Random Choice and Learning
Paulo Natenzon∗
Washington University in Saint Louis
December 2012
Abstract
We study a decision maker who gradually learns the utility of available alternatives.
This learning process is interrupted when a choice must be made. From the observer’s
perspective, this formulation yields a random choice model. We propose and analyze
one such learning process, the Bayesian Probit Process. We apply our model to the
similarity puzzle: Tversky’s similarity hypothesis asserts that if two options, x and y
have roughly equal utility and z is similar to x then making z available will reduce the
probability of choosing x more than it reduces the probability of choosing y. However,
there is also evidence that introducing a similar but inferior z may hurt y more than
x and can even increase the probability of choosing x (attraction effect). We provide
a definition of similarity based only on the random choice process and show that z is
more like x than y if and only if the correlation between z and x’s signals is larger than
the correlation between z and y’s signals. Our main result establishes that when z
and x are correlated and sufficiently close in utility, introducing z hurts y more than x
and may even increase the probability of choosing x early in the learning process, but
eventually hurts x more than y. Hence, the attraction effect arises when the decision
maker is relatively uninformed, while Tversky’s hypothesis holds when she becomes
sufficiently familiar with the options. We then show that if z is similar to x and
sufficiently inferior, the attraction effect never disappears.
∗
E-mail: [email protected]. This paper is based on the first chapter of my doctoral dissertation at
Princeton University. I wish to thank my advisor, Faruk Gul, for his continuous guidance and dedication. I
am grateful to Wolfgang Pesendorfer for many discussions in the process of bringing this project to fruition.
I also benefited from numerous conversations with Dilip Abreu, Roland Bénabou, Meir Dan-Cohen, Daniel
Gottlieb, Justinas Pelenis, and seminar participants at Alicante, Arizona State, Berkeley, D-TEA Workshop,
IESE, IMPA, Johns Hopkins, Kansas University, Haifa, Hebrew University, NYU, Princeton, the RUD
Conference at the Colegio Carlo Alberto, SBE Meetings, Toronto, UBC, Washington University in St. Louis
and Yale. All remaining errors are my own.
1
1
Introduction
The similarity puzzle is best introduced with an example. Let’s consider a consumer who is uncertain about her preference over two comparable computers: a
Mac and a PC. She hasn’t shopped for computers in a few years, she doesn’t
follow technology blogs, and she is not sure which one will turn out to be better
for her office. If pressed to make a decision, she is equally likely to choose either
one:
P{she chooses the Mac}
=1
P{she chooses the PC}
Suppose we add a third computer to the menu, and the new option is similar to
one of the existing alternatives. To fix ideas, suppose the original PC option is a
Dell, and we introduce a similar PC made by Toshiba. What should happen to
the ratio above? If the Toshiba is chosen with any positive probability when the
three options are available, it will necessarily ‘hurt’ the probability that the Mac
is chosen, or the probability that the Dell is chosen, or both. But which existing
option should the Toshiba hurt proportionally more?
The similarity puzzle arises from disparate answers to this question. On the
one hand, there is a large body of evidence and theoretical work, culminating in
Tversky’s similarity hypothesis, based on the idea that the ratio should increase
once the Toshiba is introduced. But there is also a large body of empirical
evidence showing that in many situations the ratio decreases, the most prominent
example being the attraction effect. Let’s now look at each side of this puzzle in
turn.
Tversky (1972b) proposes the following similarity hypothesis:
The addition of an alternative to an offered set ‘hurts’ alternatives that
are similar to the added alternative more than those that are dissimilar
to it.
According to this hypothesis, the consumer substitutes more intensely among
similar options, so the Toshiba should hurt the Dell proportionally more. Tversky’s similarity hypothesis summarizes a large body of work from the 1960’s and
the 1970’s which tried to understand how similarity affects choices. The similarity hypothesis is one of the underlying principles in most of the modern tools
used in discrete choice estimation (see McFadden (2001) and Train (2009)).
2
While it may be tempting to treat Tversky’s similarity hypothesis as a general
principle, it is incompatible with a large amount of empirical evidence originated
in the marketing literature, the most prominent example being the attraction
effect. The attraction effect refers to a situation in which the probability of an
existing alternative being chosen increases with the introduction of a similar but
inferior alternative.1
In our consumer example, suppose the Toshiba computer, while being almost
identical to the Dell, is clearly worse in minor ways, such as having a lower processing power, a slightly smaller screen, and costing a little more. In other words,
while the Dell and the Toshiba share many features, the Toshiba is clearly a dominated option. It would not be surprising if almost no one chose the Toshiba in
this situation. What is perhaps surprising is that in many experimental settings
the presence of the dominated option increases the probability that the dominant
option is chosen. In the example, adding a dominated Toshiba helps boost the
sales of the Dell to the detriment of the Mac.
The attraction effect stands in stark contrast to Tversky’s similarity hypothesis. In the attraction effect, the introduction of the Toshiba hurts the Mac
proportionally more. Moreover, while the probability of choosing the Mac is reduced, the probability of choosing the Dell actually increases. This constitutes
a violation of monotonicity, a very basic property for models of discrete choice.
Monotonicity (also known as regularity) states that the probability of existing alternatives cannot decrease when an option is removed from the choice set. Every
random utility model, including the multinomial logit, probit, the family of generalized extreme value models, and virtually all models currently used in discrete
choice estimation are monotonic, and therefore incompatible with the attraction
effect.
In this paper, we propose a model that allow us to understand and resolve the
similarity puzzle. The model explicitly incorporates two frictions in the decision
making process. First, the decision maker gradually learns the utility of the
available alternatives. A parameter t ≥ 0 captures the amount of information
available to the decision maker at the time a choice is made. Second, for any
1
Huber et al. (1982) and Huber and Puto (1983) provide the first experimental evidence for the attraction
effect. Since then, it has been confirmed in many different settings, including choice over political candidates,
risky alternatives, investment decisions, medical decisions and job candidate evaluation. For these and other
examples, see Ok et al. (2012) and the references therein.
3
given amount of information the decision maker has about the alternatives (for
example, if the decision maker is allowed to contemplate the alternatives for a
fixed amount of time), similarity determines how certain the decision maker is
about the relative ranking of each pair of alternatives. In the model, similarity
is captured by a parameter γij for each pair of alternatives (i, j).
Our main results show how these two new dimensions, similarity and information, interact to explain the similarity puzzle. While Tversky’s similarity
hypothesis holds for more informed choices (for a sufficiently large level of information), the opposite happens and may even lead to the attraction effect for less
informed choices (for a sufficiently small level of information).
In our model, stochastic choice arises from the decision maker’s incomplete
information about the choice alternatives. The decision maker lacks, ex-ante, any
information about the alternatives that would favor a choice of one alternative
over another. When contemplating the menu of choices, the decision maker gradually learns the utility of the alternatives. This learning process is interrupted
when a choice must be made. From the point of view of the observer, this yields
a random choice model.
A random choice rule describes frequencies of choice in each menu for a fixed
level of information precision. By indexing a family of random choice rules according to information precision, we model observable choice behavior as a random
choice process. The decision maker can ultimately be described by a rational
preference relation represented by a utility function µ. Observed choices increasingly reflect this rational preference as the decision maker’s information about
the alternatives becomes more precise.
Formally, the model can be described as follows. Let A be the universe of
choice alternatives. The decision maker has a standard rational preference relation in A. This preference is represented by the numeric utility µi of each
alternative i ∈ A. In addition, each pair of alternatives i, j is described by a
correlation parameter γij . The decision maker knows γ but does not know µ.
