Rule) and choice consistent with e x p e c t e d utility". The t e r m "bias" is used w h e n "judgments and choices are inconsistent with these normative rules in a predictable direction". Although there is s o m e debate in t h e psychology literature regarding the normative appropriateness of Bayes' Rule (e.g. Berkeley & Humphreys, 1982; Cohen, 1981 ), there is widespread a g r e e m e n t in the accounting, finance, and e c o n o m i c s literatures that t h e u s e of Bayes' Rule constitutes rational behavior. 675

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existing accounting literature. In response to this challenge, proponents of theories that assume rational behavior have offered arguments suggesting that the evidence in support of individual judgment biases is either unconvincing (because of the experimental methods used) or irrelevant (because of the belief that collective activity reduces judgment error). For example, evidence of individual judgment bias is often dismissed on the grounds that subjects w e r e not provided with performance incentives or opportunities to learn from feedback (Grether, 1978; Merton, 1987). Another criticism is that much of the evidence comes from studies in which inexperienced subjects performed unfamiliar tasks. For example, Shanteau ( 1 9 8 9 ) and Smith & Kida ( 1 9 9 1 ) suggest that one reason judgment biases are sometimes smaller in auditing studies than in psychology studies may be that many auditing studies used experienced subjects performing familiar tasks. Competitive markets have most of the properties that critics have argued are often lacking in previous individual judgment studies. In particular, markets provide monetary incentives to perform well, opportunities to learn through feedback, and familiarity with the task (assuming repeated participation). Consequently, it is often argued that individual judgment biases are not likely to persist in market settings. Despite the intuitive appeal of such arguments, it remains an open empirical question as to whether individual probability judgment biases are eliminated in market settings. In addition, those w h o dispute the relevance of individual judgment studies argue that, even if individual judgment biases persist in market settings, aggregate market outcomes could still be rational. 2 Camerer (1992) identifies four main arguments as to how markets could produce unbiased aggregate outcomes, along with related counter-arguments. According to these arguments, market outcomes will be rational if.. (1) individual judgment biases are random (the "cancellation hypothesis"), (2) the most actfve

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traders are unbiased (the "smart few hypothesis"), (3) biased waders learn from unbiased waders or buy good advice (the "learning hypothesis"), or (4) biased traders are selected out ("the evolutiothat~r hypothesis"). As was the case for the arguments against the persistence of individual judgment biases in markets, these hypotheses regarding the aggregation process remain, for the most part, open empirical questions. The issue of whether individual judgment biases matter in markets is important to accountants because investors (a primary accounting information user group) operate in a competitive market setting. Consequently, accountants have debated this issue for many years. For example, Gonedes & Dopuch ( 1 9 7 4 ) argued that, given an efficient capital market, assertions regarding market outcomes based on the results of individual judgment studies are "extremely tenuous" (p. 106). In contrast, Einhorn ( 1 9 7 6 ) observed that the individual judgment literature raises questions about the validity of the behavioral assumptions underlyIng theories of aggregate behavior, suggesting that "... the fascinating, but unanswered question remains as t o h o w sub-optimal individual behavior can lead to 'rational' behavior at the aggregate level (if indeed this exists)" (p. 198). Despite the controversy about whether individual judgment biases persist In markets and whether such biases affect aggregate outcomes, only a few accounting studies have attempted to address these issues directly. This is apparently due to the difficulties inherent In investigating these issues and questions regarding the relevance of such work given the widespread belief among accountants in market efficiency. Eger & Dickhaut ( 1 9 8 2 ) attempted to explain h o w aggregate outcomes could be rational despite persistent demonstrations of individual irrationality. They suggested that the individual judgment experiments that demonstrated irrationality may not have adequately captured probability judgments as represented in the models used in capital-market and principal-

2A "market outcome" is considered rational if it is consistent with collective behavior of rational individuals (Camerer, 1992).

BIAS IN PROBABIHTYJUDGMENTS agent research. To support this suggestion, they reported experimental results that they interp r e t e d as showing that probability judgments inferred from subjects' betting behavior demonstrated less systematic bias than direct odds estimates provided by subjects. However, as Libby ( 1 9 8 9 ) subsequently noted, the judgments inferred from subjects' bets actually varied more than the direct assessments, and thus it is not clear that the inferred judgments w e r e actually better than the direct odds estimates. Duh & Sunder ( 1986, 1987) examined whether prices observed in a series of experimental markets were closer to Bayesian price predictions (rational prices) than to price predictions obtained b y assuming traders' probability judgments were the result of one of several non-Bayesian psychological processes (biased prices). They concluded that the judgment biases predicted by the alternative psychological theories w e r e not reflected in market prices. The strong belief in the efficient market hypothesis, combined with the results of these early experimental studies in accounting, led to the belief among most accountants that market outcomes w e r e not likely to reflect the types of judgmental biases regularly observed in individual judgment studies. Therefore, studies demonstrating individual judgm e n t biases w e r e considered by many to have limited relevance for accounting settings. Recently, some accounting researchers have reconsidered w h e t h e r individual probability judgment biases can affect market outcomes. For example, Libby ( 1 9 8 9 ) interprets the early accounting studies and the more recent experimental economics studies (Camerer et aL, 1989; Camerer, 1987) that directly address the effects of individual judgment biases in markets as showing that, while the effects o f individual judgment biases may be reduced in market settings, they are "not eliminated". Also, recent evidence has led some capital markets researchers to suggest that the capital market may not be as eflicient as previously believed (Bernard, 1993) and that certain price anomalies may be the result o f individual irrationality (Abarbanell & Bernard, 1992; Debondt & Thaler, 1985, 1987, 1990; Hand, 1990, 1991). Commenting on such capital-markets work,

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Berg et aL (in press) point out that the research methods used in such studies do not provide direct evidence about individual biases, but rather only demonstrate that certain price anomalies are "consistent with" several assumed judgmental biases. They suggest that the methods used b y experimental economists can provide a m o r e direct way of studying the relation between individual and aggregate behavior. The experimental economics approach provides a means ofexamining in a controlled environment the behavior of individual traders in markets. Consistent with Berg e t aL's suggestion, this study uses an experimental economics approach to examine w h e t h e r an individual judgment bias called the base-rate fallacy (hereafter, BRF) persists in a market setting and whether this judgment bias affects market prices. As discussed in more detail later, the BRF has been shown to b e quite robust in a series of previous studies. First, an experimental setting similar to those used in the previous BRF studies was designed in order to generate a relatively strong probability judgment bias before traders entered an experimental asset market. Then the effect of this bias o n prices was examined in two kinds of markets, ( 1 ) a market in which unbiased traders had the highest e x p e c t e d payoffs and ( 2 ) a market in which biased traders had the highest e x p e c t e d payoffs. In the first kind of market ( w h e r e unbiased traders had the highest e x p e c t e d payoffs), our expectation was that competitive forces would lead to unbiased prices. In the second kind o f market ( w h e r e the biased traders had the highest expected payoffs), the expectation was that competitive forces would result in biased prices. The rationale was that in cases w h e r e the biased traders have the highest e x p e c t e d payoffs, it is difficult for prices to be unbiased because competition among the biased traders will drive prices up to the biased traders' expectations unless the biased traders either ( 1 ) b e c o m e unbiased as a result of their experience in the markets, o r ( 2 ) are inactive, and thus allow unbiased traders to set market prices. Probability judgments were collected from individual traders before each trading period ( 1 6 repetitions in each market) in order to assess h o w

BIAS IN PROBABILITYJUDGMENTS ness" s e e m e d to p r e d i c t prices b e t t e r than the Bayesian model. 5 As indicated earlier, D u b & Sunder ( 1 9 8 6 , 1 9 8 7 ) c o n c l u d e that market prices in their experimental markets w e r e closer to Bayesian p r i c e predictions than to the p r i c e predictions associated w i t h several p s y c h o logical theories, including the p r i c e predictions o b t a i n e d b y assuming traders c o m m i t t e d the BRF. A n d e r s o n & Sunder ( 1 9 9 3 ) used p r o c e d u r e s similar to C a m e r e r ( 1 9 8 7 ) to e x a m i n e the relative p e r f o r m a n c e o f professional traders and students in experimental markets. Although they c o n c l u d e that the professional traders' performance was m o r e Bayesian than that o f the students, prices in the professional-trader markets w e r e nevertheless often closer to BRF price predictions than to Bayesian p r i c e predictions. O n e p r o b l e m w i t h t h e experimental market studies d e s c r i b e d a b o v e is that t h e y represent joint tests o f ( 1 ) w h e t h e r the individual j u d g m e n t bias exists (in the specific experimental setting u s e d ) before the traders e n t e r the market, ( 2 ) w h e t h e r the individual judgm e n t bias persists in the market setting, and ( 3 ) w h e t h e r market prices reflect any existing bias. Thus, it is difficult t o interpret the results o f these studies. In particular, b e c a u s e the BRF was n e v e r s h o w n t o exist in the A n d e r s o n & Sunder ( 1 9 9 3 ) and D u b & Sunder ( 1986, 1 9 8 7 ) studies, and w a s v e r y w e a k to begin with in Camerer's ( 1 9 8 7 ) study, it is n o t clear w h e t h e r m a r k e t forces had any c o r r e c t i v e influence. 6 The p r e s e n t s t u d y handles this p r o b l e m b y incorpo-

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rating into the design a p r o b l e m structure that p r o d u c e s a relatively s t r o n g BRF bias and t h e n testing w h e t h e r market forces eliminate this individual j u d g m e n t bias and w h e t h e r any individual j u d g m e n t bias is reflected in market prices. A s e c o n d p r o b l e m with the p r e v i o u s studies is that t h e y did n o t distinguish markets in w h i c h biased traders had the highest e x p e c t e d payoffs f r o m markets in w h i c h unbiased traders had the highest e x p e c t e d payoffs. Thus, it is n o t clear w h e t h e r any j u d g m e n t biases w h i c h m a y have existed in those studies w o u l d have b e e n m o r e likely t o b e reflected in prices in markets w h e r e biased traders had the highest e x p e c t e d payoffs. W e argue that it is m o r e difficult for prices to be unbiased in markets w h e r e biased traders have the highest e x p e c t e d payoffs, b e c a u s e biased traders will bid prices up to their biased expectations. This issue was addressed in the p r e s e n t study b y designing t w o types o f markets, o n e in w h i c h unbiased traders had t h e highest e x p e c t e d payoffs and a s e c o n d in w h i c h biased traders had the highest e x p e c t e d payoffs.

