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 whether 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 where 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 when 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 where 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 where 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.

There is considerable evidence indicating that the probabil i ty 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 suppor ted by grants from the Economics Division of the National Science Foundation and the Joseph M. Katz Graduate School of Business. We 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 commen 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 judgment consis tent wi th laws of statistics and probability ( including Bayes' Rule) and choice consistent with expec ted utility". The te rm "bias" is used w h e n " judgments and choices are inconsistent with these normative rules in a predictable direction". Although there is some debate in the psychology literature regarding the normative appropriateness of Bayes' Rule (e.g. Berkeley & Humphreys , 1982; Cohen, 1981 ), there is widespread agreement in the accounting, finance, and economics li teratures that the use of Bayes' Rule const i tutes rational behavior.

675

676 A. IL GANGULY et aL

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 uncon- vincing (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 were 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 (1989) and Smith & Kida (1991) 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 proper- ties 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 who 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

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 evolu- tiothat~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 (1974) 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 (1976) 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 how 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 (1982) at tempted to explain how 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-

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

BIAS IN PROBABIHTY JUDGMENTS 677

agent research. To support this suggestion, they repor ted experimental results that they inter- p re ted as showing that probability judgments inferred from subjects' betting behavior demonstrated less systematic bias than direct odds estimates provided by subjects. However, as Libby (1989) 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 were actually bet ter 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 by 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 were 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 were not likely to reflect the types of judgmental biases regularly observed in individual judgment studies. Therefore, studies demonstrating individual judgment biases were considered by many to have limited relevance for accounting settings.

Recently, some accounting researchers have reconsidered whether individual probability judgment biases can affect market outcomes. For example, Libby (1989) 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 of 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,

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 by experimental economists can provide a more 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 et aL's suggestion, this study uses an experimental economics approach to examine whether 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 be 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 on prices was examined in two kinds of markets, ( 1 ) a market in which unbiased traders had the highest expected payoffs and (2 ) a market in which biased traders had the highest expected payoffs. In the first kind of market (where unbiased traders had the highest expected payoffs), our expectation was that competit ive forces would lead to unbiased prices. In the second kind of market (where the biased traders had the highest expected payoffs), the expectat ion was that competi t ive forces would result in biased prices. The rationale was that in cases where the biased traders have the highest expec ted payoffs, it is difficult for prices to be unbiased because competi t ion among the biased traders will drive prices up to the biased traders' expectations unless the biased traders either ( 1 ) become unbiased as a result of their exper ience in the markets, or ( 2 ) are inactive, and thus allow unbiased traders to set market prices. Prob- ability judgments were collected from individual traders before each trading period (16 repeti- tions in each market) in order to assess how

BIAS IN PROBABILITY JUDGMENTS 679

ness" s eemed to pred ic t pr ices be t te r than the Bayesian model . 5 As indicated earlier, Dub & Sunder (1986 , 1987) conc lude that market pr ices in their exper imenta l markets w e r e c loser to Bayesian pr ice predic t ions than to the pr ice predic t ions associated wi th several psychological theories, including the pr ice predic t ions ob ta ined by assuming traders c o m m i t t e d the BRF. Anderson & Sunder ( 1 9 9 3 ) used procedures similar to Camere r ( 1 9 8 7 ) to examine the relative pe r fo rmance of professional traders and students in experimental markets. Although they conclude that the professional traders' performance was more Bayesian than that o f the students, prices in the professional-trader markets were nevertheless often closer to BRF price predictions than to Bayesian pr ice predict ions.

O n e p r o b l e m wi th the exper imenta l market studies descr ibed above is that they represent joint tests o f ( 1 ) w h e t h e r the individual j udgmen t bias exists ( in the specific experi- menta l set t ing used) before the traders en te r the market, ( 2 ) w h e t h e r the individual judg- m e n t bias persists in the market setting, and ( 3 ) w h e t h e r market pr ices reflect any exist ing bias. Thus, it is difficult to in terpret the results o f these studies. In particular, because the BRF was never shown to exist in the Anderson & Sunder ( 1 9 9 3 ) and Dub & Sunder ( 1986, 1987) studies, and was very weak to begin wi th in Camerer ' s ( 1 9 8 7 ) study, it is no t clear w h e t h e r market forces had any cor rec t ive influence. 6 The presen t s tudy handles this p rob lem by incorpo-

rating into the design a p rob lem s t ruc ture that p r o d u c e s a relatively s t rong BRF bias and then testing w h e t h e r market forces el iminate this individual judgment bias and w h e t h e r any individual j udgmen t bias is reflected in market prices.

A second p rob lem with the previous studies is that they did no t distinguish markets in w h i c h biased traders had the highest expec t ed payoffs f rom markets in w h i c h unbiased traders had the highest expec ted payoffs. Thus, it is no t clear w h e t h e r any judgmen t biases w h i c h may have existed in those studies w o u l d have been m o r e likely to be reflected in pr ices in markets w h e r e biased traders had the highest expec ted payoffs. We argue that it is m o r e difficult for pr ices to be unbiased in markets w h e r e biased traders have the highest expec t ed payoffs, because biased traders will bid pr ices up to their biased expectat ions. This issue was addressed in the present s tudy by designing two types o f markets, one in w h i c h unbiased t raders had the highest expec ted payoffs and a s econd in wh ich biased traders had the highest e x p e c t e d payoffs.