Her beliefs ex-ante about the values of µi are iid Gaussian. When facing a choice
from menu b, the decision maker first observes a noisy signal about the true utilities, and updates her beliefs about µ using Bayes’ rule. The signal correlation
is given by γ. The probability of choosing option i in menu b is equal to the
probability that alternative i has the highest posterior mean belief among the
4
alternatives in b. We call the resulting random choice rule the Bayesian probit
rule. We allow the precision of the signals received to vary continuously from
pure noise to arbitrarily precise signals. The precision of the signals is captured
by the parameter t ≥ 0. A family of Bayesian probit rules ordered according to
signal precision t is a Bayesian probit process.
The key component of the model proposed in this paper is a new behavioral
definition of similarity. This new notion of similarity is subjective and identified
(revealed) from choices, much as utility is identified in classical revealed preference theory. The theory allows us to identify, from choice data, one ranking
of alternatives according to preference, and one ranking of pairs of alternatives
according to similarity.
The behavioral notion of similarity that we propose in this paper abstracts
from (and is orthogonal to) utility. Our definition is best motivated with the
following geometric example. Suppose the universe of choice alternatives is the
set of all triangles in the euclidean plane. Suppose further that your tastes over
triangles are very simple: you always prefer triangles that have larger areas.
So your preferences over triangles can be represented by the utility function that
assigns to each triangle the numerical value of its area. Among the options shown
in Figure 1, which triangle would you choose?
Figure 1 illustrates that it may be hard to discriminate among comparable
options. Since the triangles in Figure 1 have comparable areas, they are close in
your preference ranking, and you may have a hard time picking the best (largest)
one. If you were forced to make your choice with little time to examine the
options, you would have a good chance of making a mistake.
If the difference in area were more pronounced, i.e., if one triangle were much
larger than the other, you probably wouldn’t have any trouble discriminating
among the options. Likewise, a consumer presented with two products, of which
one is much cheaper and of much better quality, should seldom make a mistake.
Hence, any probabilistic model of choice should include the gap in desirability
(measured as the difference in utility) as an important explanatory variable for
the ability of subjects to discriminate among the options.
But the gap in utility alone cannot tell the whole story. Even if the utility of
a pair of objects is kept exactly the same, it is possible to increase your ability
to discriminate among a pair of options by changing some of their other features.
5
For example, suppose you still prefer larger triangles, and consider the pair shown
in Figure 2. The triangle shown on the left in Figure 2 has exactly the same area
as the triangle on the left in Figure 1, while the triangle on the right is the same
in both Figures. Hence, from the point of view of desirability, the pair shown
in Figure 1 is identical to the pair shown in Figure 2. However, note that you
are less likely to make a mistake when choosing among the options in Figure 2
(make your pick).
You probably chose the triangle on the left. You probably also found this
choice task much easier than the task in Figure 1. Why? It certainly has nothing
to do with the gap in the desirability of the two options—as we pointed out,
the pairs in both Figures exhibit exactly the same difference in area. So what
happened?
When we substituted triangle i0 for triangle i, we kept desirability (the area)
the same, but we made every other feature of triangle i0 more closely resemble
triangle j. What changed from the pair in Figure 1 to the pair in Figure 2 is
that we kept constant the features you care about (the area), while increasing
the overlap of other features. And this increased overlap clearly helped improve
your ability to discriminate among the options. In the theory that we propose
in this paper, we build on this intuition and call the pair (i0 , j) of Figure 2 more
similar than the pair (i, j) of Figure 1. Similarity is measured independently
from utility, and is another important factor determining the ability of a decision
maker to discriminate among the choice options.
The pair of triangles in Figure 2 is in fact similar according to the mathematical definition for geometric figures. In euclidean geometry, two triangles
are similar when they have the same internal angles. The concept of geometric
similarity abstracts from size, and only refers to the shape of the objects being
compared. In the same spirit, the notion of similarity that we propose in this
paper abstracts from utility. But instead of defining the similarity of two alternatives based on any of their observables attributes, we infer the level of similarity
perceived by the decision maker from her choices. Similarity in our theory is a
behavioral notion: it is subjective and inferred from choice data.
To understand the formal definition of similarity, suppose all choice data is
obtained under a fixed a level of information precision t and consider two pairs of
alternatives {i, j} and {k, `}. Under the assumptions of the model, the utility of
6
the alternatives can be inferred from pairwise choices: in any given pair, and for
any positive level of information, the more desirable alternative is chosen more
often. When each object in a pair is chosen exactly 50 percent of the time, for
every level of information precision, we infer that they are equally desirable.
Without loss of generality, suppose that i is better than j. Suppose moreover
that k is at least as good as i, and that j is at least as good as `. In other
words, the gap in desirability is wider in the pair {k, `}. Without any difference
in similarity across the pairs, the gap in desirability alone show allow the decision
maker to discriminate among k and ` at least as well as among i and j. In other
words, based solely on the gap in utility in each pair, the better option k should
be chosen from {k, `} at least as often as the better option i is chosen from {i, j}.
But it is exactly when k is not chosen from {k, `} at least as often as i is chosen
from {i, j}, that the pair {i, j} is revealed to have a larger degree of similarity.2
A second visual example helps illustrate this point.3 Suppose the universe
of choice alternatives is the set of all star-shaped figures on the euclidean plane.
Your tastes over star-shaped figures are again very simple. This time, you only
care about the number of points on a star. For example, when the options are a
six-pointed star and a five-pointed star, you always strictly prefer the six-pointed
star. With just a glance at the options in Figure 3, which star would you pick?
Again, if pressed to make a choice in a short amount of time, you may be
likely to make a mistake. The probability of making a mistake would certainly
decrease if you were given more time. You would also be more likely to make
the correct choice if one of the alternatives had a much larger number of points
than the other. Given a limited amount of time, mistakes are likely, because the
options are comparable and therefore difficult to discriminate.
Now consider the options in Figure 4. If you still prefer stars that have more
points, which of the two options would you choose? Here, similarity comes to the
rescue. The star on the left is the same in both Figures and the star on the right
has the same number of points in both Figures. Hence the gap in desirability is
the same in both Figures. But even though the difference in utility is the same,
there is a much greater overlap of features in the pair shown in Figure 4. Hence
2
The attentive reader will notice that pairwise choices may only partially reveal the similarity ranking.
As we show later, choices over menus of three options are sufficient to fully identify similarity.
3
I am grateful to David K. Levine for suggesting this example.
7
i
j
Figure 1: Which triangle has the largest area?
i’
j
Figure 2: Which triangle has the largest area?
Figure 3: Which star has more points?
Figure 4: Which star has more points?
8
it is much easier to discriminate among the options in Figure 4. According to
our definition of similarity, we infer from this pattern of choices that the pair in
Figure 4 is more similar than the pair in Figure 3.
To see how our model allows us to reconcile Tversky’s similarity hypothesis
with the attraction effect, suppose the decision maker faces a choice between two
alternatives i and j. Let k be a third alternative with the same utility as i and let
the pair i, k be more similar than the pairs i, j and j, k. In Theorem 3, we show
that introducing k reduces the probability of choosing j more than it reduces the
probability of choosing i, and may even increase the probability of choosing i,
for sufficiently low levels of information precision (i.e., at the beginning of the
random choice process). Eventually, as information becomes more precise, the
opposite occurs: the introduction of k hurts the similar i more than it hurts the
dissimilar j.
To better isolate the role of similarity, suppose that all three alternatives i,
j and k have the same utility. In Corollary 4 we show that, when the signal
correlation between i and k is taken to be arbitrarily close to one, both effects
obtained above become more extreme. For times close to zero, i and k are each
chosen with probability close to one half, while j is chosen with probability close
to zero. As time goes to infinity, or as information becomes arbitrarily precise,
i and k are each chosen with probability close to 1/4, and j is chosen with
probability close to 1/2. Hence when all three alternatives are equally desirable,
similarity generates a form of attraction effect early in the process, and that effect
tends to disappear as the decision maker learns the utilities of the alternatives
with more precision. As the correlation between i and k goes to one, these
alternatives are eventually treated as duplicates in the random choice process.