EXPERIMENTAL DESIGN This study consists o f t w o market sessions, o n e in w h i c h Bayesian ( u n b i a s e d ) traders had the highest e x p e c t e d dividend values ( m a r k e t session 1 ) a n d o n e in w h i c h BRF ( b i a s e d ) t r a d e r s had the highest e x p e c t e d dividend values

5 "Exact representativene~ '' is Camerer's interpretation of Tversky and Kahneman's "representativeness" heuristic (see footnote 3) in his experimental setting. The subsequent information his traders received took the form of a sample of three balls drawn from one of two bingo cages of known proportion of red and black balls. A sample was considered to be exactly representative of one of the two cages if the sample proportion of red (or black) balls was identical to the proportion of red (or black) balls in one of the two cages. 6 Anderson & Sunder (1993) and Duh & Sunder (1986, 1987) did not collect probability judgments from their subjects; thus, there is no way to know whether their subjects' judgments were biased before they participated in trading. Camerer (1987) did have his subjects make a series of probability judgments before trading (but not between trading periods during the experiment), but his experimental setting produced only very small pre-trading biases. For example, the deviation of average individual probability judgments from Bayesian judgments averaged across experiments ranged from +0.037 to -0.084, and averaged only -0.026 across the four possible individuating data samples (see Camerer, 1987, Table 5). These biases are very small relative to the biases reported in some previous BRF studies. For example, in the weU.known "cab problem" introduced by Kahneman and Tversky, the deviation of median and model individual probability judgments from the Bayesian posterior is typically about 0.39.

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( m a r k e t session 2). I n each m a r k e t session, 12 traders p a r t i c i p a t e d in 16 p e r i o d s of t r a d i n g in a d o u b l e - o r a l auction. T h e y t r a d e d certificates that h a d o n e - p e r i o d lives a n d paid a l i q u i d a t i n g d i v i d e n d at the e n d of the m a r k e t period. Subjects w e r e r e q u i r e d to p r o v i d e p r o b a b i l i t y j u d g m e n t s of t h e state of n a t u r e at the start of each t r a d i n g period. State probabilities probabilify

and procedures

for eUciting

judgments

Base-rate probabilities and s u b s e q u e n t information w e r e provided to subjects in a contextspecific setting that m a p p e d into the "cab p r o b l e m " w h i c h is k n o w n to generate a significant BRF.7 The setting dealt with b u y i n g and selling shares of a c o m p a n y engaged in an "ambitious project" that w o u l d result in either a 'q3uge success or a total failure" (a copy of the instructions is p r o v i d e d in the Appendix). For each o f t h e 16 t r a d i n g periods, t h e certificates a n d the related e x p e c t e d payoffs w e r e associated w i t h a different c o m p a n y . T h e d i v i d e n d payoff d e p e n d e d o n w h e t h e r the c o m p a n y s u c c e e d e d o r failed. Traders w e r e i n f o r m e d that, in t h e a b s e n c e of a n y f u r t h e r information, t h e n o r m a l c h a n c e

of success for c o m p a n i e s like the 16 u s e d i n the e x p e r i m e n t was 85% ( m a r k e t session 1 ) o r 15% ( m a r k e t session 2). H o w e v e r , b e f o r e each t r a d i n g period, traders w e r e g i v e n a n analyst's p r e d i c t i o n o f e i t h e r success o r failure for t h e c o m p a n y w h o s e certificates t h e y w o u l d b e t r a d i n g that period. T r a d e r s w e r e i n f o r m e d that the analyst u s e d o n l y company-specific informat i o n to m a k e his p r e d i c t i o n s a n d that h e was 80% accurate i n identifying firms that s u c c e e d e d a n d firms that failed. T h e y w e r e also told that the analyst's a c c u r a c y rate was d e t e r m i n e d b y t e s t i n g h i m w i t h a large s a m p l e of c o m p a n i e s , half o f w h i c h h a d s u c c e e d e d a n d half o f w h i c h had failed, s Finally, traders w e r e i n f o r m e d that the 16 c o m p a n i e s u s e d i n t h e e x p e r i m e n t consisted of 8 randomly selected companies from those that t h e analyst said w o u l d s u c c e e d a n d 8 r a n d o m l y s e l e c t e d c o m p a n i e s from those that the analyst said w o u l d f a i l 9 All o f this i n f o r m a t i o n was p r o v i d e d i n the i n s t r u c t i o n s , w h i c h w e r e read a l o u d to t h e subjects. Pilot t e s t i n g i n d i c a t e d that, like the cab p r o b l e m , o u r e x p e r i m e n t a l s e t t i n g p r o d u c e d a significant BRF bias. A l t h o u g h the e x p e r i m e n t a l s e t t i n g was con-

7 One version of the "cab problem" is as follows: "Two cab c o m p a n i e s operate in the s a m e city, the Blue a n d Green (according to the color o f the cab they run). Eighty-five p e r c e n t o f the cabs in the city are Blu~ a n d 15 p e r c e n t are Greert A cab w a s involved in a hit-and-run accident a t n i g h t in which a pedestrian w a s r u n o v a . A n eyewitness identified the cab as a Green cab. The court tested the witness's ability to distinguish between Blue a n d Green cabs u n d e r n i g h t t i m e visibility condition~ I t f o u n d that the witness w a s correct 80 p e r c e n t o f the t i m e b u t confused it w i t h the other color 2 0 p e r c e n t o f the tim~ W h a t is the p r o b a b i l i t y that the hit-and-run cab w a s Green? "" The median and modal response for this problem is typically 0.80, while the Bayesianposterior is 0.41 (Bar-Hillei, 1990). s Subjects were told that the large sample on which the analyst was tested consisted of half successful firms, and half firms that failed, to prevent subjects from concluding that the analyst's accuracy rate (80%) was more reliable for successful firms than for firms that failed. Subjects might have concluded this ff the analyst's sample included more successful firms than firms that failed. 9 Thus traders experienced an equal number of success and failure signal cases and had an equal opportunity to learn in both signal cases. In previous studies traders received certain signals more frequently than others and, therefore, because learning opportunities were not comparable, comparisons across signals were problematic. In response to a subject's question early in our second market, the experimenter reiterated the procedures used and explicitly announced that our procedures did not mean that our sample of 16 firms was drawn from a population containing half successful firms and half firms which failed. In addition, in this same market session, the base rate of success and the analyst's accuracy rate were written on a flip chart as the instructions were read to the subjects. This information was pointed out to the traders and remained in full view of all traders throughout the session. The intent was to ensure that all traders had ready access to the information necessary to respond to the task in a Bayesian manner and to eliminate, to the extent possible, any potential confusion regarding the nature of our sample and the population from which it was drawn.

BIAS IN PROBABILITY JUDGMENTS

text specific, experimental control equivalent to that achieved in previous studies that used abstract settings was maintained by generating

the signals (analyst's predictions) and associated outcomes (success or failure of a company) in advance of the experimental sessions using the s a m e p r o c e d u r e s u s e d i n t h o s e s t u d i e s . 1° The experimental setting incorporated those f e a t u r e s s u g g e s t e d i n t h e l i t e r a t u r e as w a y s t o e n s u r e t h a t B a y e s i a n r e v i s i o n is t h e n o r m a t i v e l y appropriate way for subjects to respond to our task. I n p a r t i c u l a r , t h e c r i t i c i s m t h a t t h e B a y e s i a n likelihood ratio may not be independent of either base rates or prior

probabilities (Birnbaum,

1983) was given careful consideration. This p r o b l e m w a s a v o i d e d b y m a k i n g it c l e a r t h a t t h e

analyst

"based his judgment

entirely on the

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output of a computerized analysis package that exclusively uses the accounts and other companyspecific data of each company as input and produces a measure of the project's (and therefore the company's) success potential". 1~ During the experimental sessions, traders were given the analyst's prediction for the company w h o s e certificates they w o u l d trade that period and, prior to the start of trading, asked to make a probability judgment regarding the success or failure of that company. 12 Traders were informed that at the end o f the experiment one of the 16 market periods w o u l d be randomly selected and their probability judgments for that period would be compared to the answer given by a statistician (i.e. the Bayesian posterior). 13 If their probability judgment was the same as

