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American Economic Association
Do Biases in Probability Judgment Matter in Markets? Experimental Evidence
Author(s): Colin F. Camerer
Source: The American Economic Review, Vol. 77, No. 5 (Dec., 1987), pp. 981-997
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/1810222
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Do Biases in Probability Judgment Matter in Markets?
Experimental Evidence
By COLIN F. CAMERER*
setting is designed so that prices and allocations will reveal whether traders use Bayes’
rule to integrate the prior and the sample
information, or whether they judge the likelihood of each state by the “representativeness” of the sample to the state (Amos
Tversky and Kahneman, 1982b). (Several
other non-Bayesian psychological theories
Microeconomic theory typically concerns
exchange between individuals or firms in a
market setting. To make predictions precise,
individuals are usually assumed to use the
laws of probability in structuring and revising beliefs about uncertainties. Recent evidence, mostly gathered by psychologists,
suggests probability theories might be inadequate descriptive models of individual choice.
(See the books edited by Daniel Kahneman
et al., 1982a, and by Hal Arkes and Kenneth
can be tested, too.)
Evidence of judgment bias reported by
psychologists poses an implicit challenge to
economic theory based on rationality. Sometimes that challenge is made explicit, as when
Kenneth Arrow suggested that use of the
representativeness heuristic ” typifies very
precisely the excessive reaction to current
information which seems to characterize all
the securities and futures markets” (1982, p.
5). Others have warned that judgment biases
will affect the judgments of well-trained experts who make societal decisions (about the
risk of low-probability hazards, for instance,
see Paul Slovic, Baruch Fischhoff, and Sarah
Lichtenstein, 1976).
Assertions as bold as Arrow’s are ex-
Hammond, 1986.)
Of course, individual violations of normative theories of judgment or choice may be
corrected by experience and incentives in
markets, thus producing market outcomes
which are consistent with the individualrationality assumption even if that assumption is wrong for most agents. Whether judgment and choice violations matter in markets
is a question that begs for empirical analysis.
In this paper I use experimental markets
to address this issue (see also Rong Duh and
Shyam Sunder, 1986; and Vernon Smith,
1982, for an overview). In these markets,
traders are paid dividends for holding a
one-period asset. The amount of the dividend depends upon which of two states occurred. Traders know the prior probabilities
of the states, and a sample of likelihood
information about which state occurred. The
tremely rare, because the faith that individual irrationality will not affect markets is a
strong part of the “oral tradition” in economics. This faith is often defended with
Milton Friedman’s (1953) famous claim that
theories with false assumptions (such as
strong assumptions of individual rationality)
might still predict market behavior well (see
Mark Blaug, 1980, pp. 104-14, for a cogent
discussion). Besides that “F-twist,” there is a
standard list of arguments used to defend
economic theories from the criticism that
people are not rational. (Counterarguments
are given in parentheses.)
1) In markets, agents have enough
financial incentive, and experience, to avoid
mistakes. (Incentives and experience were
provided in David Grether’s 1980 experiments on the representativeness heuristic. See
*Department of Decision Sciences, The Wharton
School, University of Pennsylvania, Philadelphia, PA
19104. I thank Mike Chemew, Marc Knez, Peter Knez,
and Lisabeth Miller for research assistance. Helpful
comments have been received from three anonymous referees, Greg Fischer, Len Green, David Grether,
Dan Kahneman, John Kagel, Peter Knez, George
Loewenstein, Charles Plott, Paul Slovic, Shyam Sunder,
Richard Thaler, Keith Weigelt, and especially Howard
Kunreuther. This research was funded by the Wharton
Risk and Decision Processes Center, the Alfred Sloan
Foundation grant no. 8551, and the National Science
Foundation grant no. SES-8510758.
981
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982 THE A MERICA N ECONOMIC RE VIEW DECEMBER 1987
also Charles Plott and Louis Wilde, 1982, p.
(Subjects see the outcomes
97.)