In Theorem 5 we tackle the attraction effect. We show that, if alternative k is
sufficiently inferior and correlated with i, the attraction effect never disappears:
starting arbitrarily early in the random choice process, introducing k increases
the probability of choosing i to a value strictly larger than one half (violating
monotonicity), and this remains true even as time goes to infinity.
1.1
Related Literature
The attraction effect and other related decoy effects are examples of choice behavior that systematically depend on the context in which the decisions are made.
9
Context dependent choice challenges the principle of utility maximization in individual decision making. In particular, violations of the weak axiom of revealed
preference break the identification of preferences and choices in standard revealed
preference theory.
To accommodate context dependence, recent advances in decision theoretic
models have relaxed the weak axiom while maintaining the assumption of deterministic choice behavior. In Manzini and Mariotti (2007), a decision maker may
exhibit context dependent behavior when the choice procedure consists of applying two or more different rationales sequentially. In Masatlioglu et al. (2012),
context-dependence arises from limited attention. For example, a consumer who
prefers the Mac may instead choose the Dell when the Toshiba is introduced,
because the presence of a third alternative induces her to fail to pay attention to
the Mac. In a similar vein, Ok et al. (2012) study context-dependent behavior
that arises when one of the alternatives in the menu serves as a reference point,
again limiting the attention of the decision maker to a subset of the choices.
By maintaining the assumption of deterministic behavior while relaxing the
weak axiom of revealed preference, this literature pushes the scope of deterministic models to their limit. But it is possible to go further. As pointed out by Gul et
al. (2012), to understand context dependent behavior, there are great advantages
in modeling choice as a stochastic phenomenon. The argument is simple: while
it may be reasonable to expect a decision maker to choose alternative i from
the set {i, j} on one occasion and to choose j from the set {i, j, k} on another
occasion, it is unlikely that we would observe these choices on every occasion.
Therefore, models which rely on deterministic violations of the weak axiom are
at best as likely to find empirical support as the standard model. In this sense, a
probabilistic model of choice may be more appropriate for measuring an individual’s tendency to choose i over j. In this paper, modeling choices as stochastic
is natural, because the similarity puzzle is formulated in probabilistic terms. All
nuanced effects of similarity on choice would disappear once we formulate the
problem deterministically.
Random choice models have a long history, having originated in psychophysics.
Until the mid twentieth century theoretical developments were limited to models
of binary choice. In his seminal work on individual choice behavior as a probabilistic phenomenon, Luce (1959) introduces the first theoretical restriction for
10
choice across menus with more than two alternatives, in the form of his random
choice hypothesis:
The ratio between the probability with which option j is chosen from a
set of options to the probability with which k is chosen from the same
set is constant across all sets that contain j and k.
which leads to the Luce model of random choice. McFadden (1974) constructs
a random utility for the Luce model, which gives rise to the widespread use of
conditional logit analysis in discrete choice estimation.
The random choice hypothesis imposes a strong restriction on the patterns of
substitutability between alternatives. Debreu (1960) provides an early example
of this limitation of the Luce rule, later referred to as the “duplicates problem”
in the discrete estimation literature. A classic version of Debreu’s example is the
red bus/blue bus paradox: suppose that the decision maker is indifferent between
taking a train and a blue bus, so that each option is chosen with probability one
half. She is also indifferent between taking the blue bus or an identical bus
that happens to be red, so she also chooses each bus with probability one half.
Intuitively, since the decision maker doesn’t care about the color of the bus,
the train should still be chosen with probability one half when both buses are
available. However, in the Luce model this pattern of choices implies that the
train is chosen with probability 1/3 when the three options are available.
Debreu’s example illustrates an extreme case in which two options are treated
like a single option by the decision maker. But even when options are not exactly
treated as duplicates, the decision maker may exhibit a pattern of substitutability
that is incompatible with the Luce rule. It is in this vein that Tversky (1972b)
proposes the similarity hypothesis:
The addition of an alternative to an offered set ‘hurts’ alternatives that
are similar to the added alternative more than those that are dissimilar
to it.
Many random choice models allow for substitutability patterns in line with
Tversky’s similarity hypothesis. Commonly used random utility models in discrete choice estimation, such as the Generalized Extreme Value family and the
multinomial probit, accommodate complex substitution patterns by allowing correlation in the error terms of random utility (see for example Train (2009)). In a
11
more theoretical vein, Tversky’s (1972a) elimination-by-aspects model and, more
recently, Gul et al.’s (2012) attribute rule are two other examples.
All the random choice models we mentioned above are random utility maximizers, and therefore they all share the property of monotonicity. A random
choice rule satisfies monotonicity if the probability of an existing alternative being
chosen can only decrease when a new alternative is added. Thus, the attraction
effect presents a challenge to random utility models. Any probabilistic model
of individual decision making that accommodates behavior compatible with the
attraction effect must therefore depart from random utility. The model proposed
in this paper departs from random utility maximization and allows violations of
monotonicity.
The majority of the recent decision theoretic developments that accommodate
behavior compatible with the attraction effect posit some form of bounded rationality or limited attention on the part of the decision maker. In contrast, our
model describes a decision maker who is fully aware of every option in her opportunity set, but only learns the utility of the available alternatives gradually. In
our model, context dependence arises when similarity allows the decision maker
to learn the relative ranking of some pairs of alternatives faster than others.
The paper is organized as follows. In the next Section we introduce the model
and provide a baseline characterization. In Section 3 we characterize the Bayesian
probit process, define our notion of similarity and show that it is captured by signal correlation. Section 4 contains our main results and addresses the similarity
puzzle. In Section 5 we compare the Bayesian probit process to existing random
utility models in the literature. In Section 6 we offer an application of the model
to a setting where noise correlation depends on observable attributes of the choice
alternatives. We conclude in Section 7. All proofs are in the Appendix.
2
Learning Process
Let A be the grand set of choice objects and let A denote the collection of all
finite subsets of A. We write A+ = A \ {∅} for the non-empty finite subsets.
Random choice rules are a generalization of deterministic choice explicitly treats
choice behavior as probabilistic.
12
2.1
Random Choice Rules
Definition. A function ρ : A × A+ → [0, 1] is a random choice rule if for all
P
b ∈ A+ we have ρ(b, b) = 1 and ρ(a, b) = i∈a ρ(i, b).
To simplify the statements below, we identify each i ∈ A with the singleton
{i}. The value ρ(i, b) specifies the probability of choosing i ∈ A, given that the
selection must be made from the choice set b ∈ A+ . The first equation in the
definition of a random choice rule is a feasibility constraint: ρ must choose among
the options available in b. The second requirement makes ρ(·, b) a probability over
b. A random choice rule ρ is monotone 4 if for every a ⊂ b ∈ A+ and i ∈ a we
have ρ(i, a) ≥ ρ(i, b). For monotone random choice rules, the probability of
choosing an alternative i can only decrease when new alternatives are added to a
menu. The three examples below illustrate classic random choice rules on which
applications have relied in the literature. All of them are monotone.
Example 1 (Rational Choice). The decision maker has a complete and transitive
preference relation < ⊂ A × A. Her random choice rule is specified as ρ(i, b) = 1
whenever i ∈ b is the unique best alternative according to <. If b has exactly
two alternatives which are best according to <, they are chosen with the same
probability. When b has more than two best alternatives according to <, we
only require each of them to be chosen with strictly positive probability, every
other alternative to be chosen with zero probability, and the resulting rule to be
monotone.5
♦
Example 2 (Random Utility). Most econometric models of discrete choice such
as logit and probit are special examples of random utility maximizers. To fix
ideas, suppose the grand set of alternatives A is finite. Let U be a collection
of strict utility functions on A. A random utility π is a probability distribution
on U. A random choice rule maximizes a random utility π if the probability of
choosing alternative i from menu b equals the probability that π assigns to utility
functions that attain their maximum in b at i.