EXPERIMENTAL DESIGN

This s tudy consists o f two market sessions, one in wh ich Bayesian (unb iased) t raders had the highest expec t ed dividend values (marke t session 1 ) a n d one in w h i c h BRF (b iased) t raders had the highest expec t ed 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.

680 A. 1L GANGULY et aL

(marke t session 2). In each market session, 12 t raders par t ic ipated in 16 per iods of t rading in a double-ora l auct ion. They t raded certificates that had one-per iod lives and paid a l iquidat ing d iv idend at the end of the market period. Subjects we re requ i red to provide probabi l i ty judgment s of the state of na tu re at the start of each t rading period.

S t a t e p r o b a b i l i t i e s a n d p r o c e d u r e s f o r e U c i t i n g

p r o b a b i l i f y j u d g m e n t s

Base-rate probabilities and subsequent informa-

t ion were provided to subjects in a context- specific setting that mapped into the "cab problem" which is known to generate a significant BRF. 7 The setting dealt with buying and selling shares of a company engaged in an "ambitious project" that would result in either a 'q3uge success or a total failure" (a copy of the instructions is provided in the Appendix). For each o f t h e 16 t rading periods, the certificates and the related expected payoffs were associated wi th a different company. The d iv idend payoff d e p e n d e d on w h e t h e r the c o m p a n y succeeded or failed.

Traders we re in formed that, in the absence of any fur ther information, the normal chance

of success for compan ies like the 16 used in the e x p e r i m e n t was 85% (marke t session 1 ) o r 15% (marke t session 2). However , before each t rading period, t raders we re g iven an analyst 's

p r ed i c t i on of e i ther success or failure for the c o m p a n y whose certificates they w o u l d be t rading that period. Traders w e r e in formed that the analyst used on ly company-specif ic informa- t ion to make his p red ic t ions and that he was 80% accurate in identifying firms that succeeded

and firms that failed. They were also told that the analyst 's accuracy rate was d e t e r m i n e d by tes t ing h im wi th a large sample of companies , half of w h i c h had succeeded and half of wh ich had failed, s Finally, t raders we re in formed that the 16 compan ies used in the e x p e r i m e n t consis ted of 8 r andomly se lec ted compan ies f rom those that the analyst said w o u l d succeed and 8 r andomly se lec ted compan ies from those that the analyst said w o u l d fa i l 9 All of this

informat ion was p rov ided in the ins t ruct ions , w h i c h were read a loud to the subjects. Pilot tes t ing indica ted that, like the cab p rob lem, our exper imen ta l set t ing p r o d u c e d a significant BRF

bias. Al though the exper imen ta l set t ing was con-

7 One version of the "cab problem" is as follows: "Two cab companies operate in the same city, the Blue a n d Green (according to the color o f the cab they run). Eighty-five percent o f the cabs in the city are Blu~ a n d 15 percent are Greert A cab was involved in a hi t -and-run accident a t n ight in which a pedestr ian was run ova . An eyewitness identified the cab as a Green cab. The court tested the witness's abi l i ty to dis t inguish between Blue a n d Green cabs under n ight t ime visibil i ty condi t ion~ I t f o u n d that the wi tness was correct 80 percen t o f the t ime bu t confused i t wi th the other color 20 percent o f the t im~ Wha t is the probabi l i ty that the hi t -and-run cab was Green? "" The median and modal response for this problem is typically 0.80, while the Bayesian posterior 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.


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 o u t c o m e s ( s u c c e s s o r fa i lure o f a c o m p a n y ) in

a d v a n c e o f t h e e x p e r i m e n t a l s e s s i o n s u s i n g t h e

s a m e p r o c e d u r e s u s e d in t h o s e s t ud i e s . 1°

T h e e x p e r i m e n t a l s e t t i n g i n c o r p o r a t e d t h o s e

f e a t u r e s s u g g e s t e d in 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 ayes i an r e v i s i o n is t h e n o r m a t i v e l y

a p p r o p r i a t e w a y f o r s u b j e c t s t o r e s p o n d t o o u r

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 Bayes i an

l i ke l i hood ra t io m a y n o t b e i n d e p e n d e n t o f e i t h e r

b a s e r a t e s o r p r i o r p r o b a b i l i t i e s ( B i r n b a u m ,

1 9 8 3 ) w a s g i v e n c a r e f u l c o n s i d e r a t i o n . T h i s

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 i t c l e a r t h a t t h e

analyst " b a s e d h i s j u d g m e n t e n t i r e l y o n t h e

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". 1~

During the experimental sessions, traders were given the analyst's prediction for the company whose certificates they would 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 would 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-outcome pairs) 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, prediction- outcome 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.


the Bayesian pos t e r io r , t hey r e c e i v e d 2000 fr. (The e x p e r i m e n t a l c u r r e n c y used was francs.) For eve ry 1% of abso lu te dev ia t ion f rom the Bayesian pos te r io r , the i r p a y m e n t was r e d u c e d b y 20 fr. At the end o f the e x p e r i m e n t , this p a y m e n t was a d d e d to any profi ts e a r n e d f rom t rad ing in the asset markets , t he deta i l s o f w h i c h are d e s c r i b e d be low.

The man ipu la t i on o f e x p e c t e d payoffs was a c c o m p l i s h e d b y r eve r s ing the base ra tes of success and failure in m a r k e t sess ions 1 and 2. That is, as exp l a ined in m o r e de ta i l la ter , r evers ing the base ra tes across the t w o m a r k e t sess ions r e su l t ed in the Bayesian t raders having the h ighes t e x p e c t e d payoffs in m a r k e t sess ion 1, and the BRF t raders having the h ighes t e x p e c t e d payoffs in m a r k e t sess ion 2.