t o The analyst's predictions, actual outcomes, and 16 specific companies (prediction-outcomepairs) used in the experiment were generated before the experimental sessions as follows. An actual outcome was determitmd first by randomly drawing a ball from a box containing 20 balls, of which 17 were successes (85%) and 3 were failures (15%). Then to determine whether the analyst's prediction for that outcome would be correct or incorrect, we drew a second ball from another box containing 20 balls, of which 16 were correct (80%) and 4 were incorrect (20%). So, for example, ff the first draw was a success outcome and the second draw was a correct prediction, this would constitute a success-success, predictionoutcome pair. We repeated this Im~cedure until we accumulated a large number of such pairs. From this large sample of prediction-outcome pairs, we randomly selected 8 cases from those for which the prediction was success and 8 cases from those for which the prediction was failure. Thus, there was a complete mapping between the actual procedures used to select the prediction-outcome pairs (i.e. the companies) and the information given to the traders in the cover story used in the experiment. The 16 prediction-outcome pairs were randomly assigned to the 16 trading periods. A consequence of using these abstract procedures was that the signals and outcomes used in our study were hypothetical. No effort was made to hide this fact from the subjects. ~A second criticism raised regarding the appropriateness of Bayesian updating in word problems such as ours is that base rates and prior probabilities are not necessarily the same thing; base rates help people set prior probabilities but need not be identical to prior probabilities (Kochler, 1989; Cohen, 1981). We avoided this problem in our task by stating the base rates in terms of prior probabilities as follows: "If you had no access to more specific information about the company, you would have estimated the chance of success for each project to be 1596 (8596), which is the normal chance of success for similar projects and similar companies." This wording also insures that the prior information is "causally relevant" to the task at hand (Cohen, 1981). ~ZSubjects responded on either a success scale or a failure scale. If they thought the company would succeed, they indicated their probability estimate of success from 0.50 to 1.00 on the success scale. If they thought the company would fail, they indicated their probability estimate of failure from 0.50 to 1.00 on the failure scale. We explained to traders that probabilities from 0 to 0.50 were crossed off both the success and failure scales because a less than O.50 chance of success (failure) would mean the trader really thought the company would fail (succeed). ~3Use of the terms "Bayesian posterior" or "correct answer" was avoided so as not to suggest to subjects that they had to know how to do a specific calculation in order to make their probability estimates. Grether (1978, 1980) criticizes payment procedures such as this on the grounds that with an incentive to behave as experts, subjects may or may not interpret this as an incentive to give the right answer. However, it is important to note that in this study this concern is limited to subjects' probability judgments and does not apply to payments in the asset market. Further, the fact that subjects in this study generally traded in accordance with their probability judgments, in conjunction with their dividend values, is consistent with the argument that they in fact were trying to give the right answers.

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t h e Bayesian p o s t e r i o r , t h e y r e c e i v e d 2 0 0 0 fr. ( T h e e x p e r i m e n t a l c u r r e n c y u s e d w a s francs.) F o r e v e r y 1% o f a b s o l u t e d e v i a t i o n f r o m t h e Bayesian p o s t e r i o r , t h e i r p a y m e n t w a s r e d u c e d b y 20 fr. At t h e e n d o f t h e e x p e r i m e n t , this p a y m e n t w a s a d d e d to a n y profits e a r n e d f r o m t r a d i n g in t h e asset markets, t h e details o f w h i c h are described below. T h e m a n i p u l a t i o n o f e x p e c t e d payoffs w a s a c c o m p l i s h e d b y r e v e r s i n g t h e b a s e rates o f s u c c e s s a n d failure in m a r k e t sessions 1 a n d 2. T h a t is, as e x p l a i n e d in m o r e d e t a i l later, r e v e r s i n g t h e b a s e rates a c r o s s t h e t w o m a r k e t sessions r e s u l t e d in t h e Bayesian t r a d e r s h a v i n g t h e h i g h e s t e x p e c t e d payoffs in m a r k e t s e s s i o n 1, a n d t h e BRF t r a d e r s h a v i n g t h e h i g h e s t e x p e c t e d payoffs in m a r k e t s e s s i o n 2.

Market procedures In e a c h m a r k e t p e r i o d , 12 t r a d e r s w e r e e a c h e n d o w e d w i t h t w o assets t h a t lived o n e p e r i o d and paid a liquidating state-dependent dividend. All t r a d i n g a n d e a r n i n g s w e r e in t e r m s o f francs, w h i c h w e r e c o n v e r t i b l e i n t o dollars at t h e r a t e o f $1.00 p e r 1000 fr. at t h e e n d o f t h e e x p e r i ment. Traders were each endowed with 2300 fr. in e a c h t r a d i n g p e r i o d , a n d 2 0 0 0 fr. w e r e s u b t r a c t e d f r o m e a c h t r a d e r ' s total francs at t h e e n d o f e a c h p e r i o d . T r a d e r s w e r e a l l o w e d to r e t a i n t h e 300 ft. d i f f e r e n c e e a c h t r a d i n g p e r i o d to h e l p offset any p o t e n t i a l losses f r o m trading. T r a d e r s v o l u n t a r i l y e x c h a n g e d assets in a double-oral-auction asset market. Buyers s h o u t e d o u t b i d s at w h i c h t h e y w e r e w i l l i n g to buy, sellers s h o u t e d o u t offers at w h i c h t h e y w e r e w i l l i n g to sell. Bids h a d t o t o p o u t s t a n d i n g b i d s a n d offers h a d to b e b e l o w o u t s t a n d i n g offers. A m a t c h i n g b i d a n d offer w a s a t r a d e w h i c h e r a s e d all p r e v i o u s b i d s a n d offers. All bids, offers, a n d t r a d e s w e r e r e c o r d e d o n a b l a c k b o a r d visible to all traders. ( N o h i s t o r y o f p r e v i o u s market period trades was displayed.) Trading p e r i o d s l a s t e d 4 minutes.

At t h e e n d o f e a c h t r a d i n g p e r i o d t h e state, S(uccess) or F(ailure), was announced and t r a d e r s c a l c u l a t e d t h e i r profits. D o l l a r profits a r e given by Profits = X[Ef - Rf + Y. Oi -- 5". Bj +

(Ec-- xs + xb)],

D(N) (1)

where X = dollar-per-franc c o n v e r s i o n rate; Ef = initial e n d o w m e n t in francs; Rf -a m o u n t o f francs r e p a i d at p e r i o d - e n d ; Oj = selling p r i c e o f ith certificate sold; Bj = p u r c h a s e p r i c e o f j t h certificate b o u g h t ; D ( N ) = d i v i d e n d s p e r certificate g i v e n state o f n a t u r e N ; E c = initial e n d o w m e n t in certificates; xs = n u m b e r o f certificates sold; a n d Xb = n u m b e r o f certificates bought. T r a d e r s c o u l d n o t sell s h o r t ( t h a t is, Ec -- Xs + xb c o u l d n o t b e n e g a t i v e ) , a n d n e t francs o n h a n d (Ef + Z Ot - 5".Bj) c o u l d n o t b e negative. D u r i n g e a c h t r a d i n g p e r i o d , t r a d e r s r e c o r d e d all t h e i r p u r c h a s e s a n d sales o f certificates a n d k e p t a r u n n i n g b a l a n c e o f certificates a n d francs o n h a n d o n a r e c o r d sheet. The amount of the dividend depended on ( 1) w h e t h e r t h e c o m p a n y s u c c e e d e d o r failed, a n d ( 2 ) a r a n d o m l y assigned t r a d e r type. T h e r e w e r e t w o t r a d e r types ( I and H), w i t h different d i v i d e n d payoffs as s h o w n in the first t h r e e c o l u m n s o f Table 1. T w o trader t y p e s w e r e used to p r o m o t e trading. Half o f t h e traders w e r e r a n d o m l y assigned to t y p e I and half w e r e assigned to t y p e II in e a c h m a r k e t p e r i o d . D i v i d e n d payoffs w e r e p r i v a t e information. T r a d e r s w e r e i n f o r m e d t h a t there would be more than one trader type in e a c h p e r i o d , b u t w e r e n o t i n f o r m e d o f t h e i d e n t i t i e s o f t h e different types. In m a r k e t s e s s i o n 2, t r a d e r s w e r e g i v e n a $10 p a r t i c i p a t i o n fee ( c o m p a r e d t o $4 in s e s s i o n 1 ) a n d t o l d that a n y c u m u l a t i v e loss f r o m t r a d i n g w o u l d b e d e d u c t e d f r o m t h e $10 fee at t h e e n d o f t h e session. 14 T h e d i f f e r e n c e in p a r t i c i p a t i o n fee w a s n e c e s s a r y b e c a u s e in m a r k e t s e s s i o n 1 Bayesian t r a d e r s w e r e e x p e c t e d t o b u y at

t4 There were two other differences between market sessions 1 and 2, neither of which had (or was expected to have) any apparent effect on traders' behavior. First, in market session 2, the 16 analysts' predictions and associated outcomes were reversed. For example, in market session 1, the first period prediction was for a success and the associated outcome was a success. This was reversed to a failure prediction and a failure outcome for the first period in market session 2. All

BIAS IN PROBABILITY JUIX~MENTS

683

TABLE 1. Model predictions: probabilities of success and expected dividend values Outcome Success

Failure

0.85

0.15

Bayesian posterior prob. of success

500 300

50 25

Bayesian expected dividend value Type I Type tI

0.15

0.85

Bayesian posterior prob. of success

50 25

Bayesian expected dividend value Type I Type II

Analyst's prediction

Analyst's prediction

Success

Failure

Success

Failure

0.96

0.59

BRF predicted prob. of success

0.80

0.20

482 289

316 188

BRF expected dividend value Type I Type II

410 245

140 80

0.41

0.04

BRF predicted prob. of success

0.80

0.20

BRF expected dividend value Type I Type II

410 245

140 80

Market session 1 Prior probability Dividend values Type I Type II Market session 2 Prior probability Dividend values Type I Type II

500 300

Bayesian prices (because they had the highest e x p e c t e d p a y o f f s ) , w h i l e in m a r k e t s e s s i o n 2 b i a s e d t r a d e r s w e r e e x p e c t e d t o b u y at b i a s e d prices (because they had the highest expected payoffs). Thus, the concern was that the biased t r a d e r s in m a r k e t s e s s i o n 2 m i g h t e x p e r i e n c e sizable t r a d i n g losses a n d t h a t lack o f r e s p o n s i b i l i t y for such losses might generate overly aggressive b i d d i n g a n d artificially h i g h p r i c e s . T h i s c o n c e r n was addressed by holding traders financially r e s p o n s i b l e f o r c u m u l a t i v e t r a d i n g l o s s e s (i.e. making clear that such losses would be deducted f r o m t h e $10 p a r t i c i p a t i o n f e e ) . t 5 Subjects were either graduate-level business students or senior undergraduate economics students and thus were familiar with the t e r m i n o l o g y a n d i d e a s i n c l u d e d in t h e e x p e r i m e n t a l s e t t i n g . I n a d d i t i o n , v i r t u a l l y all s u b j e c t s

235 138

68 36

h a d p a r t i c i p a t e d in o n e e a r l i e r d o u b l e - o r a l auction asset market with a single period l i q u i d a t i n g a s s e t as p a r t o f a n o t h e r s t u d y , a n d as such were familiar with the trading mechanism before they participated in our markets.