4
♦
Monotone random choice rules are sometimes called regular in the literature.
Alternatively, we could have imposed uniform tie-breaks for all sets with multiple best alternatives.
The resulting rule would automatically satisfy monotonicity. This alternative definition turns out to be too
restrictive for the purposes of the theory developed later on.
5
13
Example 3 (Probit Rule). The probit rule is a random utility maximizer with
Gaussian errors. Suppose the set A has n alterantives. The probit rule specifies
a random vector X : Ω → Rn with a joint normal distribution, X ∼ N (µ, Σ). In
each menu a ⊂ A, alternative i ∈ a is chosen with probability ρ(i, a) = P{Xi >
Xj , ∀j ∈ a, j 6= i}.
♦
We will sometimes restrict our attention to binary choices. A binary random
choice rule is obtained by restricting the domain of a random choice rule to sets
of two alternatives. To simplify the expression for choice probabilities in binary
menus we will write
ρ(i, j) := ρ(i, {i, j}).
Definition. A utility index µ : A → R and a function over pairs γ : A × A →
[−1, 1] are utility signal parameters on A if
(i) For each i ∈ A, γ(i, i) = 1;
(ii) For each i, j ∈ A, γ(i, j) = γ(j, i);
(iii) For each enumeration of a finite menu b = {1, 2, . . . , n} ⊂ A, the n × n
symmetric matrix whose row i and column j is given by γ(i, j) is positive
definite.
To simplify notation, we write µi = µ(i) and γij = γ(i, j).
Before introducing our model, it is useful to understand how utility signal
parameters (µ, γ) can represent the probit rule of Example 3. Let (µ, γ) be
utility signal parameters. For each enumeration of a menu b = {1, 2, . . . , n}, let
the random utility for the alternatives in b be given by the n-dimensional random
vector X, where X is joint normally distributed and has mean (µ1 , µ2 , . . . , µn )
and covariance matrix (γij ). Then by construction ρ(i, b) = P{Xi > Xj , ∀j 6= i}
defines a probit rule.
The classic interpretation in econometric work for a random utility maximizer
ρ is that it describes the frequencies of choices of a population of deterministic
utility maximizers. The randomness arises from the modeler’s imperfect knowledge of the tastes of each individual. Even though at the individual level the
choice is deterministic, due to unobservable characteristics the econometrician is
only able to predict the choices of the individual in probabilistic terms.
14
We can alternatively interpret random utility maximizers as a model of oneshot learning. When confronted with a set of alternatives, the decision maker
receives a precise signal of how those alternatives are ranked according to her
preferences. While tastes are stochastic, each choice realization is interpreted as
perfectly reflecting the ranking of alternatives at the moment in which the choice
was made.
Our model of random choice departs from random utility maximization and is
based on a gradual learning interpretation. Ultimately preferences are described
by a rational, complete and transitive ranking of all alternatives. But ex-ante
the decision maker has no information about how the alternatives in a particular
menu b are ranked. Instead, the decision maker has prior beliefs about the distribution of her preferences in the same kind of choice situation. These prior beliefs
are symmetric, i.e., all alternatives are perceived in exactly the same way ex-ante.
Before making a choice from the menu b, the individual has the opportunity to
contemplate the alternatives. This contemplation takes the form of a signal that
conveys partial information about the true utility of the alternatives. From the
point of view of the observer, this formulation yields the following random choice
rule.
2.2
Bayesian Probit Rule
The model can be described as the following modification of the probit rule. The
decision maker is described by utility signal parameters (µ, γ) and additionally by
prior parameters (m0 , s20 ) where m0 ∈ R and s0 > 0. The parameter µ describes
the true utility of each alternative and is unknown to the decision maker. The
decision maker has a prior belief about µ that is jointly Gaussian, where the
utility of each alternative is seen as an independent draw from the same Gaussian
distribution with mean m0 and variance s20 . When facing a choice from a menu
b = {1, 2, . . . , n}, the individual chooses as if observing a draw from the random
utility vector X, updating beliefs about the utility of each alternative according
to Bayes’ rule, and picking the alternative with the highest posterior mean. We
refer to the resulting random choice rule as a Bayesian probit rule with utility
signal parameters (µ, γ) and prior parameters (m0 , s20 ).
How much the decision maker learns about the alternatives before making a
choice will depend on the precision of the signals. A natural question to ask in
15
this model is how random choice behavior changes when the information about
utility is made more precise. For example, one could do comparative statics
analysis of random choice behavior when the decision maker is allowed to draw
an additional independent signal from the same distribution. Choices arising from
more precise information should exhibit fewer mistakes. More informed behavior
can be interpreted as arising, for example, from a smarter decision maker, from
a decision maker contemplating the alternatives at a closer distance, or from a
decision maker who contemplated the alternatives for a longer period of time
before making a decision.
Our model describes random choice behavior as the precision of the signal for
a random choice rule varies continuously from pure noise to arbitrarily precise
signals. To allow such comparisons, we study an entire family of random choice
rules, indexed by an ordered set. We call the resulting object a random choice
process.
2.3
Random Choice Process
To investigate the effects of increasing the precision of the utility signal over random choice behavior, we now introduce a family of random choice rules indexed
by an ordered set T. For concreteness, we can think of T = [0, ∞) as time.
Definition. A function ρ : A × A+ × T → [0, 1] is a random choice process if for
every time t ∈ T the restriction ρ(·, ·, t) is a random choice rule.
A random choice process describes the probability distribution of choices for
each finite set of alternatives b ∈ A+ and each point in time t ∈ T. Time can
be seen merely as an abstract ordering of the different choice rules that compose
the random choice process, but it can also be interpreted in the more literal
sense of a temporal dimension. For example, a random choice process can be
thought of as encoding observable behavior in experiments of individual decision
making where both the set of alternatives and the time available to contemplate
the alternatives can be manipulated.
With the literal interpretation of T as time, it is important to note that we
do not treat the entire path of each realization of a choice process as observable. While this stronger enrichment of choice data would be sufficient, and
16
is considered in theoretical work at least since Campbell (1978), and in recent
experimental work in Caplin and Dean (2010), it is not necessary here.
Henceforward, we write ρt (i, a) instead of ρ(i, a, t). We use ρt to refer to the
random choice rule obtained from the random choice process ρ by restricting its
domain to a fixed time t. A random choice process ρ is continuous if for every
i ∈ a ∈ A+ the mapping t →
7 ρt (i, a) is continuous. We are interested in the
random choice process of a decision maker whose choices are based on better
information as time increases, approaching rationality as time goes to infinity.
We say that a random choice process ρ ultimately learns if there is a random
choice rule ρ∞ satisfying the conditions of rational choice in Example 1, such
that for each i ∈ a ∈ A+ we have ρt (i, a) → ρ∞ (i, a) as t → ∞. A random choice
process ρ starts uninformed when for all menus b ∈ A+ and all alternatives
i, j ∈ b, we have ρ0 (i, b) = ρ0 (j, b). In particular, for a binary random choice
process we have ρ0 (i, j) = ρ0 (j, i) for all i, j.
Definition. A random choice process ρ is a learning process if it is continuous,
starts uninformed, ultimately learns, and for all pairs of alternatives i, j ∈ A the
mapping t 7→ ρt (i, j) is either strictly increasing, strictly decreasing, or constant.
