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201901Random Choice and Learning.pdf
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Random Choice and Learning
∗
Paulo Natenzon
†
Revised: October 2017
Abstract
Context-dependent individual choice challenges the principle of utility maximiza-
tion. We explain context-dependence as the optimal response of an imperfectly in-
formed agent to the ease of comparison of the options. We introduce a discrete-choice
model, the Bayesian probit, which allows the analyst to identify stable preferences
from context-dependent choice data. Our model accommodates observed behavioral
phenomena —including the attraction and compromise effects— that lie beyond the
scope of any random utility model. We use data from frog mating choices (Lea and
Ryan, 2015) to illustrate how our model can outperform the random utility framework
in goodness of fit and out-of-sample prediction.
∗
This work is based on the first chapter of my doctoral dissertation at Princeton University. It also
incorporates and supersedes the working paper “Preference reversal or limited sampling? Maybe t´ungara
frogs are rational after all.” I am grateful to the editor and the anonymous referees for their many useful
suggestions that significantly improved the paper. I am indebted to my advisor, Faruk Gul, for his guidance
and dedication. I am also indebted to Wolfgang Pesendorfer for many useful discussions. I thank Roland
B´enabou, Meir Dan-Cohen, Talia Dan-Cohen, Daniel Gottlieb, David K. Levine, Justinas Pelenis, Carl
Sanders, John Yiran Zhu, and audience participants in many seminars for helpful comments and suggestions.
All remaining errors are my own. I gratefully acknowledge financial support from the Weidenbaum Center
on the Economy, Government, and Public Policy at Washington University in Saint Louis.
†
Department of Economics, Washington University in Saint Louis, St. Louis, MO 63130, USA. E-mail:
pnatenzon@wustl.edu.
1
Manuscript Click here to access/download;Manuscript;2017-09-08-RCL-
PRLS-Paper-09.tex
Copyright The University of Chicago 2018. Preprint (not copyedited or formatted).
Please use DOI when citing or quoting. DOI: 10.1086/700762
Journal of Political Economy
Downloaded from www.journals.uchicago.edu by INSEAD on 09/29/18. For personal use only.
1 Introduction
Consider the following type of choice reversal. Option B is chosen in more than
50% of the choice trials in binary comparisons between A and B,
P (B, {A, B}) > .5
while option A is chosen in more than 50% of the choice trials in ternary com-
parisons among A, B and C,
P (A, {A, B, C}) > .5
Experimenters have been able to systematically generate such choice reversals in
the lab. Famous examples include decoy effects —such as the attraction and the
compromise effects— and phantom option effects (Huber, Payne and Puto (1982),
Simonson (1989), Soltani, De Martino and Camerer (2012)). These effects are
considered puzzling because they challenge the principle of utility maximization
by suggesting that individual preferences are context-dependent. Furthermore,
they are incompatible with random utility models, which are the models most
commonly used in discrete choice estimation.
Our contribution in this paper is two-fold: first, we provide a new explanation
for context-dependence in individual choice. In our model, the decision maker is
imperfectly informed about the value of the options. Context-dependence arises
from the optimal response of the decision maker to the ease of comparison of the
options, under imperfect information. Incorporating insights from psychology,
we model ease of comparison as depending on three factors: the true value of
the options, the similarity of the options, and the precision of the information
obtained by the decision maker before making a choice. Our second main con-
tribution is to provide a new discrete choice estimation framework in which the
analyst can identify preferences, similarity, and information precision from choice
data. We include an empirical application to illustrate how our new framework
can outperform random utility models in goodness of fit and out-of-sample pre-
diction, thus presenting a useful alternative for inference from context-dependent
choice data.
The following example shows how ease of comparison affects choice when the
decision maker faces uncertainty about the value of the options:
2
Copyright The University of Chicago 2018. Preprint (not copyedited or formatted).
Please use DOI when citing or quoting. DOI: 10.1086/700762
Journal of Political Economy
Downloaded from www.journals.uchicago.edu by INSEAD on 09/29/18. For personal use only.
Example 1. A consumer is about to encounter three choice options A, B, C.
The consumer has no information a priori that favors one option over another;
her prior beliefs about the utility of the options are given by any exchangeable
(i.e., invariant to permutations) and absolutely continuous distribution. Before
having a chance to contemplate the three options, she assigns probability 1/6 to
each possible strict ranking involving A, B and C, namely
B A C B C A
A B C C B A
A C B C A B
Now suppose she obtains some information about the options (for example, by
inspecting the three options), which (i) conveys very little or nothing about the
individual values of A, B and C; and (ii) conveys with a high degree of certainty
that option A is better than option C. To make it extreme, suppose she learns
that the event (A C) occurred and nothing else. Updating the prior with this
information will result in probability 1/3 given to each of
B A C
A B C
A C B
and hence option A now has 2/3 probability of being the best option. If she had
learned (C A), then C would have been the favored option. Starting from an
exchangeable prior, any information that is very imprecise about the individual
values of A, B, C but allows a precise comparison between options A and C will
make option B unlikely to be chosen.
