A Bayesian View of the Alternative-Outcomes Effect
Kuninori Nakamura (knaka@ky.hum.titech.ac.jp)
Japan Society for the Promotion of Science/
Graduate School of Decision Science & Technology,
Tokyo Institute of Technology
Kimihiko Yamagishi (kimihiko@ky.hum.titech.ac.jp)
Graduate School of Decision Science & Technology,
Tokyo Institute of Technology
the judged probability of the focal outcome. They argue that
comparing the focal outcome to the strongest alternative can
serve as a useful heuristic—that is, a short-hand process that
usually yields good results with minimal effort. Like other
recently proposed heuristics that suggest that people
sometimes use one or two simple but generally valid cues to
make seemingly complex decisions (e.g., take-the-best;
Gigerenzer & Goldstein, 1996), the comparison heuristic
suggests that people’s perceptions of certainty are often
heavily or perhaps sometimes exclusively determined by the
consideration of only two pieces of information.
There are several alternative explanations to the
alternative-outcomes effect. Dougherty & Hunter (2003)
have suggested that this effect may result from the working
memory span. Still others (Teigen, 2001) have pointed out
that this phenomenon is based on an assessment of the
presence or absence of causal factors, which could produce
or inhibit the outcome.
This article provides another explanation of the
alternative-outcomes effect. The previous explanations
presume that people commonly use heuristic short cuts to
make quick probability judgments. However, this article,
proposes that the alternative-outcomes effect is based on
rational Bayesian inference. In the present research, we
formalize the cognitive processes of the alternativeoutcomes effect as a result of Bayesian inference. In this
formalization, we show that the estimated probability for
winning the raffle corresponds to a posterior probability
estimate that amended the prior probability estimate, after
participants were given additional data on the distribution of
multiple outcomes.
Abstract
The purpose of this study is to provide a new explanation of
the alternative-outcomes effect (Windschitl & Wells, 1998).
Although the alternative-outcomes effect has been posited as
evidence for the use of heuristics, this study suggests that they
result from Bayesian inference reasoning. We reinterpret the
results of previous studies, and reanalyze existing data. Some
theoretical suggestions are also discussed.
Introduction
How do people commonly judge probabilities? Since many
heuristics and biases have been discovered that underlie the
process of making informal judgments of probability, the
basis for human probability judgments has become one of
the most important issues in judgment and decision-making
research. Many empirical studies have been conducted and
several explanations have been proposed.
One of the important findings from recent studies comes
from Windschitl and Wells’ (1998) on the alternativeoutcomes effect. They revealed that probability is sensitive
to variations in how alternative outcomes are distributed,
even when such variations have no bearing on the objective
probability of the focal outcome.
Windschitl and Wells (1998) clearly demonstrated the
alternative-outcomes effect. In one experiment they
informed the participants that they held 21 raffle tickets and
were asked to estimate their chances of winning. In the first
scenario outcome, participants were told that there were
five other raffle players, who held 14, 13, 15, 12, and 13
tickets, respectively; in the second outcome, the other five
players supposedly held 52, 6, 2, 2, and 5 tickets. Although
the objective probability of winning was in fact identical for
the two scenarios, participants informed of the first
distribution actually estimated a higher probability of
winning the raffle than those informed of the second
distribution. Windschitl and Wells called this phenomenon
the alternative-outcomes effect. Later, Windschitl and others
replicated this effect for a variety of experimental conditions
(Windschitl & Young, 2001; Windschitl, Young, & Jenson,
2002; Windschitl & Krizan, 2005).
Windschitl and Wells (1998) offered an interesting
account for the alternative-outcomes effect. According to
them, when judging the probability of a focal outcome,
people seem to compare the strength of the focal outcome
with the strength of the strongest alternative outcome. The
more this comparison favors the focal outcome, the greater
A Bayesian interpretation
In this study, we do not suggest that people use heuristics.
Rather, we propose that people estimate subjective certainty
for the focal outcome according to a Bayesian outlook. In
previous studies, it was assumed that participants’ estimates
should be based solely on the ratio of the focal outcomes to
the total outcomes. For example, when told that they held 21
raffle tickets, while the five other raffle players held 14, 13,
15, 12, and 13 tickets, respectively, the “rational” answer for
their odds of winning should be about 0.24 (=21/
(21+14+13+15+12+13)). The present research, however,
indicates that people consider both the distribution of the
focal and alternative outcomes and their prior beliefs about
the chances of their winning the raffle. Thus, our approach
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does not consider that the given distributions represent
values for the probability of winning. Rather, we assume
that the distributions serve as additional “data”, which the
participants use to support or revise their prior beliefs.