THEORE~CAL

PREDICTIONS

Given the information provided to the traders, they could use the base rate of success (85% in m a r k e t s e s s i o n 1 o r 15% i n m a r k e t s e s s i o n 2), t h e a n a l y s t ' s p r e d i c t i o n ( s u c c e s s o r f a i l u r e ) for a given company, and the analyst's accuracy rate (80%), to calculate the Bayesian posterior probability of success for the company whose c e r t i f i c a t e s t h e y w o u l d t r a d e in a n y g i v e n p e r i o d . Within a market session, the Bayesian posterior

16 periods were similarly reversed. This reversal was necessary becanse of the reversal of base rates across market session 1 (85% probability of success ) and market session 2 ( 15% probability of success ) which was done to manipulate whether Bayesian or BRF traders had the highest expected payoffs. The reversal of signals and associated outcomes holds constant the sequence of confirmatory and disconfirmatory feedback across the two market sessions. The second difference was that in market session 2 the base rate of success and analyst's accuracy rate were displayed on a flip chart (and pointed out to the traders) as the instructions were read to the subjects and remained in full view of all traders throughout the session (see footnote 9). This was done to ensure that all traders had ready access to the information necessary to respond to the task in a Bayesian manner. Although no flip chart was used in market session 1, the base rate of success and analyst's accuracy rate were announced several times during the session, repeating the wording from the instructions. This difference works directly against the prediction that prices would be more biased in market session 2 than in market session 1, and, if anything, makes the findings in market session 2 more convincing. 1Sin fact no trader lost any part of his or her $10 participation fee. The maximum net trading loss was < $1.70>, which was covered by the 300 fr. that each subject retained each trading period.

684

A. 1L GANGULYet aL

probabilities differed across trading periods only as a result of the analyst's two possible predictions (success or failure). The middle three columns of Table 1 report the Bayesian posterior probabilities for the two market sessions. Expected payoffs (dividend values), conditional on trader-type, are also reported. In contrast to applying Bayes Rule, committing the BRF means that subjects will focus o n the analyst's prediction, and ignore the base rate differences b e t w e e n the two states. ~6This yields the BRF probability predictions shown in the last three columns o f Table 1. Expected dividend values for traders committing the BRF, conditional o n trader type, are also reported. As was the case in Anderson & Sunder (1993), Camerer (1987), and Dub & Sunder (1986, 1987), the standard competitive equilibrium m o d e l was assumed to be descriptive o f the price formation process in double-oral-auction asset markets. Assuming risk neutrality, traders' reservation prices for assets are e x p e c t e d values. ( I f they are not risk neutral, their reservation prices are certainty equivalents.) While there is no theoretical assurance that competitive equilibrium o u t c o m e s will result in double-oralauction asset markets, there is a strong empirical tendency for such markets to converge to competitive equilibrium outcomes (e.~ Forsythe e t aL, 1982; Plott & Sunder, 1982). If, as in market session 1, traders with Bayesian beliefs have higher e x p e c t e d dividend values than traders with BRF beliefs, the standard competitive equilibrium model makes a straightforward prediction that prices will reflect Bayesian (unbiased) e x p e c t e d dividend values. That is, because the supply of assets is fixed, there will be excess demand at any price b e l o w the highest e x p e c t e d dividend value, and consequently prices will converge toward the Bayesian price predictions. Looking at Table 1, w e see that, in market session 1, w h e n the

analyst predicts success, the Bayesian e x p e c t e d dividend value for type I traders ( 4 8 2 fir.) is highest of the four success-signal e x p e c t e d dividend values (i.e. 482 fr. is higher than 289 fir., 410 fr., or 245 fr.). Thus, if competition a m o n g type I Bayesian traders is sufficiently strong, prices would be e x p e c t e d to converge towards 482 fir. w h e n the analyst predicts success. Using similar logic, prices w o u l d be e x p e c t e d to converge toward 316 ft. w h e n the analyst predicts failure. If, however, traders with biased beliefs have higher e x p e c t e d dividend values than traders with Bayesian beliefs as in market session 2, the competitive equilibrium model requires additional assumptions before it leads to a prediction that prices will reflect Bayesian beliefs. In fact, the prediction is that prices will reflect BRF e x p e c t e d dividend values unless biased traders either ( 1 ) b e c o m e Bayesian traders with market experience, or ( 2 ) are not active traders. That is, under the same logic used to predict Bayesian prices w h e n Bayesian traders have the highest e x p e c t e d dividend values, w e would e x p e c t prices to b e bid u p to the BRF price predictions w h e n BRF traders have the highest e x p e c t e d dividend values unless s o m e corrective force is operating. As can be seen in Table 1, this means that prices in market session 2 would b e e x p e c t e d to converge toward the BRF e x p e c t e d dividend value o f 410 fr. w h e n the analyst predicts success and toward the BRF e x p e c t e d dividend value o f 140 ft. w h e n the analyst predicts failure. There are, of course, s o m e intuitively appealing arguments in support o f the corrective forces described above. For example, biased traders could learn to b e Bayesian as a result o f o u t c o m e feedback regarding their beliefs and/or financial punishment for trading at biased prices, or at least learn to be less confident in their beliefs, and, as such, trade less actively. However,

t6Both Camerer (1987) and Dub & Sunder (1986, 1987) investigate this interpretation of the BRF. An alternative interpretation suggested by Dub & Sunder (1986) assumes that people ignore base rates and treat imperfect signals as perfect. Thus, if the analystpredicts S (F) subjects infer that the project will be a success (failure) with certainty. Because this interpretation of the BRF does not seem plausible in our setting, and because our data clearly reject it, we do not pursue this interpretation further.

BIAS IN PROBABILITYJUDGMENTS because these are not trivial requirements, the expectation was that convergence toward Bayesian price predictions would be m o r e difficult in our market session 2 w h e r e BRF traders had the highest expected values than in market session 1 w h e r e Bayesian traders had the highest e x p e c t e d values. One distinction b e t w e e n the market structure used in the present study and those used in some previous market experiments is that no single trader had a sufficient e n d o w m e n t of francs to be able to buy the entire market supply of assets at the highest expected dividend value (Bayesian or BRF) under either signal condition. (A single trader could always hold at least one-quarter of all the outstanding assets at the highest expected dividend price and in many cases substantially m o r e . ) Although, in theory, this constraint on wealth could reduce the chance that the price would be bid up to the competitive equilibrium prediction, as discussed later, actual w e a l t h constraints had little, if any, effect on our results.

RESULTS

Market session 1 Market session 1 was designed so that Bayesian traders had the highest expected dividend values. Thus, the expectation was that market prices would converge toward Bayesian price predictions over time. In addition, based o n the previous literature and pilot testing of the instrument, the expectation was that the probability judgments of a substantial portion of the traders would be biased before they entered the market. Although there was no clear theoretical basis on which to predict w h e t h e r and h o w traders' probability judgments would change with market experience, the tentative expectation was that traders' judgments would b e c o m e more Bayesian as they e x p e r i e n c e d additional trading periods. The probability judgment and price data for market session 1 are reported in Figs 1 and 2, respectively. Both figures present the data separately for the success and failure signal cases in the order of occurrence. That is, the data for

685

the 16 market periods are separated into the 8 success signal cases and 8 failure signal cases, with the first o c c u r r e n c e of a success (failure) signal identified as o c c u r r e n c e 1, the second identified as o c c u r r e n c e 2, etc. Probability judgmentg In market session 1, the first signal was a success signal and, therefore, the Bayesian posterior probability of success was 0.96 and the BRF probability judgment prediction was 0.80. The mean of the traders' probability judgments collected after the first signal was announced, but before the first period of trading began, was 0.79 (see Fig. 1), reflecting the fact that most traders' judgments were biased in the direction of the BRF prediction before t h e y began trading in the market. Moreover, it can be seen from the individual probability judgments plotted in Fig. 1 that in most of the trading periods ( b o t h success and failure signal cases) there w e r e some traders whose probability judgments w e r e closer to the BRF prediction and some whose probability judgments w e r e closer to Bayesian judgments. To test w h e t h e r traders' probability judgments became more Bayesian with trading experience, the signed deviation of the traders' mean probability judgment from the Bayesian posterior (Probdevt) in each market period was regressed against the o c c u r r e n c e number ( O c c n u m ) indicated in Fig. 1. Separate regressions w e r e run for the success and failure signal cases. The results, which are reported in Table 2A, provide evidence that, o n average, traders' judgments did move toward Bayesian judgments across trading periods. That is, the o c c u r r e n c e number coefficient is positive and statistically signficant at conventional levels in both the success and failure signal cases. Prices A Wilcoxon matched-pairs signed-rank test was performed to compare the absolute deviation of actual prices from the Bayesian price prediction to the absolute deviation of actual prices from the BRF price prediction. The deviations from the BRF price predictions are significantly smaller (p
686

A . R . GANGULY e t al.