Market procedures In each m a r k e t pe r iod , 12 t raders w e r e each

e n d o w e d wi th t w o assets tha t l ived one p e r i o d and pa id a l iquida t ing s t a t e - d e p e n d e n t d iv idend . All t r ad ing and earn ings w e r e in t e rms o f francs, w h i c h w e r e c o n v e r t i b l e in to dol lars at t he ra te o f $1.00 p e r 1000 fr. at the end o f the exper i - ment . Traders w e r e each e n d o w e d w i t h 2300 fr. in each t rad ing pe r iod , and 2000 fr. w e r e s u b t r a c t e d f rom each t r ade r ' s to ta l f rancs at the end o f each per iod . Traders w e r e a l l owed to re ta in the 300 ft. d i f fe rence each t rad ing p e r i o d to he lp offset any po ten t i a l losses f rom trading.

Trade r s vo lun ta r i ly e x c h a n g e d assets in a double-oral -auct ion asset market. Buyers shou ted ou t b ids at w h i c h t hey w e r e wi l l ing to buy, se l lers shou t ed o u t offers at w h i c h they w e r e wi l l ing to sell. Bids had to t op ou t s t and ing b ids and offers had to b e b e l o w ou t s t and ing offers. A m a t c h i n g b id and offer was a t r ade w h i c h e ra sed all p r ev ious b ids and offers. All bids, offers, and t r ades 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 vis ible to all t raders . ( N o h i s to ry o f p r e v i o u s m a r k e t p e r i o d t r ades was d i sp layed . ) T rad ing p e r i o d s las ted 4 minutes .

At the e n d o f each t rad ing p e r i o d the state, S (ucces s ) o r F(a i lure) , was a n n o u n c e d and t raders ca l cu l a t ed the i r profits. Dol lar profi ts a re

g iven by

Profits = X[Ef - Rf + Y. Oi -- 5". Bj + D(N) (Ec-- xs + xb)], ( 1 )

w h e r e X = dol lar -per- f ranc c o n v e r s i o n rate; Ef = init ial e n d o w m e n t in francs; Rf -- a m o u n t o f f rancs r e p a i d at pe r iod -end ; Oj = sel l ing p r i c e o f i th cer t i f icate sold; Bj = p u r c h a s e p r i c e o f j t h cer t i f ica te b o u g h t ; D ( N ) = d iv idends p e r cer t i f icate g iven s ta te o f n a t u r e N ; E c = init ial e n d o w m e n t in cert if icates; xs = n u m b e r o f cer t i f icates sold; and Xb = n u m b e r o f cer t i f icates

bought . T rade r s c o u l d no t sel l sho r t ( t ha t is, Ec -- Xs

+ xb c o u l d no t b e nega t ive) , and ne t francs on hand (Ef + Z Ot - 5". Bj) c o u l d no t b e negat ive. Dur ing each t rad ing per iod , t r aders r e c o r d e d all the i r pu rchase s and sales o f cer t i f icates and kep t a r unn ing ba lance o f cer t i f icates and francs on hand o n a r e c o r d sheet .

The a m o u n t of the d i v i d e n d d e p e n d e d o n ( 1 ) 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, and ( 2 ) a r a n d o m l y ass igned t r ade r type. The re w e r e two t rader types ( I and H), w i th different d iv idend payoffs as shown in the first three columns of Table 1. Two t rader types w e r e used to p r o m o t e trading. Half of the t raders w e r e randomly assigned to type I and half w e r e assigned to type II in each m a r k e t per iod . D iv idend payoffs w e r e p r iva te informat ion. Traders w e r e i n fo rmed tha t t h e r e w o u l d b e m o r e than one t r ade r type in each pe r iod , b u t w e r e no t i n fo rmed o f the iden t i t i e s o f t he di f ferent types.

In m a r k e t sess ion 2, t r aders w e r e g iven a $10 pa r t i c ipa t ion fee ( c o m p a r e d to $4 in sess ion 1 ) a n d to ld that any cumula t ive loss f rom t rad ing w o u l d b e d e d u c t e d f rom the $10 fee at the e n d o f t he session. 14 The d i f fe rence in pa r t i c ipa t ion fee was neces sa ry b e c a u s e in marke t sess ion 1 Bayesian t r ade r s w e r e e x p e c t e d to 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

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

683

Outcome Analyst's prediction Analyst's prediction

Success Failure Success Failure Success Failure

Market session 1 Bayesian posterior

Prior probability 0.85 0.15 prob. of success 0.96 0.59

Bayesian expected Dividend values dividend value Type I 500 50 Type I 482 316 Type II 300 25 Type tI 289 188

Market session 2 Bayesian posterior

Prior probability 0.15 0.85 prob. of success 0.41 0.04

Bayesian expected Dividend values dividend value Type I 500 50 Type I 235 68 Type II 300 25 Type II 138 36