I Red of three draws with replacement)
2) Random mistakes of individuals will
cancel out. (The biases found by psychologists are generally systematic -most people
err in the same direction.)
3) Only a small number of rational
agents are needed to make market outcomes
rational, if those agents have access to
enough capital or factors of production. (Institutional constraints may prevent those
agents from making markets rational; see
Thomas Russell and Richard Thaler, 1985.)
4) Agents who are less rational may
learn implicitly from the actions of more
2 Black
CgeX
1J I
1-6
/ 10 bulls
Numbered 1-10
Inital Cage
710
1 Black l
C
Y
FIGURE 1
rational agents. (This argument requires that
“more rational” agents are identifiable, perence, and learning opportunity than in previhaps by their more vigorous trading.)
ous judgment experiments.
5) Agents who are less rational may
learn explicitly from more rational agents by
I. Experimental Design
buying advice or information. (Institutional
constraints, and the well-known problems of
In the experiments, each of 8 or 10 traders
adverse selection and moral hazard, may
is endowed with two assets that live one
limit the extent of information markets.)
period and pay a liquidating state-dependent
6) Agents who are less rational may be
dividend.
driven from the market by bankruptcy, either
by natural forces or at the hands of more
A. State Probabilities
rational competitors. (A new supply of agents
who are less rational, or inexperienced, may
The state is represented by which one of
be constantly entering the market.)
two bingo cages (X or Y) is chosen (Figure
Most of these arguments, though not all of
1). A third bingo cage containing 10 balls is
them, are put to the test in the market
used to determine whether cage X or cage Y
experiments described below. Subjects trade
has been chosen. The X cage contains 1 red
for up to 7 hours, observing nearly 100 realiand 2 black balls. The Y cage contains 2 red
zations of the state variable, and every trade
balls and 1 black one. The prior probabilities
earns them a (small) dollar profit or loss
of X and Y are .6 and .4.1 Figure 1 is shown
(argument 1). The representativeness heurison a blackboard for all subjects to see,
tic is systematic in direction (argument 2).
throughout the experiment.
Subjects trade with one another in a “douAfter either X or Y is chosen (but not
ble-oral” auction with no constraints on bidannounced), a sample of three balls is drawn
ding or offering activity (argument 3), so
from the chosen cage, with replacement, and
they can learn implicitly from others’ trading the sample is announced before trading bebehavior (argument 4).
gins. Since the cages X and Y contain differMany of the standard arguments are not
ent populations of balls, which are known
tested in the experiments: There is no exto traders, they can use Bayes’ rule to calcuplicit market for advice (argument 5); sublate P(X/sample) from the prior P(X) and
jects cannot sell short (argument 3); and
bankruptcy is unlikely, though conceivable
(argument 6). The first two arguments are
1Unequal priors were chosen because priors of .5 and
being tested in further work. Even with these
.5 might have made it too easy for subjects to intuit the
limits, the market experiments provide a
Bayesian posteriors. Experiments with equal priors are a
greater combination of incentives, experinatural direction for future work.
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VOL. 77 NO. 5 CAMERER: BIASES IN PROBABILITYJUDGMENT 983
TABLE 1-BAYESIAN EXPECTED DIVIDEND VALUES
Bayesian Posterior P( X/sample)
.923 .750 .429 .158
Bayesian Expected Valuesa
Type
No. of Traders Dividend No. of Reds
(Experiment No.) X Y 0 1 2
3
I 5 (1,3-5,11x-15xh) 4 (2,9r,10h) 500 200 477 425 329 247
II 5 (1,3-5,llx-15xh) 4 (2,9r,10h) 350 650 373 425 521 603
I 5 (6-7) 4 (8,12x) 525 225 502 450 354 272
II 5 (6-7) 4 (8,12x) 180 480 203 255 351 433
Note: All dividends were actually 80 francs higher, for both types of traders and in both
states, in experienced subjects experiments llx, 13x-15xh. (Therefore, all Bayesian
expected values are 80 francs higher, too.) In all analyses prices are adjusted for this
80-franc difference.