2.4
Bayesian Probit Process
This subsection completes the description of our model. The Bayesian probit
process is a random choice process ρ in which every component random choice
rule ρt is a Bayesian probit rule with the same utility signal parameters (µ, γ).
As t increases, ρt is based on more precise information. To introduce the formal
description, consider for a moment the case of discrete time t ∈ {1, 2, 3, . . . }. In
this case ρ1 is just the Bayesian probit rule described in Subsection 2.2 above.
Given ρt , we let ρt+1 correspond to the random choice rule obtained from ρt
as if the decision maker could observe an additional independent utility signal
realization from the same distribution. The continuous time case is analogous to
the discrete case when we let the time interval go to zero.
Utility signals as a diffusion process
The decision maker’s prior beliefs about the utility of each alternative are identically and independently distributed Gaussian with mean m0 ∈ R and variance
17
s20 > 0. This prior reflects her information about the alternatives before starting
the learning process. Ex-ante, all alternatives look the same, reflecting the fact
that no signal has been observed. When the grand set of choice objects A has
enough variety, we can think of the prior as the actual distribution of the utility parameter when the decision maker faces choice problems with alternatives
from A. With this interpretation, the decision maker has correct beliefs, given
the information available, and assuming that each finite menu is composed of an
independent random sample of objects from A.
Let {1, 2, . . . , n} be an enumeration of the alternatives in a menu b ∈ A+ . We
model choices from b at each point in time as being governed by a covert utility
signal process. The signal is interpreted as representing the decision maker’s
noisy perception of the desirability of each alternative over time. We model the
signal process X as a Brownian motion with drift, given by
dX(t) = µ dt + Λ dW (t),
X(0) = 0
(1)
where X(t) is an n-dimensional vector, µ ∈ Rn is a constant drift vector, Λ is a
constant n × n matrix with full rank and W (t) = (W1 (t), . . . , Wn (t)) is a Wiener
process.
The signal starts at X(0) = 0 almost surely. For each fixed time t > 0,
the vector X(t) has a joint Gaussian distribution with mean µt and covariance
matrix tΛΛ0 . Each coordinate of X is a one-dimensional signal that corresponds
uniquely to one of the alternatives in the menu. When we refer to the signal X
for menu b ∈ A+ , we have in mind an enumeration of the alternatives in b that
corresponds to the ordering of the coordinates in X.
We will assume that the signal processes that the decision maker observes
when facing different menus are consistent: the drift and covariance parameters
are always taken form the corresponding utility signal parameters (µ, γ) on A. For
example, when one of the alternatives is deleted from menu b, the corresponding
coordinate of X is removed, but otherwise the signal process remain the same.
2.5
Baseline characterization
We start by characterizing a baseline Bayesian probit process in which choices
evolve as if pairs of alternatives did not vary in similarity.
18
We now introduce some additional notation. When ρ is a random choice rule,
the number ρ(i, a) is a probability value between zero and one. To simplify the
statements below, it will sometimes be convenient to express this quantity in the
scale [−∞, ∞] by using the standard Gaussian cumulative distribution function
Φ. Given a random choice rule ρ, we define


 −∞,
ρ̂(i, a) :=
Φ−1 (ρ(i, a)) ,


+∞,
ρ̂ : A × A+ → [−∞, ∞] by
if ρ(i, a) = 0
if ρ(i, a) ∈ (0, 1)
(2)
if ρ(i, a) = 1.
Definition. The random choice rule ρt satisfies the Gaussian triangle condition
when, for all i, j, k ∈ A,
ρ̂t (i, k) = ρ̂t (i, j) + ρ̂t (j, k).
(3)
A random choice process ρ satisfies the Gaussian triangle condition when it is
satisfied by every ρt .
The Gaussian triangle condition is a cross-restriction on how the random
choice rule is able to discriminate among pairs of alternatives. It says that if we
know how well the random choice rule discriminates among the pairs (i, j) and
(j, k), then we should also know how it discriminates between i and k. Note that
the condition is expressed using the Gaussian transformation ρ̂ defined in (2).
This condition arises naturally in a model with Gaussian noise. Analogous conditions would hold for other distributions, e.g., the logit model.
Definition. A random choice process ρ has a Gaussian learning rate when for
all alternatives i, j, k, l ∈ A and all times s, t ∈ T
ρ̂t (i, j) × ρ̂s (k, l) = ρ̂s (i, j) × ρ̂t (k, l).
(4)
To understand the substance of this last property, suppose for a moment that
t > s and note that when the denominators are different from zero we can rewrite
condition (4) as
ρ̂t (i, j)
ρ̂t (k, l)
=
.
ρ̂s (i, j)
ρ̂s (k, l)
The left hand side ratio is a measure of how the discrimination between alternatives i and j evolved from time s to time t. The Gaussian learning rate condition
requires that this ratio be constant across pairs of alternatives: if we know how
19
the discrimination between i and j changed from s to t, and we know how alternatives k and l were discriminated at time s, then we must know how those two
alternatives are discriminated at t.
Finally, we introduce a simple notion of time change. A function f : T → T
is a time change when it is strictly increasing and continuous. We say that ρ
a Bayesian learning process up to a time change when there is a time change f
such that ρ0t = ρf −1 (t) is a Bayesian learning process. Of course, every Bayesian
learning process ρ can be shown to be Bayesian learning process up to a time
change by taking f (t) = t.
Theorem 1 (Baseline Model). A binary random choice process ρ is a simple
Bayesian probit process (up to a time change) if and only if it is a learning process,
has a Gaussian learning rate and satisfies the Gaussian triangle condition.
3
Revealed Similarity
In classical revealed preference theory, choice data takes the form of a deterministic choice function. A deterministic choice function picks one or more elements
from a given menu of alternatives with probability one. The Weak Axiom of Revealed Preference (WARP) imposes consistency on choice data across different
menus of alternatives. If an element j is picked from menu a when k was also
available, then WARP does not allow k to be picked in any other menu that contains j, unless j is picked as well. The theory allows us to tease out a preference
relation from choice data that satisfies WARP.
Our choice data takes the form of a random choice process ρ. Since the choice
data is stochastic, a random choice process finely captures the tendency of a
decision maker to pick an alternative j over an alternative k. Our theory goes
beyond classical revealed preference to explain comparability: how easy is it for
a decision maker to choose the best alternative out of the pair j, k?
We define comparability as how easy it is to compare two options. Comparability depends on preference. Consider a choice from the pair j, k where j is
actually preferred to k. Comparability is directly proportional to how likely the
decision maker is to choose j. In other words, the less likely the decision maker
is to make a mistake and pick k, the easier it is to compare the pair j, k.
Let ρ be a random choice rule. We define a preference relation % on A from
20
the binary random choices in ρ in the following manner. Let i % j and say that
alternative i is at least as good as j if
ρ(i, j) ≥
1
≥ ρ(j, i).
2
Write ∼ for the symmetric part of % as usual. When ρ satisfies the assumptions
of Theorem 1, % is complete and transitive and represented by µ. Hence under
those assumptions the random choice rule reveals the preference ranking %.
Analogously, from a given ρ we can define a similarity relation %̇ over pairs
of alternatives on A × A. A pair of alternatives i, j ∈ A is more similar than a
pair i0 , j 0 ∈ A, denoted as (i, j)%̇(i0 , j 0 ), when i ∼ i0 , j ∼ j 0 and
ρ(i, j)ρ(j, i) ≤ ρ(i0 , j 0 )ρ(j 0 , i0 )
(5)
and we say that i is more similar to j than to j 0 when we take i = i0 above.