BRF predicted prob. of success 0.80 0.20

BRF expected dividend value

Type I 410 140 Type II 245 80

BRF predicted prob. of success 0.80 0.20

BRF expected dividend value

Type I 410 140 Type II 245 80

Bayes i an p r i c e s ( b e c a u s e t h e y h a d t h e h i g h e s t

e x p e c t e d payof fs ) , w h i l e in m a r k e t s e s s ion 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 a t b i a s e d

p r i c e s ( b e c a u s e t h e y h a d t h e h i g h e s t e x p e c t e d

payof fs ) . Thus , t h e c o n c e r n w a s tha t t h e b i a s e d

t r a d e r s in m a r k e t s e s s ion 2 m i g h t e x p e r i e n c e

s izable t rad ing losses and tha t lack o f respons ib i l i ty

fo r s u c h lo s se s m i g h t g e n e r a t e o v e r l y a g g r e s s i v e

b i d d i n g a n d ar t i f ic ia l ly h i g h p r i ce s . Th is c o n c e r n

w a s a d d r e s s e d b y h o l d i n g t r ade r s f inancia l ly

r e s p o n s i b l e fo r c u m u l a t i v e t r a d i n g losses (i .e.

m a k i n g c l e a r t h a t s u c h losses w o u l d b e d e d u c t e d f r o m t h e $10 p a r t i c i p a t i o n fee ) . t 5

S u b j e c t s w e r e e i t h e r g r a d u a t e - l e v e l b u s i n e s s

s t u d e n t s o r s e n i o r u n d e r g r a d u a t e e c o n o m i c s

s t u d e n t s a n d t h u s w e r e fami l ia r w i t h t h e

t e r m i n o l o g y a n d ideas i n c l u d e d in t h e e x p e r i -

m e n t a l se t t ing . In add i t ion , v i r t u a l l y all s u b j e c t s

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 -

a u c t i o n asse t m a r k e t w i t h a s ing le p e r i o d

l i q u i d a t i n g asse t as p a r t o f a n o t h e r s tudy , and as

s u c h w e r e fami l i a r w i t h t h e t r a d i n g m e c h a n i s m

b e f o r e t h e y p a r t i c i p a t e d in o u r marke t s .

T H E O R E ~ C A L P R E D I C T I O N S

G i v e n t h e i n f o r m a t i o n p r o v i d e d t o t h e t r ade r s ,

t h e y c o u l d u s e t h e b a s e r a t e o f s u c c e s s ( 8 5 %

in m a r k e t s e s s i o n 1 o r 15% in m a r k e t s e s s i o n

2) , t h e ana lys 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 )

fo r a g i v e n c o m p a n y , and t h e analys t ' s a c c u r a c y

r a t e ( 8 0 % ) , t o c a l c u l a t e t h e Bayes i an 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 fo r t h e c o m p a n y w h o s e

ce r t i f i ca t e s t h e y w o u l d t r a d e in any g i v e n p e r i o d .

W i t h i n a m a r k e t sess ion , t h e Bayes i an p o s t e r i o r

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 GANGULY et 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 repor t the Bayesian poster ior 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 on the analyst's prediction, and ignore the base rate differences be tween the two states. ~6 This yields the BRF probabili ty predictions shown in the last three columns of Table 1. Expected dividend values for traders committing the BRF, conditional on trader type, are also reported.

As was the case in Anderson & Sunder (1993) , Camerer (1987), and Dub & Sunder (1986, 1987), the standard competi t ive equilibrium model was assumed to be descriptive of the price formation process in double-oral-auction asset markets. Assuming risk neutrality, traders' reservation prices for assets are expected values. ( I f they are not risk neutral, their reservation prices are certainty equivalents.) While there is no theoretical assurance that competi t ive equili- br ium outcomes will result in double-oral- auction asset markets, there is a strong empirical tendency for such markets to converge to competitive equilibrium outcomes (e.~ Forsythe et aL, 1982; Plott & Sunder, 1982).

If, as in market session 1, traders wi th Bayesian beliefs have higher expec ted dividend values than traders with BRF beliefs, the standard competi t ive equilibrium model makes a straight- forward predict ion that prices will reflect Bayesian (unbiased) expected dividend values. That is, because the supply of assets is fixed, there will be excess demand at any price be low the highest expec ted dividend value, and consequently prices will converge toward the Bayesian price predictions. Looking at Table 1, we see that, in market session 1, when the

analyst predicts success, the Bayesian expected dividend value for type I traders (482 fir.) is highest of the four success-signal expec ted dividend values (i.e. 482 fr. is higher than 289 fir., 410 fr., or 245 fr.). Thus, if compet i t ion among type I Bayesian traders is sufficiently strong, prices would be expec ted to converge towards 482 fir. when the analyst predicts success. Using similar logic, prices would be expec ted to converge toward 316 ft. when the analyst predicts failure.

If, however, traders with biased beliefs have higher expec ted dividend values than traders with Bayesian beliefs as in market session 2, the competi t ive equilibrium model requires additional assumptions before it leads to a predict ion that prices will reflect Bayesian beliefs. In fact, the predict ion is that prices will reflect BRF expec ted 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 when Bayesian traders have the highest expec ted dividend values, we would expec t prices to be bid up to the BRF price predictions when BRF traders have the highest expec ted dividend values unless some correct ive force is operating. As can be seen in Table 1, this means that prices in market session 2 would be expec ted to converge toward the BRF expec ted dividend value of 410 fr. when the analyst predicts success and toward the BRF expec ted dividend value of 140 ft. when the analyst predicts failure. There are, of course, some intuitively appealing arguments in support o f the correct ive forces described above. For example, biased traders could learn to be Bayesian as a result o f ou tcome 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 analyst predicts 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.


because these are not trivial requirements, the expecta t ion was that convergence toward Bayesian pr ice predictions would be more difficult in our market session 2 where BRF traders had the highest expected values than in market session 1 where Bayesian traders had the highest expected values.