aIn francs.
the likelihood functions P(sample/X) and
P(sample/Y) (which are determined by the
cage contents). The top line of Table 1 gives
the Bayesian posteriors for all three-ball
samples. The possible samples are characterized by the number of reds only, since
the order of draws should not matter and the
data suggest the order did not matter to
subjects. (In some experiments, like John
Hey’s 1982 experiments on price search,
order does seem to matter. Subjects were
paid in his 1987 experiments and order still
mattered.)
B. Market Procedure
Subjects were undergraduate men, and
some women, recruited from quantitative
methods and economics classes at the Wharton School. These students have all taken
statistics and economics courses. Experiments 1 to 10 used subjects who had not
been in any previous market experiments.
Five experiments used “experienced” subjects who had been in experiments 1 to 10;
these experiments are numbered llx to 15xh
(the “x” reminds the reader that subjects
were experienced). Experiments were conducted in one 3-hour session (experiments 1
and 2, 6, 9 and 10, llx to 15xh) or two
2-hour sessions held on consecutive evenings
(experiments 3 to 5, 7 and 8).
All trading and earnings are in terms of
francs, which are converted to dollars at the
end of the experiment at a rate of $.001
dollars per franc ($.0015 in experiment 1).2
Traders are endowed with 10,000 francs and
two certificates in each trading period, and
10,000 francs is subtracted from their total
francs at the end of each period. In some
experiments a known fixed cost (around
5,000 francs) was subtracted from their total
earnings at the end of the experiment.
Traders voluntarily exchange assets in a
“double-oral auction”: Buyers shout out bids
at which they will buy, sellers shout out
offers at which they will sell. Bids must top
outstanding bids and offers must undercut
outstanding offers. A matching bid and offer
is a trade, which erases all previous bids and
offers. All bids, offers, and trades in a period
are recorded by the experimenter on a transparency visible to subjects. (No history of
previous periods of trading is posted.) Trading periods last 4 minutes in 10-subject experiments, 3 minutes in 8-subject experiments.
At the end of each trading period the state
(X or Y) is announced and traders calculate
2In practice, using francs makes traders more precise
in their trading than they would be with dollars, for
example, traders routinely haggle over 5-franc dif-
ferences between bids and offers, which represent half a
penny. Francs may also alleviate competition among
traders for relative status in dollar earnings, because
traders’ dollar conversion rates (while identical) are
privately known.
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984 THE AMERICAN ECONOMIC REVIEW DECEMBER 1987
their profits. Dollar profits are given by
(1) PROFITS
Xs
Xb
=X Ef-R + , 0i- L Bj +D(S)
L ~ =lI j=l
x (EC-xS+ Xb)-F
surance that competitive equilibrium will
result. However, simple models of the double-oral auction as a dynamic game with
incomplete information are beginning to
establish the theoretical tendency of doubleoral auctions to converge to competitive
equilibrium (Daniel Friedman, 1984; Robert
Wilson, 1985; see David Easley and John
Ledyard, 1986). The empirical tendency to
converge is well-established (for example,
Smith, 1982), even in designs meant to inhibit
convergence (Smith and Arlington
Ef= initial endowment in francs,
Rf= amount of francs repaid at per-Williams, in press).
where X= dollar-per-franc conversion rate,
iod-end,
E= initial endowment in certificates,
x= number of certificates sold,
D. Competing Theories
In each experiment, traders are randomly
assigned to either of two “types,” which
Bj= purchase price of jth certificate differ in the dividends they receive in the two
bought,
states X and Y (see Table 1). The dividends
D(S) = dividends per certificate in state
are chosen so that competing theories predict different patterns of prices and allocaS,
F = fixed cost per experiment in
tions (see Table 2). Each theory will now be
francs.
described briefly.