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A. IL GANGULYet aL TABLE 2. Changes in probability judgments and prices across trading periods

A: Probability judgments: Probdevt = a + b Occnum + et t-statistic a'

b'

Market session 1 Analyst'sprediction: Success Failure

--0229 - O.375

0.016 0.028

Market session 2 Analyst's prediction: Success Failure

0.342 0.286

-0.012 0.O00

B: Price biases: B'aaye~-- a'/(l - b'), where a' and b' are estimates from Pt - P ~

for b'

R2

3.79*** 2.73**

0.71 O.55

2.33** 0.03

0.47 O.O0

-- a + b(P~-t - Pu,yes) + et

B'Bayes~

t-Statistic

B'BaF:~

t-statistic

Market session 1 Analyst's prediction: Success Failure

- 108.1 -- 103.5

- 18.51"** 0.91

-36.1 72.5

-6.18"** 0.64

Market session 2 Analyst'sprediction: Success Failure

115.6 264.5

12.43"** 2.12"

-59.4 192.5

--6.11"** 1.54

Significantat *10%, **5%, and ***1% level, two-tailed t-test. ~¢B'Bay~ represents the estimated bias relative to the Bayesian prediction. :~B'BaF represents the estimated bias relative to the BRF prediction, obtained by substituting P'~es with PBr.~ in the regression equation. 2, prices n e v e r w e n t a b o v e t h e BRF p r e d i c t i o n in a n y success signal case, a n d w e r e closer to t h e BRF prices t h a n the Bayesian p r i c e i n all failure signal cases. 17 Despite t h e a p p a r e n t s u p e r i o r i t y of the BRF m o d e l in e x p l a i n i n g t h e p r i c e s o b s e r v e d i n m a r k e t session 1, e x a m i n a t i o n o f Fig. 2 suggests that t h e r e was s o m e m o v e m e n t t o w a r d Bayesian prices o v e r time as e x p e r i e n c e in the m a r k e t increased. In particular, i n t h e failure signal case, prices crossed o v e r the BRF p r i c e p r e d i c t i o n a n d c o n t i n u e d to m o v e t o w a r d the Bayesian p r i c e p r e d i c t i o n . T h e s e p r i c e m o v e m e n t s are c o n s i s t e n t w i t h t h e fact that p r o b a b i l i t y judgm e n t s m o v e d t o w a r d t h e Bayesian p o s t e r i o r o v e r t i m e ( s e e Fig. 1). T o e x a m i n e the p r i c e

m o v e m e n t s , w e e m p l o y e d t h e f o l l o w i n g partial adjustment model used in Camerer (1987): Pt-PBayes

=a+b(Pt-I

--enayes)+et,

(2)

w h e r e P t is the average p r i c e i n m a r k e t p e r i o d t, PBayes is t h e Bayesian p r i c e p r e d i c t i o n a n d et is a r a n d o m e r r o r term. This specification implies that t h e d e v i a t i o n f r o m e q u i l i b r i u m is r e d u c e d b y a fraction (1 - b ) each trade, w i t h t h e s p e e d o f a d j u s t m e n t inversely r e l a t e d t o h o w close b is to 1. O n e advantage o f this specificat i o n is that it yields a n e s t i m a t e d bias relative to t h e Bayesian p r e d i c t i o n , B ' n a y e s = a ' / ( 1 -b ' ) , w h e r e a ' a n d b ' are t h e o r d i n a r y least squares estimates o f a a n d b i n e q u a t i o n ( 2 ) . T a b l e 2B r e p o r t s t h e e s t i m a t e d bias relative

~7Although individual buyers were apparently wealth-constrab. :d in some success signal cases, all buyers were operating well within their wealth constraint in all failure signal cases. Thus, failure of prices to converge to the Bayesianprediction cannot be attributed to wealth constraints in the failure signal cases, and are also unlikely to be explained by the wealth constraints in the s u c c e s s signal cases.

BIAS IN PROBABILITY JUDGMENTS

689

Rather, what appears to be going on in market to the Bayesian price predictions (B'eayes) and the estimated bias relative to the BRF price session 1 is that the n u m b e r of traders with prediction (B'BRF), along with the associated t- Bayesian beliefs is simply too small, relative to statistics, m The estimated bias relative to either the market size, to generate the level of price price prediction should b e zero if the equili- competition n e e d e d to drive prices to the brium price estimated from the partial adjust- Bayesian prediction. To examine this possibility, ment model is equal to that price prediction. traders w e r e classified as either Bayesian or BRF For the success signal cases, both B'eayes and traders based on their probability judgments, B'BP.r are negative and significantly different with probability judgments closer to the Bayesian from zero, indicating that both the Bayesian posterior than the BRF prediction classified as price prediction and the BRF price prediction Bayesian. There were, on average, 4 Bayesian are significantly higher than the estimated traders per market period, verus 8 BRF traders equilibrium price. For the failure signal cases, per market period. But the theoretical price neither B'B~yes nor B'Bm~is significantly different predictions do not depend on probability judgfrom zero, indicating that both the Bayesian merits alone, but rather on relative expected model and the BRF model provide reasonable dividend values. A Bayesian with a type II dividend predictions o f the estimated equilibrium price. structure may have an e x p e c t e d dividend value Overall, the results for market session 1 that is closer to the BRF price prediction than suggest that, although both probability judg- to the Bayesian price prediction. Taking this into ments and market prices are generally closer to account, the average n u m b e r of traders per the BRF predictions than Bayesian predictions, period in market session 1 with expected there is some m o v e m e n t towards Bayesian dividend values closer to the Bayesian price probability judgments and prices across periods, prediction than to the BRF price prediction is especially in the failure signal cases. One 2.5 (versus 9.5 with e x p e c t e d dividend values possible interpretation o f these results is that closer to the BRF price prediction). These the observed market prices are really Bayesian results, along with separate calculations for the prices with risk aversion. The structure~ of success and failure signal cases, are presented market session 1 is such that type I traders in the top half o f Table 3. The amount of overall with Bayesian expectations have the highest market activity accounted for by these Bayesian e x p e c t e d dividend values so that Bayesian and BRF trader groups is also reported in Table traders could still be driving prices to the 3, with activity defined as the percentage of total Bayesian price prediction (adjusted downward bids and asks accounted for by each group to reflect traders' risk aversion). However, as (Camerer et aL, 1989). As the results reported in Table 3 show, not will b e seen later, prices in market session 2 lie considerably above Bayesian e x p e c t e d dividend only w e r e there very few Bayesian traders in values, consistent with risk-seeking behavior. market session 1 ( o n average 2.5 per market Therefore, because it is unreasonable to assume period), but these unbiased traders accounted that traders in market session 1 w e r e strongly for only 35% of the overall market activity. That risk averse, while traders in market session 2, is, even though the Bayesian traders w e r e drawn from the same sample population, w e r e disproportionately active (i.e. 21% of the total strongly risk seeking, it appears unlikely that risk traders accounted for 35% of the market aversion explains the prices observed in market activity), the BRF traders still accounted for nearly two-thirds of the overall market activity. session 1.

m The standard error of B' is calculated from a Taylor series approximation involvingthe variances of a' and/7' and their covariances~B'BRFis obtained by replacingPBa~ with PB~ in equation (2) and using the resulting a' and b' to calculate B'Bal, = a ' / ( 1 -- b ' ) .

A. R. GANGULY et aL

690

TABLE 3. Number of Bayesian and BRF traders and percentage of market activity Success signals

Failure signals

Overall market (all 16 periods)

Number of traders#

Percentage of activity,

Number of traders

Percentage of activity

Number of traders

Percentage of activity

Market session 1 Bayesian BRF

2.4 9.6

30 70

2.6 9.4

40 60

2.5 9.5

35 65

Market session 2 Bayesian BRF

8.0 4.0

54 46

3.8 8.2

22 78

5.9 6.1

38 62

t Number of traders represents the average number of traders per market period with expected dividend values closer to the price predictions of the model indicated (Bayes or BRF). Percentage of activity represents the percentage of total bids and asks accounted for by the indicated group of traders. Thus, t h e a c t i v i t y level o f t r a d e r s w i t h e x p e c t e d d i v i d e n d values c l o s e r to t h e Bayesian p r i c e prediction may simply have been too low to d r i v e p r i c e s c l o s e r to Bayesian prices. Interestingly, c o n s i s t e n t w i t h t h e fact t h a t p r i c e s m o v e d closer to Bayesian prices in the failure signal cases t h a n in t h e success signal cases ( s e e Fig, 2), t h e Bayesian traders a c c o u n t e d for a larger p e r c e n t a g e o f t h e a c t i v i t y in t h e failure signal c a s e s ( 4 0 % ) t h a n in t h e s u c c e s s signal c a s e s ( 3 0 % ) . M a r k e t session 2 M a r k e t session 2 w a s d e s i g n e d so that t h e traders whose probability judgments were b i a s e d as p r e d i c t e d b y t h e BRF h a d t h e h i g h e s t e x p e c t e d d i v i d e n d values. Thus, t h e e x p e c t a t i o n w a s t h a t m a r k e t p r i c e s w o u l d m o v e t o w a r d BRF p r i c e s o v e r time. T h e p r o b a b i l i t y j u d g m e n t a n d p r i c e d a t a for m a r k e t s e s s i o n 2 a r e r e p o r t e d in Figs 3 a n d 4, r e s p e c t i v e l y , in t h e s a m e f o r m a t u s e d t o r e p o r t t h e d a t a for m a r k e t s e s s i o n 1. Probability j u d g m e n t s In m a r k e t session 2, t h e first signal w a s a failure signal and, t h e r e f o r e , t h e Bayesian p o s t e r i o r p r o b a b i l i t y o f s u c c e s s w a s 0.04 a n d t h e BRF p r o b a b i l i t y j u d g m e n t p r e d i c t i o n w a s 0.20. T h e m e a n o f t h e t r a d e r s ' p r o b a b i l i t y j u d g m e n t s c o l l e c t e d after t h e signal w a s a n n o u n c e d , b u t b e f o r e t h e first p e r i o d o f t r a d i n g began, w a s 0.29 ( s e e Fig. 3), w h i c h is c l e a r l y c l o s e r to t h e BRF p r e d i c t i o n . Thus, as w a s t h e c a s e for m a r k e t session 1, s o m e t r a d e r s '