To interpret this notion of similarity, note that i and i0 are equally ranked
according to preference, and also j and j 0 are equally ranked according to preference. Insofar as preference is concerned, the alternatives in the pair (i, j) are
compared exactly as the alternatives in the pair (i0 , j 0 ). Expression (5) says that
the random choice rule discriminates the pair (i, j) at least as well as the pair
(i0 , j 0 ). A strict inequality in (5) therefore indicates that (i, j) must be easier to
compare.
For a random choice process ρ satisfying the assumptions of Theorem 1, there
is never strict inequality in (5). In this sense, Theorem 1 characterizes a baseline
case in which all pairs of alternatives are equally similar. The characterization
is useful because it allows us to identify the Gaussian triangle condition as the
property responsible for all alternatives being equally similar.
To understand the role of the Gaussian triangle condition, let i ∼ i0 and j 0 ∼ j.
This implies that ρ(i, i0 ) = ρ(j, j 0 ) = 1/2 and therefore ρ̂(i, i0 ) = ρ̂(j, j 0 ) = 0. By
the Gaussian triangle condition we have
ρ̂(i, j) = ρ̂(i, i0 ) + ρ̂(i0 , j) = ρ̂(i, i0 ) + ρ̂(i0 , j 0 ) + ρ̂(j 0 , j) = ρ̂(i0 , j 0 )
and therefore ρ(i, j) = ρ(i0 , j 0 ), so that the inequality in (5) is never strict.
Hence, in order to study the consequences of varying degrees of similarity
for random choice behavior, we need to relax the Gaussian triangle condition.
In the next subsection, we will require the Gaussian triangle condition to fail.
21
Moreover, in order to identify the similarity relation, we will require that the
grand set of alternatives A have a pivotal alternative and enough variety in the
way that the Gaussian triangle condition is violated by triples that include the
pivotal alternative.
3.1
A distinct alternative
In order to facilitate the introduction of the next assumption, we will restate the
Gaussian triangle condition in a more convenient form. Recall that the Gaussian
triangle condition requires that for all triples of alternatives i, j, k ∈ A,
ρ̂(i, k) = ρ̂(i, j) + ρ̂(j, k).
Therefore, when ρ(i, k) 6= 1/2, so that ρ̂(i, k) 6= 0, we can rewrite this condition
as a ratio
ρ̂(i, j) + ρ̂(j, k)
=1
ρ̂(i, k)
or equivalently in the following convenient form
2
ρ̂t (i, j) + ρ̂t (j, k)
1−
= 0.
ρ̂t (i, k)
Definition. A random choice rule ρ has a distinct alternative k ∗ when, for every
pair i 6∼ j ∈ A and every ε > 0, there exist a pair of alternatives j 0 , j 00 ∈ A such
that j 0 ∼ j ∼ j 00 , ρt (j, k ∗ ) = ρt (j 0 , k ∗ ) = ρt (j 00 , k ∗ ) and
2
2
ρ̂t (i, k ∗ ) + ρ̂t (k ∗ , j 00 )
ρ̂t (i, k ∗ ) + ρ̂t (k ∗ , j 0 )
1−
<ε−1<1−ε<1−
.
ρ̂t (i, j 00 )
ρ̂t (i, j 0 )
To understand the role of the distinct alternative k ∗ in the condition above,
note that it requires that for every pair of alternatives i, j in which i is not equally
preferred to j, we can find two alternatives j 0 , j 00 equally preferred to j, such that
the triples i, j 0 , k ∗ and i, j 00 , k ∗ sufficiently violate the Gaussian triangle condition
in opposite directions. So this condition embeds both an assumption about the
existence of a distinct or pivotal alternative k ∗ , and also a condition about a rich
variety of options in the set A. Substituting the distinct alternative assumption
for the Gaussian triangle condition, one can obtain a Bayesian Probit Process in
which γ is not orthogonal.
Suppose M is a property of a random choice rule. We say that a random
choice process ρ eventually has property M when there is a time T ∈ T such that
for all times t > T the random choice rule ρt has property M .
22
Proposition 2 (Ranking stabilization). For every menu b ∈ A+ and every pair
of alternatives i j in b, the Bayesian learning process eventually chooses i more
often than j in b.
4
Similarity Puzzle
In order to state the main results, throughout this section we assume that ρ is
a Bayesian Probit Process with a distinct alternative. We use % to denote the
preference relation over alternatives corresponding to ρ, and we use %̇ to denote
the similarity relation over pairs corresponding to ρ. Choice alternatives are
labeled 1, 2 and 3.
In the Theorems below, alternatives 1 and 2 are equally desirable: 1 ∼ 2. So
alternatives 1 and 2 are chosen with probability one-half from menu {1, 2} at any
˙ (1, 2) and (1, 3) ˙ (2, 3), i.e., the pair (1, 3) is
date. We also assume that (1, 3) more similar than pairs (1, 2) and (2, 3). In other words, alternative 3 is more
similar to alternative 1 than to alternative 2.
Theorem 3 (Similarity Puzzle). There are times T1 , T2 > 0 such that introducing
3 hurts 2 more than it hurts 1 for all times before T1 , and the reverse occurs for
all times greater than T2 .
Early in the learning process, the decision maker is relatively unfamiliar with
the options. The relative ranking of similar options will be learned faster than
dissimilar ones. Theorem 3 shows that this can go against Tversky’s similarity
hypothesis early in the learning process, but that as the decision maker becomes
more familiar with the options this “attraction effect” disappears. The following
Corollary illustrates both effects in their most extreme form.
Corollary 4. Let 1 ∼ 2 ∼ 3 and ε > 0. If the pairs (1, 2) and (2, 3) are equally
similar and the pair (1, 3) is taken sufficiently similar, we have
(i) for all t sufficiently small, ρt (1, 123), ρt (3, 123) > 1/2 − ε; and
(ii) for all t sufficiently large, ρt (1, 123), ρt (3, 123) ∈ [1/4, 1/4 + ε).
Item (i) in the Corollary says that, when the decision maker is uninformed,
she will choose one of the alternatives of the very similar pair with probability
close to one. To understand this result, consider what happens at times close to
23
zero, when utility signals are very imprecise. As we saw in Proposition 7, in the
limit as t goes to zero utility becomes irrelevant, and choice probabilities depend
solely on similarity. When the similarity of the pair (1, 3) is extreme, the decision
maker rapidly learns wether 1 is a better alternative than 3 or vice versa, before
learning much about anything else, and therefore alternative 2 will seldom be the
most attractive.
Item (ii) says that, in the limit as t goes to infinity, the alternatives in the very
similar pair (1, 3) are treated as a single option. In general, any small difference
in the utility of the alternatives will eventually be learned by the decision maker,
and the best alternative will be chosen with probability close to one. When the
three alternatives have the exact same utility, it is similarity, rather than utility,
that determines how ties are broken: alternative 2 is chosen with probability
close to one-half and an alternative in the very similar pair (1, 3) is be chosen
with probability close to one-half.
Our final result shows that the initial positive effect of similarity can persist
indefinitely when the similar options are not equally desirable. Moreover, similarity can cause the probability of an existing option to increase, leading to a
violation of monotonicity and to the classic attraction effect. Recall that alternatives 1 and 2 are equally desirable, and therefore chosen from the menu {1, 2}
with probability one-half at any date.
Theorem 5 (Attraction effect). For every ε > 0, if alternative 3 is sufficiently
inferior, then alternative 1 is chosen from menu {1, 2, 3} with probability strictly
larger than one-half for all t > ε.
Hence adding a sufficiently inferior alternative 3 to the menu {1, 2} will boost
the probability of the similar alternative 1 being chosen —the phenomenon known
as the attraction effect. The Theorem shows that the attraction effect appears
arbitrarily early in the choice process, and lasts indefinitely. For a concrete
illustration of how the attraction effect arises in the context of our model, consider
the following numerical example.
Example 4 (Attraction Effect). The set of alternatives is given by A = {1, 2, 3}.