Bayesian. If traders use Bayes’ rule to
Traders may not sell short (that is, EC -X
+ xb cannot be negative), and net francs on
calculate posterior probabilities given the
hand (Ef + Yi – -B1) cannot be negative.sample data, prices should converge to the
Bayesian expected values given in Table 2,
C. Market Equilibrium
assuming risk neutrality. (Tests and controls
for risk neutrality are described below.) In
Assuming risk neutrality, traders’ reservathe experiments described by the top panel
of Table 2, for instance, type I traders should
tion prices for assets are expected values. (If
they are not risk neutral, their reservation
pay up to 477 if the sample is 0 reds, 425 if 1
prices are certainty equivalents.) Since each
red, 329 if 2 reds, and 247 if 3 reds. Type II
trader’s endowment of francs is large enough
traders should pay up to 373, 425, 521, and
to buy virtually the entire market supply of
603, respectively. Therefore, if the sample is
assets, and the supply is fixed (by the initial
0 reds, then type I traders should buy from
endowment, and the short-selling restriction),
type II traders at a price of 477. If the
there is excess demand at any price less than
sample is 2 or 3 reds, the type II traders
the highest expected value. Thus, in competishould buy all the units, at prices of 521 or
tive equilibrium, prices should be bid up to
603, respectively. If the sample is 1 red, then
the largest expected value of any trader. One
type I and type II traders both have a Bayesirrational trader who pays too much can
ian expected value of 425 francs, so we extherefore create a market price that is too
pect half the units will be held by each of the
high. The empirical question is whether such
two types of traders. (Trades might take
traders exist, and whether the experience
place because of uncontrolled differences in
and financial discipline of a market makes
risk tastes, but units are still equally as likely
them more rational over the course of an
to end up in the hands of type I and type II
experiment.
traders.) In experiments 6 to 8 and 12x,
Of course, the double-oral auction is not
dividends were chosen so that the Bayesian
Walrasian, so there is no theoretical asexpected values of the type I and type II
0i= selling price of ith certificate sold,
Xb= number of certificates bought,
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VOL. 77 NO. 5 CA MER ER: BIA SES IN PROBA BILITY JUDGMENT 985
TABLE 2-PRICE AND ALLOCATION PREDICTIONS OF COMPETING THEORIES
Predictions Expressed as: Price P (Type Holding Assets)
Number of Reds in Sample
Theory 0 Reds 1 Red 2 Reds 3
Reds
Experiments 1-5, 9r-11x, 13x-15xh
Bayesian 477 (I) 425 (1,II) 521 (II) 603 (II)
Exact Representativeness 477 (I) P > 425 (I) P > 521 (II) 603 (II)
Conservatism P < 477 (I) P > 425 (II) P < 521 (II) P < 603 (II)
Overreaction P > 477 (I) P > 425 (I) P > 521 (II) P > 603 (II)
Base-Rate Ignorance 467 (I) 450 (II) 550 (II) 617 (II)
Experiments 6-8, 12x
Bayesian 502 (I) 450 (I) 354 (I) 433 (II)
Exact Representativeness 502 (I) P > 450 (I) P > 354 (II) 433 (II)
Conservatism P < 502 (I) P < 450 (I) P > 354 (I) P < 433 (II)
Overreaction P > 502 (I) P > 450 (I) P > 354 (II) P > 433 (II)
Base-Rate Ignorance 492 (I) 425 (II) 380 (II) 447 (II)
traders were (nearly) equal when a 2-red
sample was drawn.