judgments were biased before they began t r a d i n g in m a r k e t s e s s i o n 2. Also, it c a n b e s e e n f r o m t h e i n d i v i d u a l p r o b a b i l i t i e s p l o t t e d in Fig. 3 that in m o s t t r a d i n g p e r i o d s t h e r e are s o m e traders making biased judgments and some m a l d n g Bayesian j u d g m e n t s . As w a s d o n e for m a r k e t s e s s i o n 1, t h e d e v i a t i o n s for e a c h t r a d i n g p e r i o d o f t h e m e a n p r o b a b i l i t y j u d g m e n t from t h e Bayesian p o s t e r i o r w e r e r e g r e s s e d against t h e o c c u r r e n c e n u m b e r i n d i c a t e d in Fig. 2, w i t h t h e r e s u l t s r e p o r t e d in t h e b o t t o m half o f T a b l e 2A. O n average, t h e r e w a s s o m e significant m o v e m e n t in p r o b a b i l i t y j u d g m e n t s t o w a r d s t h e Bayesian p o s t e r i o r in t h e s u c c e s s signal cases. H o w e v e r , t h e r e w a s n o c h a n g e a c r o s s p e r i o d s for failure signal cases. Prices A W i l c o x o n m a t c h e d - p a i r s i g n e d - r a n k test of deviations was performed to determine w h e t h e r t h e Bayesian m o d e l o r BRF m o d e l p r o v i d e d t h e b e t t e r e x p l a n a t i o n for p r i c e s in m a r k e t s e s s i o n 2. As e x p e c t e d , t h e BRF m o d e l significantly o u t p e r f o r m e d t h e Bayesian m o d e l (p
~9In market session 2, any binding wealth constraints actually work against finding BRF prices and in favor of finding Bayesian prices, because biased traders are constrained from buying up all of the available supply of units at biased prices.

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BIAS IN PROBABILITYJUDGMENTS Estimated biases obtained from the partial adjustment price equations are r e p o r t e d in the b o t t o m half of Table 2B. Consistent with o u r expectations, the estimated bias from the BRF price prediction is smaller than the estimated bias from the Bayesian price prediction in b o t h the success ( - 5 9 . 4 versus 115.6) and failure (192.5 versus 264.5) signal cases, with the estimated equilibrium price being significantly a b o v e the Bayesian price prediction in b o t h cases. The estimated equilibrium price is significantly b e l o w the BRF price prediction for the success s i ~ cases but not significantly different from the BRF prediction for the failure signal cases.~ Thus, although the BRF m o d e l does not provide a c o m p l e t e l y accurate price predictor, it clearly provides a better characterization of prices than the Bayesian model. Because, in market session 2, the m e a n o b s e r v e d market prices e x c e e d the Bayesian price in all 16 periods and actually m o v e away from the Bayesian price o v e r time, these prices cannot b e the result o f Bayesian beliefs with risk aversion. The only w a y these results could conceivably be interpreted as supporting the Bayesian m o d e l w o u l d b e to assume that traders started out as risk-seeking traders and b e c a m e m u c h m o r e risk-seeking as they gained market experience. Such an assumption would, of course, b e the exact opposite o f the assumption necessary to explain the market session 1 results. A m u c h simpler and m o r e compelling explanation for the results is that in market session 2 m a n y traders c o m m i t t e d the BRF and competition a m o n g these biased traders drove prices towards the BRF price prediction. The individual trader data r e p o r t e d in the b o t t o m half of Table 3 support this explanation. In market session

693

2, the average n u m b e r of traders with e x p e c t e d values closer to the BRF price prediction than the Bayesian price prediction was 6.1, and these traders accounted for 62% of overall market activity. 2° Interestingly, consistent with the fact that prices w e r e the m o s t biased in the failure signal cases (see Fig. 4), there w e r e on average 8.2 biased traders p e r period in the failure signal cases and these biased traders accounted for 78% of overall market activity. The overshooting of the BRF price prediction in the last 6 failure signal cases in market session 2 can also b e explained, at least partially, on these same grounds. On average, 5.2 traders had e x p e c t e d dividend values that w e r e above the BRF price prediction (i.e. e x p e c t e d values even m o r e biased than assumed in the BRF price predictions), and these traders appear to have generated strong competitive pressures which pushed prices above the BRF prediction.

DISCUSSION Markets in which Bayesian traders have the highest e x p e c t e d dividend value w o u l d appear to have the best chance of converging to Bayesian predicted prices. If competition a m o n g a subset o f traders holding Bayesian beliefs is strong enough, prices should b e driven to the predicted Bayesian equilibrium even though the majority of traders hold biased beliefs. However, if the n u m b e r of traders with Bayesian beliefs is small relative to the size of the market, competition m a y not be strong enough to drive prices to the predicted equilibrium. This appears to underlie the results r e p o r t e d for market session 1 where, on average, only 2.5 traders

In fact, for each of the last 6 market periods in market session 2, at least 1 trader had reached a binding constraint, being unable to purchase more units at the going market price by the end of the trading period. So that, if anything, wealth constraints might have kept prices down (i.e. closer to Bayesianprices) in market session 2. 20Based on probability judgments alone, there were on average 9.8 traders per market period in market session 2 whose probability judgments were d o ~ r to the BRF prediction, compared with 2.2 traders whose probability judgments were closer to the Bayesian posterior.

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had expected dividend values closer to the Bayesian price prediction than to the BRF price prediction. Thus, although prices moved in the direction of the Bayesian price prediction, they nevertheless remained closer to the BRF price prediction. The requirements for market prices to converge to the Bayesian prediction are considerably more demanding in markets where biased traders have the highest expected dividend values, as it takes only a subset of biased traders to drive prices away from the Bayesian prediction and toward the higher expected dividend values of the biased traders. In market session 2, a majority of the traders were biased (i.e. committed the BRF), and, on average, half of the traders had expected dividend values closer to the BRF price predictions than to the Bayesian price predictions. Further, these biased traders accounted for nearly two-thirds of the overall market activity. The result was that market prices were driven closer to the BRF price prediction and away from Bayesian price prediction.21 Although Dub & Sunder's (1986, 1987) results differ from the results of the present study in that they found Bayesian prices while this study finds biased prices, their results are nevertheless generally consistent with the hypothesis that market prices more easily converge to Bayesian price predictions w h e n Bayesian traders have higher expected dividend values than BRF traders. Dub and Sunder indicate that their conclusion that the Bayesian model dominates the BRF model is based primarily on results from market periods in which high-frequency signals occurred. High-frequency signals are

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signals associated with the higher base-rate outcome, and by definition, o c c u r more frequently than signals associated with the lower base-rate outcome. However, their design was such that, in all cases where the high-frequency signal occurred, the expected payoffs of Bayesian traders exceeded the expected payoffs of BRF traders (referred to as NBR2 traders in their studies). 22 Thus, consistent with the hypothesis proposed in the present study, prices were driven to Bayesian prices w h e n traders with Bayesian beliefs had the highest expected payoffs. Duh and Sunder find, however, that for the limited number of market periods in which low-frequency signals occurred, prices do not consistently converge to Bayesian prices. They attribute this result to traders' limited exposure to low-frequency signals in their markets. However, this result may also be due in part to the fact that, in four of the eight markets reported in Duh and Sunder (1987), BRF traders had higher expected payoffs than Bayesian traders when low-frequency signals occurred. Interestingly, prices were above Bayesian price predictions in all four of these markets, and considerably above Bayesian prices in two of them. In this respect, Dub and Sunder's findings are largely consistent with the results of the present study. As suggested earlier, a potential explanation for w h y Bayesian prices were the predominant finding in Camerer's ( 1 9 8 7 ) asset market study is that probability judgment biases may not have been very strong to begin with in that study. This possibility was ruled out in the present study by using the kind of experimental setting known to produce a strong BRF bias, and collecting

21Costs to traders for deviating from the Bayesianposterior were relatively severe, as the trading profits (i.e. X[ZOt - xs + Xb)]) of traders whose expected dividend values were closer to the Bayesian price prediction exceeded those of traders whose expected dividend values were closer to the BRF price prediction by $7.06 (44¢ on average per period) in market session 1 and $4.80 (30¢ on average per period) in market session 2 (a profit differential of 75% and 110%, respectively). Based on probability judgments alone, trading profits of traders whose probability judgments were closer to the Bayesianposterior exceeded those of traders whose probabilityjudgments were closer to the BRFprediction by $2.94 ( 18¢ on averageper period) in market session 1 and $2.35 ( 15¢ on averageper period) in market session 2 (a profit differential of 30% and 38%, respectively).

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22This is true for all four markets reported in Dub & Sunder (1986), and all eight reported in Dub & Sunder (1987). The 1987 study reports results for the four markets originally reported in the 1986 study and for four new markets.