One distinction be tween the market structure used in the present study and those used in some previous market experiments is that no single trader had a sufficient endowment 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 more . ) Although, in theory, this constraint on wealth could reduce the chance that the price would be bid up to the competit ive equilibrium prediction, as discussed later, actual wea l th 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 expectat ion was that market prices would converge toward Bayesian price predictions over time. In addition, based on the previous literature and pilot testing of the instrument, the expecta t ion was that the probability judgments of a substantial port ion of the traders would be biased before they entered the market. Although there was no clear theoretical basis on which to predict whether and how traders' probability judgments would change with market experience, the tentative expectat ion was that traders' judgments would become more Bayesian as they exper ienced additional trading periods.

The probability judgment and price data for market session 1 are repor ted 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

the 16 market periods are separated into the 8 success signal cases and 8 failure signal cases, with the first occurrence of a success (failure) signal identified as occurrence 1, the second identified as occurrence 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 ey 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 (both success and failure signal cases) there were some traders whose probability judgments were closer to the BRF prediction and some whose probability judgments were closer to Bayesian judgments.

To test whether traders' probability judgments became more Bayesian with trading experience, the signed deviation of the traders' mean probability judgment from the Bayesian poster ior (Probdevt) in each market period was regressed against the occurrence number (Occnum) indicated in Fig. 1. Separate regres- sions were run for the success and failure signal cases. The results, which are reported in Table 2A, provide evidence that, on average, traders' judgments did move toward Bayesian judgments across trading periods. That is, the occurrence 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<O.O01), indicating that the BRF model outperforms the Bayesian model as a predictor of prices. Further, looking at Fig.

686 A . R . GANGULY e t al.

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TABLE 2. Changes in probability judgments and prices across trading periods

A: Probability judgments: Probdevt = a + b Occnum + et

t-statistic a' b' for b' R 2

Market session 1 Analyst's prediction:

Success --0229 0.016 3.79*** 0.71 Failure - O. 375 0.028 2.73* * O. 55


Success 0.342 -0.012 2.33** 0.47 Failure 0.286 0.O00 0.03 O.O0

B: Price biases: B'aaye~ -- a'/(l - b'), where a' and b' are estimates from Pt - P ~ -- a + b(P~-t - Pu,yes) + et

B'Bayes~ t-Statistic B'BaF:~ t-statistic


Success - 108.1 - 18.51"** -36.1 -6.18"** Failure -- 103.5 0.91 72.5 0.64


Success 115.6 12.43"** -59.4 --6.11"** Failure 264.5 2.12" 192.5 1.54

Significant at *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, pr ices never w e n t above the BRF p red ic t ion in any success signal case, and were closer to the BRF prices than the Bayesian pr ice in all failure signal cases. 17

Despite the apparent super ior i ty of the BRF mode l in expla in ing the pr ices observed in marke t session 1, examina t ion of Fig. 2 suggests that there was some m o v e m e n t toward Bayesian pr ices over t ime as expe r i ence in the marke t increased. In part icular , in the failure signal case, pr ices crossed over the BRF pr ice p red ic t ion and c o n t i n u e d to move toward the Bayesian pr ice predict ion. These pr ice m o v e m e n t s are cons i s ten t wi th the fact that probabi l i ty judg- m e n t s m o v e d toward the Bayesian pos te r ior over t ime (see Fig. 1). To examine the pr ice

movemen t s , w e emp loyed the fol lowing partial ad jus tment mode l used in Camerer (1987) :

P t - P B a y e s = a + b ( P t - I - - e n a y e s ) + e t , ( 2 )

w h e r e Pt is the average pr ice in market pe r iod

t, PBayes is the Bayesian pr ice p red ic t i on and et is a r a n d o m error term. This specification implies that the devia t ion f rom equ i l ib r ium is r e duc e d by a fract ion (1 - b ) each trade, wi th the speed o f a d j u s t m e n t inversely re la ted to h o w close b is to 1. O n e advantage of this specifica- t ion is that it yields an es t imated bias relat ive to the Bayesian predic t ion , B'nayes = a ' / ( 1 - -

b ' ) , w h e r e a ' and b ' are the ord inary least squares est imates of a and b in equa t ion (2) .

Table 2B repor ts the es t imated bias relative

~7 Although 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 Bayesian prediction 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.


to the Bayesian price predictions (B'eayes) and the estimated bias relative to the BRF price prediction (B'BRF), along with the associated t- statistics, m The estimated bias relative to either price prediction should be zero if the equilibrium price estimated from the partial adjustment model is equal to that price prediction. For the success signal cases, both B'eayes and B'BP.r are negative and significantly different from zero, indicating that both the Bayesian price predict ion and the BRF price prediction are significantly higher than the estimated equilibrium price. For the failure signal cases, nei ther B'B~yes nor B'Bm~ is significantly different from zero, indicating that both the Bayesian model and the BRF model provide reasonable predictions of the estimated equilibrium price.

Overall, the results for market session 1 suggest that, although both probability judgments and market prices are generally closer to the BRF predictions than Bayesian predictions, there is some movement towards Bayesian probability judgments and prices across periods, especially in the failure signal cases. One possible interpretation of these results is that the observed market prices are really Bayesian prices with risk aversion. The structure~ of market session 1 is such that type I traders with Bayesian expectations have the highest expec ted dividend values so that Bayesian traders could still be driving prices to the Bayesian price prediction (adjusted downward to reflect traders' risk aversion). However, as will be seen later, prices in market session 2 lie considerably above Bayesian expec ted dividend values, consistent with risk-seeking behavior. Therefore, because it is unreasonable to assume that traders in market session 1 were strongly risk averse, while traders in market session 2, drawn from the same sample population, were strongly risk seeking, it appears unlikely that risk aversion explains the prices observed in market session 1.