Exact Representativeness. If subjects
traders will hold units in 1-red periods. (Recall that the Bayesian theory predicts types I
and II are equally likely to hold units in
take the representativeness of the sample to
1-red periods.)
the cage contents as a psychological index of
Base-Rate Ignorance. If subjects judge
P(state/sample) by the representativeness of
samples to states, their judgments may ignore
differences in the prior probabilities (or
“base rates”) of states (Tversky and Kahneman, 1982b). In our setting it is difficult to
integrate this aspect of representativeness
with other aspects, like the psychological
power of exact representativeness, because
the two aspects often work in opposite directions. In 1-red samples, for instance, exact
representativeness predicts P( X/1 red) will
be overestimated, while ignorance of the
higher base rate of X implies P(X/1 red)
will be underestimated. Since predictions of
a theory that integrates representativeness
with base-rate ignorance are ambiguous, I
define base-rate ignorance as using Bayes’
rule with erroneous priors P(X) = P(Y) =
.5. Predictions of this theory are shown in
the cage’s likelihood, non-Bayesian expected
values might result. Representativeness is a
vague notion, but we can distinguish some
precise variants of it. For instance, subjects
might think P(X/sample) =1, if the sample
resembles the X-cage contents more closely
than the Y-cage contents. Or they might
think P( X/sample) = 1, if the sample exactly matches the X-cage contents. These
extreme hypotheses are clearly ruled out by
the data presented below.
More reasonably, subjects may be intuitively Bayesian for most samples, but overestimate a cage’s likelihood when a sample
resembles the cage exactly. This “exact representativeness” theory predicts that subjects
will judge P( X/1 red) to be greater than the
Bayesian posterior .75 because a 1-red sample exactly matches the X-cage’s contents.
Similarly, P(Y/2 red) will be judged to be
greater than .57; other probabilities will be
Bayesian. Of course, there are other possible
interpretations but since they are either imprecise or clearly incorrect, only exact repre-
sentativeness will be considered carefully.
Under exact representativeness, prices will
be higher than Bayesian in 1- and 2-red
periods (as shown in Table 2) and type I
Table 2.
Of course, ignoring base rates completely
is rather implausible. For example, in an
experiment with a prior probability of .001,
it seems unlikely that subjects will act as if
the prior is .5. If priors are simply underweighted, but not ignored, the data will
show some statistical support for the complete base-rate ignorance theory. The theory
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986 THE A MERICAN ECONOMIC REVIEW DECEMBER 1987
should be considered an extreme benchmark
that helps us judge whether priors are underweighted at all.
Conservatism. Subjects may be “conservative” in adjusting prior probabilities
for sample evidence (for example, Ward
(francs)
e
p
t
e
a
Edwards, 1968).
Overreaction. Subjects may adjust prior
probabilities too much, as if overreacting to
sample evidence. The overreaction theory
k 0 red I red 2 reds 3 reds
6
1
1
12
5
makes the same prediction as representativeness in 1- and 2-red periods, but it predicts
bias in 0- and 3-red periods where representativeness does not. Note that the conFIGURE 2
servatism and overreaction theories make exactly opposite predictions. This implies quite
a challenge for the Bayesian theory: Prices
(lower) end of the confidence interval. Bayesmust be exactly at the Bayesian prediction,
ian expected values are shown by dashed
or insignificantly different from it, for bothlines, and the direction of the exact repretheories to be falsified.
sentativeness prediction is shown by an arrow
number
II. Results
Fifteen experiments have been conducted
ten with inexperienced subjects, five with
experienced subjects- excluding two inconclusive pilot experiments. For the sake of
brevity, many details of the analyses are
omitted and can be found in working papers
available from the author.
There are two kinds of data which distinguish between theories: prices at which
trades occurred, and the number of units of
the asset that traders held at the end of
trading periods.
A. Trade Prices
The mean prices across experiments 1 to 8
are summarized by a time-series of 90 percent confidence intervals, shown in Figure
2.3 The upper (lower) solid line is the upper
3 Confidence intervals were constructed by first
calculating mean prices in each period of each experiment, then separating the time-series of mean prices for
each different sample. Data from experiments 6 to 8
were normalized so that the Bayesian predictions in
those experiments were the same as in experiments 1 to
5. This yields groups of data such as 8 mean prices from
the first 0-red period in each of the 8 experiments
Of
marked “R.” Each of the four panels represents a different sample. From left to right,
observations within a panel represent data
from the first time that sample was drawn,
the second time the same sample was drawn,
and so forth.