The revision of prior beliefs according to new data can be
formalized easily via a Bayesian approach, wherein the
probability estimate for the focal outcome may be viewed as
a posterior probability. More specifically, the posterior
probability p(H|Y) of winning the raffle, Y, after seeing the
actual distribution, H, can be obtained from Bayes’ theorem:
p (H | Y
) = p (H )pp(Y(Y) | H ) .
Application to other phenomena
A Bayesian approach to probability judgment can explain
not only the alternative-outcomes effect but also other
related studies, such as the ratio bias (Denes-Raaji &
Epstein, 1994). The ratio bias refers to people's tendency to
select betting options that display a greater absolute number
but a smaller probability of winning (e.g., 9/100) than an
option with a smaller absolute number but better odds (e. g.,
1/10). To explain this bias, we consider the ratios (“9/100”
or “1/10”) as “data”, which confirm or refute the prior belief.
We also assume that the prior distribution is a beta
distribution:
(1)
In this equation, p(H) is the prior belief about winning the
raffle; p(Y) is the prior belief about the shape of the
distribution; and p(Y|H) is the conditional probability of Y
occurring, given H. According to Bayes' theorem, the
posterior probability is proportional to the product of the
prior probability and the likelihood of H. That is,
(2)
p H | Y ∝ p (H ) p (Y | H ) .
(
f ( x) =
b −1
L
x )= ∏
n!
i
x
∏
i
p
xi
i
. (4)
where xi is frequency of the outcome, pi is the probability of
xi occurring ,and n is the total number of outcomes, evident
when all the frequencies are summed,
(6)
Denes-Raj, V., & Epstein, S. (1994). Conflict between
intuitive and rational processing: When people behave
against their better judgment. Journal of Personality and
Social Psychology, 66, 819–829.
Dougherty, M. R. P., & Hunter, J. E. (2003). Probability
judgment and subadditivity: The role of working memory
capacity and constraining retrieval. Memory and
Cognition, 31, 968–982.
Gigerenzer, G., & Goldstein, D. G. (1996). Reasoning the
fast and frugal way: Models of bounded rationality.
Psychological Review, 103, 650-669.
Teigen, K. H. (2001). When equal chances=good chances:
Verbal probabilities and the equiprobability effect.
Organizational Behavior and Human Decision Processes
85, 77-108.
Windschitl, P., & Krizan, Z. (2005). Contingent approaches
to making likelihood judgments about polychotomous
cases: The influence of task factors. Journal of
Behavioral Decision Making, 18, 281–303.
Windschitl, P. D., & Wells, G. L. (1998). The alternativeoutcomes effect. Journal of Personality and Social
Psychology, 75, 1411–1423.
Windschitl, P. D., & Young, M. E. (2001). The influence of
alternative outcomes pn gut-level perceptions of certainty.
Organizational Behavior and Human Decision Processes,
85, 109–134.
Windschitl, P. D., Young, M. E., & Jenson, M. E. (2002).
Likelihood judgment based on previously observed
outcomes: The alternative-outcomes effect in a learning
paradigm. Memory and Cognition, 30, 469–477.
We also assume that p(Y|H) is a multinomial function,
2
(5)
References
)
x
.
Thus, the certainty of winning the bet, after seeing the ratio,
can be calculated as the posterior mean:
a + nhit .
(7)
p( H | Y ) =
a + b + ntotal
In this formalization, a qualifying condition that satisfies
p{H | (9,100 )} ≥ p{H | (1,10 )} surely exists.
We can assume that p(H) remains the same across the set of
experimental conditions in the studies of the alternativeoutcomes effect. After all, the posterior probability of
winning the raffle is proportional to the conditional
probability of seeing the distribution of the focal and
alternative outcomes:
(3)
p H | Y ∝ p (Y | H ) .
p(Y | H ) = f (x1
B ( a , b)
B(a, b ) = ∫10 π a−1 (1 − π ) dπ .
)
(
π a −1 (1 − π ) b−1
∑x .
i
Experimental results and discussion
The above formalization indicates that the probability
estimate for the focal outcome is proportional to p(Y|H). To
test this prediction, we performed an experiment in which
participants were required to estimate their subjective
certainty about winning at a fictitious slot machine. This slot
machine showed 6 characters of the alphabet (A, B, C, D, E,
F), but only A was a winning letter. Participants were shown
155 multinomial outcomes, which represented the results of
turns at the slot machine. Then they were asked to estimate
the subjective probability of a focal outcome. We calculated
estimated means for each stimulus. We classified the stimuli
according to frequency of focal outcome, and calculated the
likelihood of the distribution of the alternative outcome for
each condition. In these calculations, we set the value of
each of the xi at 1/6. The results indicate that the estimated
probabilities for the focal outcomes were proportional to the
likelihood of Y (rs = 0.54-0.92, all ps < .05). Thus, the
results support predictions based on a Bayesian approach.
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