Rather, what appears to be going on in market session 1 is that the number of traders with Bayesian beliefs is simply too small, relative to the market size, to generate the level of price competi t ion needed to drive prices to the Bayesian prediction. To examine this possibility, traders were classified as either Bayesian or BRF traders based on their probability judgments, with probability judgments closer to the Bayesian posterior than the BRF prediction classified as Bayesian. There were, on average, 4 Bayesian traders per market period, verus 8 BRF traders per market period. But the theoretical price predictions do not depend on probability judg- merits alone, but rather on relative expected dividend values. A Bayesian with a type II dividend structure may have an expected dividend value that is closer to the BRF price prediction than to the Bayesian price prediction. Taking this into account, the average number of traders per period in market session 1 with expected dividend values closer to the Bayesian price prediction than to the BRF price prediction is 2.5 (versus 9.5 with expected dividend values closer to the BRF price prediction). These results, along with separate calculations for the success and failure signal cases, are presented in the top half of Table 3. The amount of overall market activity accounted for by these Bayesian and BRF trader groups is also repor ted in Table 3, with activity defined as the percentage of total bids and asks accounted for by each group (Camerer et aL, 1989).

As the results repor ted in Table 3 show, not only were there very few Bayesian traders in market session 1 (on average 2.5 per market period), but these unbiased traders accounted for only 35% of the overall market activity. That is, even though the Bayesian traders were disproportionately active (i.e. 21% of the total traders accounted for 35% of the market activity), the BRF traders still accounted for nearly two-thirds of the overall market activity.

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

690 A. R. GANGULY et aL

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

Overall market Success signals Failure signals (all 16 periods)

Number Percentage Number Percentage Number Percentage of traders# of activity, of traders of activity of traders of activity

Market session 1 Bayesian 2.4 30 2.6 40 2.5 35 BRF 9.6 70 9.4 60 9.5 65

Market session 2 Bayesian 8.0 54 3.8 22 5.9 38 BRF 4.0 46 8.2 78 6.1 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 he ac t iv i ty level o f t r ade r s w i th e x p e c t e d d i v i d e n d values c lose r to the Bayesian p r i c e p r e d i c t i o n m a y s imply have b e e n t o o l o w to d r ive p r i ces c lose r to Bayesian pr ices . In teres t - ingly, cons i s t en t w i t h the fact tha t p r i ces m o v e d c loser to Bayesian pr ices in the failure signal cases than in the success signal cases ( s ee Fig, 2), the Bayesian t raders accoun ted for a larger pe rcen tage o f t he ac t iv i ty in t he fai lure signal cases ( 4 0 % ) than in t he success signal cases ( 3 0 % ) .

Marke t session 2 Market sess ion 2 was des igned so that the

t r aders w h o s e p r o b a b i l i t y j u d g m e n t s w e r e b i a sed as p r e d i c t e d b y the BRF had the h ighes t e x p e c t e d d i v i d e n d values. Thus, t he e x p e c t a t i o n was tha t ma rke t p r i ce s w o u l d m o v e t o w a r d BRF p r i ce s o v e r t ime. The p r o b a b i l i t y j u d g m e n t and p r i c e da ta for marke t sess ion 2 are r e p o r t e d in Figs 3 and 4, respec t ive ly , in the same format u s e d to r e p o r t t he da ta for m a r k e t sess ion 1.

Probabil i ty j u d g m e n t s In marke t sess ion 2, the first s ignal was a fa i lure signal and, there fore , the Bayesian p o s t e r i o r p robab i l i t y o f success was 0 .04 and the BRF p robab i l i t y j u d g m e n t p r e d i c t i o n was 0.20. T h e m e a n o f the t raders ' p r o b a b i l i t y j udgmen t s c o l l e c t e d after the signal was announced , bu t be fo re the first p e r i o d o f t r ad ing began, was 0.29 ( s ee Fig. 3), w h i c h is c lea r ly c lose r to the BRF p red ic t ion . Thus, as was the case for marke t sess ion 1, s o m e t r ade r s '

j u d g m e n t s w e r e b i a sed b e f o r e t hey began t rad ing in marke t sess ion 2. Also, i t c an b e seen f rom the ind iv idua l p robab i l i t i e s p l o t t e d in Fig. 3 that in mos t t r ad ing p e r i o d s t h e r e are s o m e t r ade r s mak ing b i a sed j u d g m e n t s and s o m e maldng Bayesian judgments .

As was d o n e for m a r k e t sess ion 1, the dev ia t ions for each t rad ing p e r i o d o f t he mean probabi l i ty judgment f rom the Bayesian pos te r io r w e r e r eg re s sed against t h e o c c u r r e n c e n u m b e r i nd i ca t ed in Fig. 2, w i th the resul t s r e p o r t e d in t he b o t t o m half o f Table 2A. O n average, t h e r e was s o m e significant m o v e m e n t in p robab i l i t y j u d g m e n t s t o w a r d s the Bayesian p o s t e r i o r in t he success signal cases. Howeve r , t h e r e was no change across p e r i o d s for fa i lure signal cases.