The decision maker’s prior beliefs are i.i.d. standard normally distributed. Alternatives 1 and 2 are equally desirable with utility µ1 = µ2 = 3. Alternative 3
is less desirable with µ3 = −3. The inferior alternative 3 is similar to alternative
24
1 with γ13 = 1/2, while γ12 = γ23 = 0.
When choosing from the menu {1, 2} the decision maker can never discriminate among the two options: each option is chosen with probability one-half at
any given time. Figure 5 describes what happens once the inferior but similar
alternative is introduced.
Choice Probability
1
0.0001
0.01
1
100
10 000
1
2
0
1
1
2
0.0001
0.01
1
Time Hlog scaleL
100
10 000
0
Figure 5: The attraction effect
Figure 5 shows time in logarithmic scale, to better visualize the start of the
choice process. Note that while the graph shows 10000 units of time, the first
half covers the first unit of time. The top curve (solid line) is the probability that
alternative 1 is chosen; the middle curve (dotted line) corresponds to the choice
probability for alternative 2; and the bottom curve (dashed line) corresponds to
the choice probability of alterative 3.
At the beginning of the choice process, when choices are based on relatively
little information about the alternatives, the probability of choosing the inferior
alternative 3 rapidly decreases, while the probability of choosing alternative 1 is
boosted up. Similarity allows the decision maker to learn the relative ranking
of alternatives 1 and 3 faster than the relative ranking of the other pairs. This
leads to the attraction effect: the addition of the inferior but similar alternative
3 causes the probability of choosing alternative 1 to increase.
25
The attraction effect is sustained as information gets more precise. Since
alternatives 1 and 2 are equally desirable, eventually each one is chosen with
probability converging to one-half. But while the choice probability of alternative
1 tends to one-half from above, the choice probability of alternative 2 tends to
one-half from below. The choice probability of alternative 1 very early jumps
above one-half, violating monotonicity, and remains there for the entire length of
the choice process. By Theorem 5, we know this violation of monotonicity can
happen arbitrarily early in the choice process, if the added alternative 3 is taken
sufficiently inferior.
5
♦
Naı̈ve probit versus Bayesian probit
Proposition 6 (Equivalence of binary choices). A binary random choice process
ρ is a Bayesian probit process if and only if it is a naı̈ve probit process. In the
affirmative case, they can share the same utility signal structure.
Binary choices alone do not allow us to distinguish a Bayesian probit process
from a naı̈ve probit process, in either the baseline characterization or the main
characterization above. We now show that while behavior is indistinguishable
when restricted to binary choices, the overlap between the two models disappears
when choice from menus with three or more alternatives are considered. To see
this it is sufficient to consider ternary choice. The next result shows that in
menus of three alternatives, the Bayesian probit process and the naı̈ve probit
process exhibit radically different choice behavior from the very beginning. Let
ρ be a Bayesian probit process and let ρ̈ be a naı̈ve probit process sharing the
utility signal parameters (µ, γ). Let b = {1, 2, 3}.
Proposition 7 (Ternary choice at time zero). In the limit as t → 0+, the naı̈ve
probit rule and the Bayesian probit rule choose alternative 1 with probability
1
1
(1+γ23 −γ12 −γ13 )
ρ̈0 (1, b) = +
arctan √2(1−γ )2(1−γ )−(1+γ −γ −γ )2
12
13
23
12
13
4 2π
1
1
(1+γ12 )(1+γ13 )−γ23 (1+γ12 +γ13 +γ23 )
ρ0 (1, b) = +
arctan √(3+2γ +2γ +2γ )(1+2γ γ γ −γ 2 −γ 2 −γ 2 )
12
13
23
12 13 23
12
13
23
4 2π
and analogous expressions hold for alternatives 2 and 3.
Corollary 8. The Bayesian probit process and the naı̈ve probit process are indistinguishable if and only if they share utility signal parameters and γ is orthogonal.
26
Proposition 7 and its corollary show that when we observe choice behavior
from menus with more than two alternatives, the Bayesian probit process and
the naı̈ve probit process exhibit incompatible behavior, except for the knife-edge
case of a common parameterization with orthogonal noise. Figure 5 illustrates
how choices for both processes start at time zero from menu b = {1, 2, 3}. The
Figure shows how the probability of choosing alternative 1 changes as a function
of the correlation γ12 between alternatives 1 and 2. The remaining correlation
parameters γ13 and γ23 are fixed at zero. The solid line corresponds the Bayesian
probit rule, and the dashed line corresponds to the naı̈ve probit rule. Note that
the curves intersect when γ12 = 0, where noise is orthogonal and both processes
start with a uniform distribution, i.e., each alternative is chosen with probability
1/3. As can be seen in Proposition 7, choices at time zero do not depend on µ.
Proposition 7 and its corollary relied on choice behavior at the start of the
learning process to distinguish between the Bayesian probit and the naı̈ve probit.
We can in general distinguish them from observed choice behavior, except in
the knife-edge case of orthogonal utility signals. Note that since each random
choice rule ρ̈t in the naı̈ve probit is a random utility maximizer, the naı̈ve probit
process must satisfy monotonicity. Our main results in the previous section
demonstrated another distinction of the Bayesian probit from the naı̈ve probit
in terms of observable choices, namely, the violation of monotonicity.
6
Application: Relating Noise to Observables
In experimental settings where the attraction and compromise effects are observed, choice alternatives are often described by two attributes. For example, in
Simonson (1989) cars are described by ‘ride quality’ and ‘miles per galon’, brands
of beer are described by ‘price of a six-pack’ and ‘quality rating’ and brands of
mouthwash by ‘fresh breath effectiveness’ and ‘germ-killing effectiveness’. Assume that each alternative xi is described by a vector (xi1 , xi2 ) ∈ R2 . The extra
structure allows us to build a model in which the noise correlation depends explicitly on the observable characteristics. This model allows us to distinguish
violations of monotonicity based on the location of the new alternatives in the
attribute space. For example, it allows us to distinguish the attraction effect
from the compromise effect.
27
0.5
0.4
0.3
0.2
0.1
-1.0
0.5
-0.5
1.0
Figure 6: Probability of alternative 1 being chosen from menu b = {1, 2, 3} as a function
of the correlation γ12 between the utility signals of alternatives 1 and 2, in the limit as
t is taken arbitrarily close to zero. The solid line corresponds to the Bayesian learning
model. The dashed line corresponds to the naı̈ve learning probit. The correlations γ13 and
γ23 are fixed equal to zero. Since choices at time zero are determined solely by γ, and the
parameterization is symmetric from the point of view of alternatives 1 and 2, the graph
simultaneously describes the choice probabilities for alternative 2. When γ12 is zero the
utility signal structure is simple, choices at time zero correspond to the uniform distribution
for both models, and therefore the two lines cross at exactly one third. As γ12 reaches one,
alternatives 1 and 2 become maximally similar and the naı̈ve learner chooses each of them
with probability 1/4. In contrast, the Bayesian learner chooses alternatives 1 and 2 with
probability 1/2 and alternative 3 with probability zero.
28
Example 5 (Parameterization of covariance matrix in multinomial probit).
Hausman and Wise (1978) propose the following parameterization of the covariance matrix for a multinomial probit, when each choice alternative xi is described by two positive coordinates (xi1 , xi2 ) ∈ R2 . In their specification, the
random utility for alternative xi is given by
U (xi ) = β1 xi1 + β2 xi2
where β1 , β2 are independent and normally distributed with the same variance
σ 2 > 0. This can be interpreted as a consumer with a linear utility over object in
a two-dimensional commodity space, where the linear weights of each dimension
are random. The common weights will cause the utility for different alternatives
to be correlated. Specifically we have
xi1 xj1 σ 2 + xi2 xj2 σ 2
p
p
Corr(U (xi ), U (xj )) =
σ 2 (xi1 + xi2 ) σ 2 (xj1 + xj2 )
xi1 xj1 + xi2 xj2
=
||xi ||||xj ||
= cos(xi , xj )
so that the correlation coefficient for the utilities of alternatives xi and xj corresponds to the cosine of the angle formed by the two alternatives in the twodimensional commodity space.