Prices converge, from below, toward the
Bayesian levels. These data clearly rule out
many non-Bayesian theories of probability
judgment (like the two extreme brands of
representativeness mentioned above). How-
ever, prices do not converge exactly to the
Bayesian expected values. There is some evidence of exact representativeness, because
prices drift above the Bayesian expected values in 1- and 2-red periods. However, the
confidence intervals are wide, and the degree
of bias is rather small. Indeed, since prices
should only converge to Bayesian predictions if the hypotheses of risk neutrality,
numbered 1 to 8. The mean of those means, and its
standard error (the standard deviation divided by 81/2)
are used to calculate the 90 percent confidence interval.
A second confidence interval was calculated using mean
prices from the second 0-red period in each of the 8
experiments, and so on. Not all experiments have the
same number of 0-red periods, so the number of observations in each confidence interval gradually decreases. The procedure was stopped just before there
was only one experiment left with an Nth observation
of a particular sample.
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VOL. 77 NO. 5 CAMERER: BIASES IN PROBABILITYJUDGMENT 987
for (2), it works well empirically and there is
no well-established theory of price convergence which suggests it is wrong.
650R
Call the bias for the t th price B,; it equals
Pt- PBayes. If we define equilibrium as a bias
(fra ncs
that does not change each period, we impose
Bt =Bt_1 = B on (2) and get
(3) B=a+bB+e,.
r
red
t
Figure13
red
2
shows
reds
3
reds
confidence
number
of
perie
6
ds
FIGURE 3
Since E(e,) = 0, a little alg
we can estimate the degree of equilibrium
bias B consistently by the estimator B’=
a’/(l- b’), where a’ and b’ denote ordinary
least squares estimators of a and b in (2).
The standard error of B’ can be calculated
from a Taylor series approximation involvcompetitive equilibrium, and Bayesian updating are all true simultaneously, it is rather ing the variances of a’ and b’ and their
remarkable that prices converge as closely to covariance.
Regressions were first run separately for
the Bayesian predictions as they do.
Figure 3 shows confidence intervals from
each period, effectively allowing a and b to
vary each period. The simple specification
experiments with experienced subjects. Prices
(2) fit fairly well: The convergence rate b
begin closer to the Bayesian expected value,
was typically estimated precisely, and residuand have less tendency to drift above it in land 2-red periods. The confidence intervals
als were uncorrelated and roughly homoare also wide, because they summarize a
skedastic. An F-test (Jan Kmenta, 1971, p.
small number of experiments.4
373) was used to test whether adjacent periWe can define bias in prices as a deviation
ods could be pooled at the 10 percent level.
from the Bayesian prediction. If the BayesPeriods were pooled, starting with the last
ian theory is true, biases will be around zero.
period, until the F-test was violated.
To conduct statistical tests on price biases,
The estimate B’ resulting from the last
the time-series of prices in each experiment
group of poolable periods in each experimust be independent. Since prices are typiment are shown in Table 3. Also reported is
the t-statistic testing the hypothesis that B =
cally autocorrelated, the equilibrium degree
of bias is estimated from a simple partial
0, which is simply B’ divided by its (apadaptation model (a first-order autoregresproximated) standard error. Sample sizes
sion),
are shown in parentheses next to each experiment number. T-statistics marked with
(2) Pt PBayes a + b(Pt-l PBayes) + e,
asterisks are unreliable because the assumption of normality of residuals was violated at
where P1 is the t th observation of price
the 1 and
percent level, by the studentized range
PBayes is the Bayesian prediction. This
specification implies that the deviation from
equilibrium is reduced by a fraction 1 – b
each trade. If b is close to 1, convergence is
very slow; if b is close to 0, convergence is
fast. While there is no theoretical rationale
5I thank Dave Grether for correcting a mistake in
earlier estimates of V(B’). The Taylor series approximation of a’/(l – b’) around its true value a/(I – b) is
a/(1- b)+(a’- a)/(1- b)+ a(b’- b)/(l- b)2, plus
some higher-order terms. Using this expression to
calculate (approximately) V(a/(I – b)), or E[(a’/(1 4Intervals flare out in Figure 3 when the number of
different experiments used to construct them drops
steeply and standard errors increase dramatically.