Prices A W i l c o x o n ma tc he d -pa i r s igned-rank tes t o f dev ia t ions was p e r f o r m e d to d e t e r m i n e w h e t h e r the Bayesian m o d e l o r BRF m o d e l p r o v i d e d the b e t t e r exp lana t ion for p r i ces in m a r k e t sess ion 2. As e x p e c t e d , t he BRF m o d e l s ignif icantly o u t p e r f o r m e d the Bayesian m o d e l (p<O.O01) . Moreover , m a r k e t p r i ces ac tua l ly m o v e d away f rom Bayesian p r i c e s t o w a r d BRF p r i ce s in the success signal cases, and m o v e d b e y o n d BRF p r i ce s in t h e fai lure signal cases ( s e e Fig. 4). That is, p r i ces b e c a m e m o r e b iased w i t h t r ad ing e x p e r i e n c e , w i th m e a n p r i ce s in the last four success signal p e r i o d s and the last s ix fai lure signal p e r i o d s be ing c lose r to the BRF p r e d i c t i o n than the Bayesian p red ic t ion . ~9

~9 In 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.

BIAS IN PROBABILITY J U D G M E N T S 691

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Estimated biases obtained from the partial adjustment pr ice equations are repor ted in the bo t t om half of Table 2B. Consistent wi th our expectations, the estimated bias from the BRF price predict ion is smaller than the est imated bias f rom the Bayesian price predict ion in bo th the success ( - 5 9 . 4 versus 115.6) and failure (192.5 versus 264.5) signal cases, with the est imated equilibrium price being significantly above the Bayesian price predict ion in both cases. The estimated equilibrium price is significantly be low the BRF price predict ion for the success s i ~ cases but not significantly different f rom the BRF predict ion for the failure signal cases.~ Thus, although the BRF model does not provide a comple te ly accurate pr ice predictor, it clearly provides a be t ter characterization of prices than the Bayesian model.

Because, in market session 2, the mean observed market prices exceed the Bayesian pr ice in all 16 per iods and actually move away from the Bayesian price over time, these prices cannot be the result o f Bayesian beliefs with risk aversion. The only way these results could conceivably be interpreted as supporting the Bayesian model would be to assume that traders started out as risk-seeking traders and became much more risk-seeking as they gained market experience. Such an assumption would, of course, be the exact opposi te of the assumption necessary to explain the market session 1 results.

A much simpler and more compell ing explanation for the results is that in market session 2 many traders commit ted the BRF and competi t ion among these biased traders drove prices towards the BRF price prediction. The individual t rader data repor ted in the bo t tom half of Table 3 support this explanation. In market session

2, the average number of traders with expec ted values closer to the BRF price predict ion than the Bayesian price predict ion was 6.1, and these traders accounted for 62% of overall market activity. 2° Interestingly, consistent with the fact that prices were the mos t biased in the failure signal cases (see Fig. 4), there were on average 8.2 biased traders pe r per iod in the failure signal cases and these biased traders accounted for 78% of overall market activity. The overshoot- ing of the BRF price predict ion in the last 6 failure signal cases in market session 2 can also be explained, at least partially, on these same grounds. On average, 5.2 traders had expec ted dividend values that were above the BRF price predict ion (i.e. expec ted values even more biased than assumed in the BRF price predictions), and these traders appear to have generated strong compet i t ive pressures which pushed prices above the BRF prediction.

DISCUSSION

Markets in which Bayesian traders have the highest expec ted dividend value would appear to have the best chance of converging to Bayesian predic ted prices. If compet i t ion among a subset o f traders holding Bayesian beliefs is strong enough, prices should be driven to the predic ted Bayesian equilibrium even though the majority of traders hold biased beliefs. However, if the number of traders with Bayesian beliefs is small relative to the size of the market, compet i t ion may not be strong enough to drive prices to the predicted equilibrium. This appears to underlie the results repor ted 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 Bayesian prices) in market session 2.

20 Based on probability judgments alone, there were on average 9.8 traders per market period in market session 2 whose probability judgments were do~r to the BRF prediction, compared with 2.2 traders whose probability judgments were closer to the Bayesian posterior.

694 & IZ GANGULY e t aL

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 when Bayesian traders have higher expected dividend values than BRF traders. Dub and Sunder indicate that their conclusion that the Bayesian model domi- nates the BRF model is based primarily on results from market periods in which high-frequency signals occurred. High-frequency signals are

signals associated with the higher base-rate outcome, and by definition, occur 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 when 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 why Bayesian prices were the predominant finding in Camerer's (1987) 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

21 Costs to traders for deviating from the Bayesian posterior were relatively severe, as the trading profits (i.e. X[ZOt - ZBj + D ( N X E c - 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 Bayesian posterior exceeded those of traders whose probability judgments were closer to the BRF prediction by $2.94 ( 18¢ on average per period) in market session 1 and $2.35 ( 15¢ on average per period) in market session 2 (a profit differential of 30% and 38%, respectively).

22 This 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.


probabil i ty judgments to verify the existence of the bias. 23 One possible reason why 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 poster ior probabilities. In the present study this anchor was the analyst's accuracy rate (80%). Of course, using the analyst's accuracy rate to estimate how likely it is that the firm will actually succeed or fail means that the base rates of success and failure will be ignored, or at least underweighted. In balls-and-bingo-cage settings, however , a strong anchor for estimating probabilities that ignore the base rate may result only in cases where the sample drawn is highly "representat ive" of the underlying population. For less representat ive samples, the anchor may be weaker or nonexistent. Indeed the available evidence supports this interpretation in that it is in cases of highly representative samples that probabil i ty 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 to r t s the results of only two 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 more easily learn to make better probability judgments in different settings. In addition, the degree to which biased probabil i ty judgments are reflected in prices might vary across market institutions as prices themselves have been shown to vary across institutions (Holt, in press). Thus, future work

could examine whe the r the present results generalize to other types of market institutions. Finally, future work using exper imental asset markets could investigate the effects of specific psychological biases suggested in the capital markets literature as possible explanations for apparent pr ice anomalies. This approach would provide a useful bridge be tween the individual judgment literature and the capital markets literature.