BIAS IN PROBABILITYJUDGMENTS probability judgments to verify the existence of the bias. 23 One possible reason w h y experimental settings such as the one used in this study induce a relatively strong bias is that they provide an "anchor" (Tversky & Kahneman, 1974) that subjects use as a basis for estimating posterior probabilities. In the present study this anchor was the analyst's accuracy rate (80%). Of course, using the analyst's accuracy rate to estimate h o w likely it is that the firm will actually succeed or fail means that the base rates of success and failure will b e ignored, or at least underweighted. In balls-and-bingo-cage settings, however, a strong anchor for estimating probabilities that ignore the base rate m a y result only in cases w h e r e the sample drawn is highly "representative" of the underlying population. For less representative samples, the anchor may b e w e a k e r or nonexistent. Indeed the available evidence supports this interpretation in that it is in cases of highly representative samples that probability judgment biases (Grether, 1992) and price biases (Camerer, 1987, 1990) are most pronounced in balls-and-bingo-cage settings. A limitation of this study is that it r e t o r t s the results of only t w o market sessions. Future studies could investigate the generalizability of the present results to abstract settings such as balls.and-urn settings or other context-specific settings. Perhaps traders could m o r e easily learn to make better probability judgments in different settings. In addition, the degree to w h i c h biased probability judgments are reflected in prices might vary across market institutions as prices themselves have b e e n s h o w n to vary across institutions (Holt, in press). Thus, future w o r k

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could examine w h e t h e r the present results generalize to other types of market institutions. Finally, future w o r k using experimental asset markets could investigate the effects of specific psychological biases suggested in the capital markets literature as possible explanations for apparent price anomalies. This approach would provide a useful bridge b e t w e e n the individual judgment literature and the capital markets literature. Although the prevailing belief in both the e c o n o m i c s and accounting literature has b e e n that individual judgment biases probably do not influence market outcomes, this study and a growing amount of other evidence calls this belief into question ( s e e Berg et aL, in press; and Camerer, 1992, for reviews of s o m e of the other experimental evidence). Recent theoretical developments in the finance literature suggest that traders with biased beliefs about asset prices (referred to as noise traders) can endure in markets and influence prices ( D e Long et aL, 1989, 1990, 1991; Schleffer & Summers, 1990). Given such developments, recent experimental findings in accounting regarding individual investors' judgments (e.g. Lewis et aL, 1992; Lipe, 1994; Maines, 1990; Moser, 1989) and the growing empirical literature on price anomalies (Bernard, 1993) take o n r e n e w e d importance. Thus, there n o w seems to be reason for optimism that a combination of theoretical, experimental, and capital markets research will lead to significant i m p r o v e m e n t in our understanding of h o w individual judgments affect market outcomes.

23Economists have criticized the use of such context-specific settings on the grounds that: (1) subjects are often not told the truth about the random process examined, (2) there is often no incentive to provide correct answers, and (3) it is difficult to control information when giving verbal descriptions (Grether 1978, 1980). The first two of these criticisms were addressed in the present study by simply telling subjects the truth about the random process used and embedding the problem in a market environment with financial incentives. The last criticism was addressed by carefully wording the instructions, taking into account the common criticisms of the tasks used in previous BRF studies.

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A . R . GANGULY et aL BIBLIOGRAPHY Abarbanell, J. & Bernard, V., Tests of Analysts' Overreactionf~nderreaction to Earnings Information as an Explanation for Anomalous Stock Price Behavior, Journal of Finance (July 1992) pp. 1181-1207. Ajzen, I., Intuitive Theories of Events and the Effects of Base-rate Information on Prediction,Journal of Personality and Social Psychology (1977) pp. 303-314. Anderson, M. J. & Sunder, S., Professional Traders as Intuitive Bayesians, Working paper, Michigan State University (1993). Bar-Hillel, M., The Base-rate Fallacy In Probability Judgments, Acta Psycbologtca (1980) pp. 211-233. Bar-Hillel, M., Representativeness and Fallacies of Probability Judgments, Acta Psychologtca (1984) pp. 91-107. Bar-Hillel, M., Back to Base Rates, in Hogarth, 1Z M. (ed.), Insights in Decision Making: A Tribute to HillelJ. Einhorn, pp. 200-216 (Chicago: University of Chicago Press, 1990). Berg, J., Dickhaut, J. & McCabe, K., The Individual vs. The Aggregate, in Ashton, R. H. & Ashton, A. H. ( eds),Judgement and Decision Making Reaearch in Accounting and Auditing ( New York: Cambridge University Press, in press). Berkeley, D. & Humphreys, P., Structuring Decision Problems and the "Bias Heuristic", Acta Psychologica (1982) pp. 201-252. Bernard, V. L, Stock Price Reactions to Earnings Announcements: A Summary of Recent Anomalous Evidence and Possible Explanations, in Thaler, IZ H. (ed.), Advances in Behavioral Finance, pp. 303-340 (New York: Russel Sage Foundation, 1993). Birnbaum, M. H., Base Rates in Bayesian Inference: Signal Detection Analysis of the Cab Problem,American Journal of Psychology (1983) pp. 85-94. Camerer, C. F., Do Biases in Probability Judgment Matter in Markets? Experimental Evidence, American Economic Review (1987) pp. 981-997. Camerer, C. F., Do Markets Correct Biases in Probability Judgment? Evidence from Market Experiments, in Green, L & Kagel, J. H. (eds),Advances in BebavloralEconomtc$, pp. 126-172 (Notwod, NJ: Ablex, 1990). Camerer, C. F., The Rationality of Prices and Volume in Experimental Markets, Organizational Behavior and Human Decision Processes (1992) pp. 237-272. Camerer, C., Loewenstein, G. & Weber, M., The Curse of Knowledge in Economic Settings: An Experimental Analysis, Journal of Political Economy (1989) pp. 1232-1254. Christensen-Szalansid,J. J. J. & Beach, k R., Experience and the Base-rate Fallacy, OrganizationalBehavior and Human Performance (1982) pp. 270-278. Cohen, L J., Can Human Irrationality be Experimentally Demonstrated?, Behavioral and Brain Sciences (1981) pp. 317-331. Debondt, W. F. M. & Thaler, R. H., Does the Stock Market Overreact?Journal of Finance (July 1985) pp. 793-805. Debondt, W. F. M. & Thaler, R. H., Further Evidence of Investor Overreaction and Stock Market Seasonality, Journal of Finance (july 1987) pp. 557-582. Debondt, W. F. M. & Thaler, R. H., Do Security Analysts Overreact?, American Economic Review (May 1990) pp. 52-57. De Long, J. B., Shleifer, A., Summers, L H. & Waldmann, R.J., The Size and Incidence of the Losses from Noise Tracfing,Journal of Finance (1989) pp. 681-696. De Long, J. B., Shleifer, A., Summers, L H. & Waldmann, 1~ J., Noise Trader Risk in Financial Markets, Journal of Political Economy (1990) pp. 703-738. De Long, J. B., Shleifer, A., Summers, L H. & Waldmann, R. J., The Survival of Noise Traders in Financial Markets, Journal of Bt~ine$$ (1991) pp. 1-19. Duh, R. R. & Sunder, S., Incentives, Learning and Processing of Information in a Market Environment: An Examination of the Base-rate Fallacy, in Moriarty, S. (ed.), Laboratory Market Research, pp. 50-79 (Norman, OK: University of Oklahoma, Center for Economic and Management Research, 1986). Dull, R. R. & Sunder, S., Economic Agent as an Intuitive Bayesian, Working paper, University of Minnesota ( 1987 ). Eger, C. & Dickhaut, J., An Examination of the Conservative Information Proc&~Mng Bias in an Accounting Framework,Journal of Accounting Research ( 1982 ) pp. 711-723. Einhorn, H. J., A Synthesis: Accounting and Behavioral Science, Journal of Accounting Research (Supplement 1976)pp. 196-206. Forsythe, R., Palfrey, T. R. & Plott, C. R., Asset Valuation in an Experimental Market, EconomeW/ca (1982) pp. 537-567.

BIAS IN PROBABIIXI~ JUDGMFaN~ Gonedes, N. & Dopuch, N., Capital Market Equilibrium, Information Production, and Selecting Accounting Techniques: Theoretical Framework and Review of Empirical Work, Studies on Financial Accounting Object/yes (1974) pp. 48-129. Grether, D. M., Recent Psychological Studies of Behavior Under Uncertainty, American Economic Review (1978) pp. 70-77. Grether, D. M., Bayes' Rule as a Descriptive Model: The Representativeness Heuristic, Quarterly Journal of Economics (1980) pp. 537-557. Grether, D. M., Testing Bayes Rules and the Representativeneas Heuristic: Some Experimental Evidence, Journal of Economic Behavior and Organization (1992) pp. 31-57. Hammerton, M., A Case of Radical Probability Estimation,Journal of Experimental Psychology (1973) pp. 252-254. Hand, J. It. M., A Test of the Extended Functional Fixation Hypothesis, Accounting Review (1990) pp. 739-763. Hand, J. R. M., Extended Functional Fixation and Security Returns Around Earnings Announcements: A Reply to Ball and Kothari, Accounting Review (1991) pp. 739-746. Holt, C. A,, Industrial Organizations: A Survey of the Laboratory Research, in Kagel, J. & RotlL A. (eds), Handbook of Experimental Economics (Princeton, NJ: Princeton University Press, in press). Kahneman, D. & Tversky, A., On the Psychology ofPrediction, PsychologtcalReview (1973) pp. 237-251. Koehler, J. J., The Normative Status of Base Rates in Probabllistic Judgment, Working paper, Stanford University ( 1989 ). Lewis, B. L, Schipper, K. & Zmijewski, M., The Effect of Missing Information on Security Valuation, Working paper, University of Colorado (1992). Libby, R., Experimental Research and the Distinctive Features of Accounting Settings, in Frecka, T. J. (ecL), The Stare'of Accounting Research As We Enter the I990s, pp. 126--147 (University of Illinois Golden Jubilee Sympo6ium, 1989). Hpe, M. G., Individual Investors' Risk Judgments and investment Decisions: The Impact of Accounting and Market Data, Working paper, University of Colorado (1994). Lyon, D. & Slovic, P., Dominance of Accuracy Information and Neglect of Base Rates in Probability Estimation, Acta Psychologtca (1976) pp. 287-298. Maines, L A., The Effect of Forecast Redundancy on Judgments of a Consensus Forecast's Expected Accuracy,Journal of Accounting Research (1990) pp. 29-47. Merton, R. C., On the Current State of the Stock Market Rationality Hypothesis, in Domtmsch, It., Fischer, S. & Boasons,J. ( eds), M ~ i c s and Ftnanc~ Essays in Honor of Franco Modtgliant, pp. 93124 (Cambridge, MA: MIT Press, 1987). Moser, D. V., The Effects of Output Interference, Availability, and Accounting Information on Investors' Predictive Judgments, Accounting Review (1989) pp. 433-448. Plott, C. R. & Sunder, S., Effmiency of Experimental Security Markets with Insider Information,Journal of Polittcal Ecotmmy (1982) pp. 663-698. Schleifer, A. & Summers, L. H., The Noise Trader Approach to Finance,Journa/ofEconomtc Perspectives (1990) pp. 19-33. Shantean, J., Cognitive Heuristics and Biases in Behavioral Auditing: Review, Comments and Observations, Accountin~ Organ/xat/ons and Society (1989) pp. 165-177. Smith, J. F. & Kida, T., Heuristics and Biases: Expertise and Task Realism in Auditing, Psychological Bulletin (1991) pp. 472-489. Smith, V., Schatzberg, J. & Waller, W., Experimental Economics and Auditing, Auditing: A Journal of Practice and Theory (Fall 1987) pp. 71-93. Swieringi, P~ J. & Weick, IL E., An Assessment of Laboratory Experiments in Accounting, Journal of Accounting Research (Supplement 1982) pp. 56-101. Tversky, A. & Kahneman, D., Judgment under Uncertainty: Heuristics and Biases, Sc/ence (1974) pp. 1124-1131. Tversky, A. & Kahneman, D., Causal Schemas in Judgment Under Uncertainty, in Kahneman, D., Slovic, P. & Tversky, A. (eds),Judgment Under Uncertaingg: Heuristics and Biases, pp. 49-72 (New York: Cambridge University Press, 1982).