b’) – a/(I – b))21 yields V(a’)/(l – b)2 + a2V(b’)/(l
– b)4 + 2aCOV(a’, b’)/(l – b)3. Evaluating this ex-
pression at a’ and b’ gives approximations of V( B’).
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988 THE AMERICAN ECONOMIC REVIEW DECEMBER 1987
TABLE 3-ESTIMATES OF BIAS IN EQUILIBRIUM PRICES, AND TESTS OF
THE BAYESIAN HYPOTHESIS AGAINST COMPETING HYPOTHESES
0-Red Periods
Bias Significance Levels, Bayesian vs. Base-Rate
Experiment (n) B t-Statistic Conservatism Overreaction Ignorance
Inexperienced subjects
1 (24) -28.31 2.31 .01 .99 .06
2 (46) 19.28 10.95 .999 .000 .000
3 (54) -11.09 – 3.67* .000 .999 .999
4 (57) 4.97 4.56* .999 .000 .000
5 (10) – 22.35 – 6.57 .000 .000 .999
6 (29) 15.36 5.67* .999 .000 .999
7
(70)
32.44
.65*
.76
.24
.57
8 (7) – 37.90 – 2.20 .99 .01 .12
9r (9) 10.01 8.70 .000 .999 .999
lOh
(10)
mean
Experienced subjects
-2.54
–
-1.16*
2.95
.12
.49
.88
.51
.999
.57
llx (53) -44.12 12.15 .000 .999 .999
12x (34) 15.17 1.78 .96 .04 .01
13x
(8)
76.50
.49
.69
.31
.51
14x (18) 11.61 3.64 .999 .000 .999
15xh (16) 4.92 1.41 .92 .08 .35
mean
2.14
.71
.29.57
1-Red Periods
Exact Representativeness, Base-Rate
Bayesian vs. Overreaction Conservatism Ignorance
1 (13) 5.00 2.63* .005 .005 .999
2 (40) 56.34 7.94* .000 .000 .000
3
(40)
1.18
.29*
.46
.46
.999
4 (25) 49.81 18.94 .000 .000 .000
5 (37) 31.19 10.23 .000 .000 .000
6 (28) 51.80 9.10 .000 .999 .999
7 (16) 23.12 4.65 .001 .999 .985
8
(57)
93.83
.18
.43
.57
.51
9r (8) 51.15 6.21 .000 .000 .000
lOh (50) 54.63 14.12* .000 .000 .000
mean
39.92
.09
.30
.45
llx (44) – 2.76 – 2.08 .98 .98 .999
12x (7) 32.18 3.82 .001 .999 .999
13x
(24)
.96
.49
.31
.31
.999
14x (33) 27.88 1.89* .03 .03 .21
15xh (8) 29.77 3.49 .005 .005 .001
mean
17.61
.27
.47
.64
(continued)
test. Other diagnostic tests and estimates of
b are reported in working papers.
Roughly speaking, biases are distributed
around zero in 0-, 2-, and 3-red periods.
Biases are positive in 1-red periods of every
experiment except llx, generally with large
t-statistics. Biases are also positive in 2-red
periods with experienced subjects, but not
with inexperienced subjects.