1
Random utility models can incorporate the ease of comparing options in bi-
nary choices, but they cannot account for the intuitive behavior involving three
options presented in Example 1. In the random utility framework, the utility of
the options is given by random variables U
A
, U
B
, U
C
: Ω → R on a probability
space (Ω, P), and the decision maker chooses the option with the highest random
utility realization in each choice trial. In simpler formulations (such as logit) these
1
The optimality of choosing among the pair of options that is easy to compare in this example is analogous
to the optimality of switching doors in the solution to the Monty Hall problem (Selvin, 1975).
3
Copyright The University of Chicago 2018. Preprint (not copyedited or formatted).
Please use DOI when citing or quoting. DOI: 10.1086/700762
Journal of Political Economy
Downloaded from www.journals.uchicago.edu by INSEAD on 09/29/18. For personal use only.
variables are independent. More flexible formulations (such as nested logit) allow
U
A
, U
B
, U
C
to be correlated. No matter if the variables are correlated or inde-
pendent, for every state of nature ω ∈ Ω in which {U
A
> U
B
and U
A
> U
C
}
we have, in particular, that {U
A
> U
B
}. Therefore, the probability of these two
events must satisfy
P{U
A
> U
B
and U
A
> U
C
} ≤ P{U
A
> U
B
}
and the probability of choosing A can only decrease when option C is intro-
duced. This property, called monotonicity, is satisfied by every random utility
model, no matter what distributional assumptions we make on U
A
, U
B
, U
C
. In
particular, neither independent shocks nor correlated shocks to utility are able
to accommodate the choice reversal in (1).
To explain how our new model departs from random utility, we can interpret
the classic random utility framework as an “as-if” model of optimizing behavior
subject to informational constraints. In this interpretation, the random variables
U
A
, U
B
, U
C
represent a noisy signal about the true value of each option. The
decision maker observes a signal realization for each option in the choice problem
and chooses the option with the highest signal realization. This “as-if” story
in which the value of the options is perceived with noise is the interpretation
originally proposed for random utility models in psychology (Thurstone, 1927a,b).
Our new model, the Bayesian probit, is parameterized like the classic multi-
nomial probit model: signals have a joint Gaussian distribution. Like the multi-
nomial probit, the signals for different options may be correlated. But our model
departs from the random utility framework in two ways. First, the decision maker
has a prior. She sees the value of the options in every choice problem as random
draws from the same population. Second, the decision maker does not choose
the option with the highest signal realization. Instead, she chooses the option
with the highest expected value, conditional on the information conveyed by the
signals. As a consequence, in our model, the presence of option C offers addi-
tional information that can influence the probabilities of choosing A versus B.
Example 1 above is an extreme case in which alternative B is never chosen. The
Bayesian probit model captures this extreme example when the signals for A and
C are perfectly correlated. But our model can also accommodate less extreme
cases often found in experimental data.
4
Copyright The University of Chicago 2018. Preprint (not copyedited or formatted).
Please use DOI when citing or quoting. DOI: 10.1086/700762
Journal of Political Economy
Downloaded from www.journals.uchicago.edu by INSEAD on 09/29/18. For personal use only.
The experimental literature contains many examples of choice reversals among
humans (see Section 2). In recent years, biologist have also found evidence of
choice reversals among a diverse array of species. It turns out, for example,
that the same experimental designs that generate choice reversals among human
subjects work equally well for monkeys, frogs, birds, bees, and even slime mold.
Throughout the paper, we use a dataset from Lea and Ryan (2015) involving
mate choices by female t´ungara frogs in the lab for illustrative purposes. Lea
and Ryan’s dataset, which contains two kinds of choice reversals, is simple, yet
impressively thorough, in so far as the authors tested all binary comparisons. It
is therefore well suited to the task of illustrating the strength of our model with a
concrete example. At the same time, the effectiveness of our model in explaining
behavior among these non-humans suggests its broad reach and applicability to
contexts that might be of interdisciplinary interest. Indeed, where Lea and Ryan
conclude that the frogs are acting irrationally, thereby defying some basic tenets
of natural selection, we show that the frogs may have rational preferences after
all.
The plan of the paper is as follows. Section 2 reviews the evidence of context-
dependent individual choice in many settings, and presents the dataset that acts
as our exemplary case. Section 3 provides our new explanation for context-
dependence as an optimal response of decision makers to imperfect information
about the values of the options and to their ease of comparison. In Section 4
we define the primitives of our model and we propose non-parametric definitions
of easier to compare, revealed preference and revealed similarity. Section 5 intro-
duces our parametric model, the Bayesian probit, explains how the parameters
of the model are identified from choice data, and provides testable implications.
Section 6 contains the main theoretical results, showing how the Bayesian probit
accommodates many types of apparently irrational choice reversals from the max-
imization of a single, stable utility function under informational constraints. In
Section 7 we fit the model to the frog mating choice data and perform pseudo out-
of-sample prediction exercises. These provide an illustration of how our model
can outperform the random utility framework in goodness of fit and prediction.
Section 8 concludes by suggesting two possible extensions of our framework.
5
Copyright The University of Chicago 2018. Preprint (not copyedited or formatted).
Please use DOI when citing or quoting. DOI: 10.1086/700762
Journal of Political Economy
Downloaded from www.journals.uchicago.edu by INSEAD on 09/29/18. For personal use only.
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