Although the prevailing belief in both the economics and accounting literature has been that individual judgment biases probably do not influence market outcomes, this study and a growing amount of o ther evidence calls this belief into question (see Berg et aL, in press; and Camerer, 1992, for reviews of some of the other experimental evidence). Recent theoretical developments in the finance li terature suggest that traders wi th biased beliefs about asset prices ( referred to as noise t raders) can endure in markets and influence prices (De 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 li terature on price anomalies (Bernard, 1993) take on renewed importance. Thus, there now seems to be reason for opt imism that a combinat ion of theoretical, experimental , and capital markets research will lead to significant improvement in our under- standing of h o w individual judgments affect market outcomes.

23 Economists 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.


B I B L I O G R A P H Y

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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 money 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 number 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 money in the experiment m an unlikely event m these losses will be subtracted from the 10,000 franc participation fee.

The experiment will 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 on 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 how 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 company 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 igh 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 ix teen companies until completion of the project and found out whether the analyst was right or wrong 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, we 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 company on the


"Phase I Answer Sheet" for that period. Please refer now to your sample "Phase I Answer Sheet", and note that y o u are to r e s p o n d o n o n e o f t h e t w o sca les o n t h i s s h e e t , n o t b o t h . After you have indicated your probabil i ty judgment on the "Phase I Answer Sheet", you will go on to Phase II for that period.

Payment scheme for Phase I At the end of the experiment , we' l l pick at random one of your sixteen "Phase I Answer Sheets"

and compare your answer with the answer that a statistician has given us for that period. If your answer on the randomly selected sheet is identical to the statistician's, you will be paid 2000 francs. For every one-percent by which your answer is off-target, you lose 20 francs from the 2000. For example, if you answer is different f rom the statistician's answer by 4 percentage points, then you get [2000 -- (4 X 20)] or 1920 francs. Please note that it would be in your interest to try to do your best each t ime you respond, since any one of your sixteen answers can be selected at random for payment.

Phase II Phase II of each period will start after you have made your probabili ty judgment in Phase I. In

this phase of each period, you will participate in a market in which each of you may buy or sell s tock certificates of the company for which you made your probabil i ty judgment in Phase I.

All that we 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 be paid a dividend in francs for each stock certificate you hold. How much the dividend will be depends on two things:

( i ) Whether the company failed or succeeded. The certificate will always pay a very high dividend if the company succeeded, and a very low dividend if it failed and filed for bankruptcy.

(i i) Your randomly assigned "trader type" for that period. Your "Record Worksheet" (descr ibed later) for each per iod will indicate your 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 know your possible dividend payoffs in each period. There will be different types of traders in each market period, and the dividend payoffs they get (under each of the two conditions: failed or succeeded) may be different from what you get.

Market organization Each market period will last 4 minutes. At the beginning of each period, we will give you 2300

francs and two stock certificates of the company for which you made a probabili ty judgment in Phase I. At the end of each period, we 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 buy one 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 new bid to buy must be higher than any outstanding bid, and each new offer to sell must be lower than any outstanding offer. If a bid or offer is accepted, a trade has been made for a single certificate. Any ties in bids, offers, or acceptances will be resolved by random choice. Except for bids, offers, or acceptances you are not to speak to any other subject. There may be many bids and offers that are not accepted, but you are free to keep trying to do as well as you can for yourself.


• Y o u r r e c o r d s

Refer n o w to your sample "Record Worksheet". At the top, you are told your Trader Number, the Period Identification, your Trader Type for the current per iod and the dividends you will get ff the company fails or succeeds. Directly be low this is a table for you to record your transactions. In row O, you will find, already entered, your initial endowment of 2300 francs and 2 certificates. Each subsequent row correslxmds to a single transaction. In the first two columns you will record the pr ice at which you bought or sold a certificate. If, for example, you sold a certificate for 100 francs, you would enter + 100 under "SELL". This hypothetical sale has been recorded in row 1 of your sample Record Worksheet. Since you received 100 francs w h e n you sold the certificate, the balance of "Francs on Hand" (Column 3) is increased to 2400. In addition, since you sold a certificate, the balance of "Certificates on Hand" (Column 4) is reduced to 1. Row 2 on your sample Record Worksheet shows the effect of buying a certificate for 75 francs. The balance of "Francs On Hand" and "Certificates On Hand" are revised accordingly.

The section at the bo t tom of the "Record Worksheet" is for calculating your end-of-period wealth in francs. When the per iod ends, you will be told whe ther the company had actually failed or succeeded. You enter the number of certificates you own in "A", and the realized dividend for you for the given per iod (and for the actual o u t c o m e - fared or succeeded) in "B". You multiply these two and write that down in "C". Next, you wri te down your final "Francs On Hand" in "D". You add "C" and "D" and subtract 2000 ( t ax) f rom it, and write down 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 be calculated over all periods and added to your 10,OOO francs participation fee. Should you incur a net loss over all periods, this loss will be subtracted from your participation fee.

Are there any questions before we proceed?

the effects of biases in probability judgments on market prices

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