The right-hand columns of Table 3 test
the hypothesis that prices are Bayesian
against each of the competing theories. The
tests of the Bayesian theory against exact
representativeness, conservatism, and overreaction are one-tailed t-tests of the null hypothesis B = 0 against one-sided alternative
hypotheses (which vary depending upon the
theory and the sample). Since the base-rate
ignorance theory predicts a point estimate of
the bias rather than a direction, the significance level of the Bayesian hypothesis against
the base-rate ignorance alternative was esti-
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VOL. 77 NO. 5 CAMERER: BIASES IN PROBABILITYJUDGMENT 989
TABLE 3 -(CONTINUED)
2-Red Periods
Exact Representativeness, Base-Rate
Bayesian vs. Overreaction Conservatism Ignorance
1 (61) – 27.00 -4.84 .999 .000 .999
2
(24)
60.25
.11
.46
.54
.50
3 (22) 99.00 6.27 .000 .999 .000
4 (83) 77.39 7.74 .000 .999 .000
5 (52) – 53.31 6.55 .76 .24 .64
6 (77) -1.74 -.02* .51 .51 .52
7 (16) 98.24 18.16 .000 .000 .000
8 (16) 45.45 3.35 .001 .001 .000
9r (24) -17.23 -4.55 .999 .000 .999
10h (27) -7.57 -.95 .67 .33 .999
mean
27.35
.44
.36
.44
llx (18) 49.28 7.55 .000 .999 .000
12x (8) 20.80 12.16* .000 .000 .000
13x (15) 17.22 14.35 .000 .999 .000
14x (11) 22.62 18.26 .000 .999 .000
15xh (17) 12.47 .80 .21 .79 .62
mean
24.48
.04
.78
.12
3-Red Periods
Base-Rate
Bayesian vs. Conservatism Overreaction Ignorance
1
(40)
2.47
.40
.65
.35
.04
2 (17) – 209.34 – 3.51 .000 .999 .85
3 (41) 41.88 4.12* .999 .000 .000
4
(48)
11.26
.64*
.74
.26
.41
5 (26) – 10.57 .70 .24 .76 .90
6
(32)
31.55
1.29*
.90
.10
.24
7 (22) 20.04 3.24* .999 .001 .000
8 (28) – 26.46 -.79 .29 .71 .70
9r
(29)
14.01
.29
.61
.39
.48
10h
(7)
2.61
.02
.51
.49
.50
mean
12.23
.59
.41
.41
(2
deleted)
9.64
llx (37) – 22.51 – 6.52 .000 .999 .999
12
x
(9)
24.98
.49
.69
.31
.39
13x (35) 11.65 .65 .74 .26 .40
14x (28) 114.21 .13 .55 .45 .50
15xh (9) – 38.00 – 3.74 .000 .999 .999
mean
4.70
.40
.60
.66
Notes:
*
denotes
studentiz
unreliable.
Biases
are
tru
greater
than
the
maximu
experiment
mated from likelihood ratios.6 Significance
levels were estimated by assuming the t-statistics were normally distributed (a reason-
6P(data/Bayesian) and P(data/Base-rate Ignorance)
were calculated assuming the estimate B’ was normally
distributed with standard deviation s( B’). Assuming
one of the two theories is true, and they are equally
likely; Bayes’ rule can then be used to calculate
P(Bayesian/data).
7).
able approximation for most of the sample
sizes in Table 3). Levels less than .001 or
above .999 are reported as .000 or .999.
The significance levels of tests against most
of the alternative theories are roughly 50
percent, suggesting departures from the
Bayesian predictions are not systematic.
However, the Bayesian theory can be strongly
rejected against the alternative of exact representativeness in most 1-red periods and
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990 THE AMERICAN ECONOMIC REVIEW DECEMBER 1987
many 2-red periods. Of course, the statistical
significance of a bias is simply a measure of
whether it could be due to chance. Whether
the biases are economically significant is discussed in the conclusion.
units. This fraction is quite stable across
experiments, and is about the same in early
periods (the first half of the periods) and late
periods. With experienced subjects, about 90
percen