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APPENDIX INSTRUCTIONS This is an experiment in the economics of decision making. If you follow the instructions carefully and make good decisions you will earn m o n e y which will be paid to you in cash at the end of the experiment. At each stage of the experiment, you will be rewarded for making good decisions. The reward will be in the form of payment to you of a certain specified n u m b e r of "francs" (the experimental currency we'll use). The francs you earn at each stage will be totaled, and final payment will be made to you at the end of the entire experiment by converting francs into dollars at the rate of 1000 francs -- $1.00. In addition, you will be paid 10,000 francs for participating in the experiment. However, should you lose m o n e y in the experiment m an unlikely event m these losses will be subtracted from the 10,000 franc participation fee. The experimentwill consist of sixteen periods. Each period will consist of two phases: Phase I and Phase II. Phase I In Phase I of each period you will make decisions based o n the facts given below. These facts will remain the same for all the periods throughout the entire experiment. You are thinking of buying and selling shares of some companies that belong to the same industry. Each of these companies has an ambitious project, which may turn out to be a huge success or a total failure. The success or failure of the project in each case will result in the success or failure of the company. Before buying or selling the shares, you want to assess whether the project will succeed or fail. If you had no access to more specific information about the company, you would have estimated the chance of success for each project to be 15%, which is the normal chance of success for similar projects in similar companies. Each period, you will be given specific information regarding each company in the form of an analyst's judgment. We asked an analyst to go through the accounts and other company-specific data of each company, and then to predict which projects (and therefore, which companies) would succeed and which would fail. The analyst based his prediction entirely on the output of a computerized analysis package that exclusively uses the accounts and other company-specific data of each company as input and produces a measure of the project's (and therefore the company's success) potential. In order to know h o w good the analyst is in nlaking these predictions, we have contacted an agency that specializes in independently rating analysts' ability. On a test carried out with a large number of companies half of which had failed, our analyst (using the same computer package) had an 80% success rate. In other words, the analyst identified a failed c o m p a n y correctly 80% of the time, and a successful company also correctly 80% of the time. The companies in the test sample given to the analyst were similar to the companies about which you will be making your decisions. We have randomly selected e i g h t of the companies that the analyst said would fail and eight of those that the analyst said would succeed. We followed the affairs of these s i x t e e n companies until completion of the project and found out whether the analyst was right or w r o n g in each case. To repeat, our sample of sixteen companies includes a random sample of eight companies that the analyst said would succeed and eight companies that the analyst said would fail. For Phase I of each of the sixteen periods, w e will give you the analyst's prediction regarding the company. Based on that information, and on all other information that we have given you, you will make a probability judgment regarding the success or failure of the c o m p a n y on the

BIAS IN PROBABILITYJUDGMENTS

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"Phase I Answer Sheet" for that period. Please refer n o w to y o u r sample "Phase I Answer Sheet", and note that y o u a r e t o r e s p o n d o n o n e o f t h e t w o s c a l e s o n t h i s s h e e t , n o t b o t h . After you have indicated y o u r probability judgment on the "Phase I Answer Sheet", you will go on to Phase II for that period.

Payment scheme f or Phase I At the end of the experiment, we'll pick at random one of y o u r sixteen "Phase I Answer Sheets" and c o m p a r e y o u r answer with the answer that a statistician has given us for that period. If your answer o n the randomly selected sheet is identical to the statistician's, you will be paid 2000 francs. For every one-percent b y which y o u r answer is off-target, you lose 20 francs from the 2000. For example, if you answer is different from the statistician's answer by 4 percentage points, then y o u get [2000 -- (4 X 20)] or 1920 francs. Please note that it would b e in y o u r interest to try to do y o u r best each time you respond, since any one of y o u r sixteen answers can be selected at r a n d o m for payment. Phase II Phase II of each period will start after you have made y o u r probability judgment in Phase I. In this phase o f each period, you will participate in a market in w h i c h each of you may buy or sell stock certificates of the c o m p a n y for which you m a d e y o u r probability judgment in Phase I. All that w e told you in Phase I about a particular company, and the analyst's judgment regarding that particular company, will remain the same in Phase II. At the end of each market period, you will b e paid a dividend in francs for each stock certificate you hold. H o w m u c h the dividend will b e depends on two things: (i) Whether the company failed or succeeded. The certificate will always pay a very high dividend if the c o m p a n y succeeded, and a very low dividend if it failed and filed for bankruptcy. (ii) Your randomly assigned "trader type" for that period. Your "Record Worksheet" (described later) for each period will indicate y o u r trader type and the dividend payoff you can get for each share you hold at the end of that period. This is private information; only you will k n o w y o u r possible dividend payoffs in each period. There will be different types of traders in each market period, and the dividend payoffs they get ( u n d e r each of the two conditions: failed or s u c c e e d e d ) may b e different from what you get.

Market organization Each market period will last 4 minutes. At the beginning of each period, w e will give you 2300 francs and two stock certificates of the c o m p a n y for w h i c h you made a probability judgment in Phase I. At the end of each period, w e will charge you a "tax" of 2000 francs. Any trader with sufficient funds wishing to purchase a stock certificate, is free to make a verbal bid to b u y o n e certificate at a specified price, and any trader with a certificate is free to accept the bid. Likewise, any trader with a certificate wishing to sell one certificate, is free to make a verbal offer to sell a certificate at a specified price, and any trader with sufficient funds is free to accept the offer. Each n e w bid to buy must be higher than any outstanding bid, and each n e w offer to sell must b e lower than any outstanding offer. If a bid or offer is accepted, a trade has b e e n m a d e for a single certificate. Any ties in bids, offers, or acceptances will be resolved b y r a n d o m choice. Except for bids, offers, or acceptances you are not to speak to any other subject. T h e r e m a y b e m a n y bids and offers that are not accepted, but you are free to keep trying to do as well as you can for yourself.

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Refer n o w to your sample "Record Worksheet". At the top, you are told y o u r Trader Number, the Period Identification, y o u r Trader Type for the current period and the dividends you will get ff the c o m p a n y fails or succeeds. Directly b e l o w this is a table for you to record your transactions. In r o w O, you will find, already entered, your initial e n d o w m e n t of 2300 francs and 2 certificates. Each subsequent r o w correslxmds to a single transaction. In the first two columns you will record the price at which you bought o r sold a certificate. If, for example, you sold a certificate for 100 francs, you w o u l d enter + 100 u n d e r "SELL". This hypothetical sale has b e e n r e c o r d e d in r o w 1 of your sample Record Worksheet. Since y o u received 100 francs w h e n you sold the certificate, the balance of "Francs on Hand" ( C o l u m n 3) is increased to 2400. In addition, since you sold a certificate, the balance o f "Certificates on Hand" ( C o l u m n 4 ) is reduced to 1. Row 2 o n y o u r sample Record Worksheet shows the effect of buying a certificate for 75 francs. The balance of "Francs On Hand" and "Certificates O n Hand" are revised accordingly. The section at the b o t t o m o f the "Record Worksheet" is for calculating y o u r end-of-period wealth in francs. W h e n the period ends, you will b e told w h e t h e r the c o m p a n y had actually failed or succeeded. You enter the n u m b e r of certificates you o w n in "A", and the realized dividend for you for the given period (and for the actual o u t c o m e - fared or s u c c e e d e d ) in "B". You multiply these t w o and write that d o w n in "C". Next, you write d o w n y o u r final "Francs On Hand" in "D". You add "C" and "D" and subtract 2000 ( t a x ) from it, and write d o w n the total in "F". If the total in "F" is positive, it represents a profit. If it is negative, it represents a loss. Your net profits will b e calculated o v e r all periods and added to y o u r 10,OOO francs participation fee. Should you incur a net loss over all periods, this loss will be subtracted from y o u r participation fee. Are there any questions before w e proceed?