♦
Our behavioral notion of similarity translates immediately to this particular
setting under the simple parameterization of Example 5. Since similarity is captured by noise correlation, two alternatives are maximally similar when they lie
on the same ray departing from the origin. In this case they form an angle of
zero radians, and the correlation γij is equal to the cosine of zero, which is one.
This corresponds to the intuition that when alternatives clearly dominate each
other in all attributes, comparison is facilitated. On the other hand, if alternatives form an angle close to ninety degrees, the cosine of their angle is close to
zero, and noise is uncorrelated. Intuitively, since the alternatives do not share
any physical attributes, their comparison is hard.
29
γij = cos θ
i
j
θ
Figure 7: The noise correlation parameter γij equals the cosine of the angle formed by i and
j in R2 .
7
Conclusion
We proposed a model of random choice in which the decision maker initially has
a coarse and symmetric perception of the utility of the alternatives and before
making a choice engages in a learning process about his own ranking of the available options. The learning process is interrupted when a choice must be made.
We characterized the random choice behavior of such a decision maker when information about utility follows a diffusion process. We showed conditions that
allowed us to identify a preference ranking over the alternatives and, in addition,
a similarity ranking over pairs of alternatives. Both rankings are subjective and
revealed by the choice process.
In the model, the intensity with which a random choice rule discriminates
between two alternatives in a menu depends on three factors: the relative ranking of the alternatives according to preference; the similarity of the alternatives;
and the overall precision of the information available to the decision maker. We
modeled choices as a random choice process in which the precision of the information can be interpreted as the amount of time available to contemplate the
options. Discrimination improves when alternatives are more distant according
to preference, when the alternatives are more similar, and with the amount of
30
time the decision maker is allowed to contemplate the alternatives.
The model captures similarity in the form of utility signal correlation. Signal
correlation causes the learning process to be context dependent, and therefore
choices are also context dependent. By identifying a notion of similarity that is
independent of preference, this model allowed us to solve the similarity puzzle of
random choice.
By allowing context dependence through noise correlation, the model has
the potential to address other empirical regularities found in the literature and
related to context dependence, such as the status quo bias and the compromise
effect. Applications of the model to these phenomena is left for future work.
References
Campbell, Donald E., “Realization of Choice Functions,” Econometrica, 1978,
46 (1), pp. 171–180.
Caplin, Andrew and Mark Dean, “Search, Choice, and Revealed Preference,”
Theoretical Economics, Forthcoming 2010.
Debreu, Gerard, “Review: Individual Choice Behavior,” The American Economic Review, 1960, 50 (1), 186–188.
Gul, Faruk, Paulo Natenzon, and Wolfgang Pesendorfer, “Random
Choice as Behavioral Optimization,” Working Paper, Princeton University
February 2012.
Hausman, Jerry A. and David A. Wise, “A Conditional Probit Model
for Qualitative Choice: Discrete Decisions Recognizing Interdependence and
Heterogeneous Preferences,” Econometrica, 1978, 46 (2), pp. 403–426.
Huber, Joel and Christopher Puto, “Market Boundaries and Product
Choice: Illustrating Attraction and Substitution Effects,” The Journal of Consumer Research, 1983, 10 (1), pp. 31–44.
, J. W. Payne, and C. Puto, “Adding asymmetrically dominated alternatives: Violations of regularity and the similarity hypothesis,” Journal of
Consumer Research, 1982, 9 (1), 90–98.
31
Keller, Godfrey, “Brownian Motion and Normally Distributed Beliefs,” Working Paper, University of Oxford December 2009.
Luce, R. Duncan, Individual Choice Behavior: a Theoretical Analysis, Wiley
New York, 1959.
Manzini, Paola and Marco Mariotti, “Sequentially Rationalizable Choice,”
American Economic Review, 2007, 97 (5), 1824–1839.
Masatlioglu, Y., D. Nakajima, and E.Y. Ozbay, “Revealed attention,” The
American Economic Review, 2012, 102 (5), 2183–2205.
McFadden, Daniel, “Conditional logit analysis of qualitative choice behavior,”
in Paul Zarembka, ed., Frontiers in Econometrics, New York: Academic Press,
1974, chapter 4, pp. 105–42.
, “Economic Choices,” The American Economic Review, 2001, 91 (3), 351–378.
Ok, E.A., P. Ortoleva, and G. Riella, “Revealed (P)reference Theory,”
Working Paper, August 2012.
Simonson, Itamar, “Choice Based on Reasons: The Case of Attraction and
Compromise Effects,” The Journal of Consumer Research, 1989, 16 (2), pp.
158–174.
Train, Kenneth, Discrete Choice Methods with Simulation, 2nd ed., Cambridge
University Press, 2009.
Tversky, A., “Choice by elimination,” Journal of mathematical psychology,
1972, 9 (4), 341–367.
, “Elimination by aspects: A theory of choice,” Psychological Review, 1972, 79
(4), 281–299.
8
Appendix: Proofs
Proof of Theorem 1 — Necessity
Suppose ρ is the Bayesian learning process with prior parameters (m0 , s20 ) and
utility signal structure (µ, γ). Consider an enumeration of a finite menu b =
32
{1, 2, . . . , n}. Prior beliefs about µ are jointly Gaussian with mean m(0) =
(m0 , m0 , . . . , m0 ) and covariance matrix s(0) = s20 I, where I is the n × n identity
matrix. An application of Bayes’ rule gives that posterior beliefs about µ after
observing the signal process X up to time t > 0 are joint Gaussian with mean
m(t) and covariance s(t) given by:
−1
s(t) = s(0)−1 + t(ΛΛ0 )−1
m(t) = s(t) s(0)−1 m(0) + (ΛΛ0 )−1 X(t)
where it is immediate that s(t) is a deterministic function of time, while m(t)
is random and has a joint Gaussian distribution itself. The parameters of the
posterior mean m(t) are given by
E[m(t)] = s(t) s(0)−1 m(0) + t(ΛΛ0 )−1 µ
Var[m(t)] = ts(t)(ΛΛ0 )−1 s(t)0
The following Lemma is easy and the proof omitted. For a discussion of the
single dimensional case see Keller (2009).
Lemma 9. Beliefs converge to the true value: E[m(t)] → µ, Var[m(t)] → 0 and
s(t) → 0.
Now consider a binary menu enumerated as b = {1, 2}. When facing the
alternatives in menu b, the decision maker’s choices at t > 0 depend on the
realization of the random vector m(t) = (m1 (t), m2 (t)). Alternative 1 is chosen
if and only if m2 (t) − m1 (t) < 0. Since m(t) is Gaussian, m2 (t) − m1 (t) is also
Gaussian with mean and variance given by
2
ts0 (µ2 − µ1 ) 2ts40 (1 − γ12 )
,
ts20 + 1 − γ12 (ts20 + 1 − γ12 )2
hence
ρt (1, 2) = P{m2 (t) − m1 (t) < 0} = Φ
√
(µ1 − µ2 )
tp
2(1 − γ12 )
!
where again Φ denotes the cdf of the standard Gaussian distribution. When the
utility signal structure (µ, γ) is simple this reduces to
√ (µi − µj )
ρt (i, j) = Φ
t √
2
33
From this expression it is straightforward to verify that ρ is continuous, monotonic, starts uninformed, ultimately learns, and has a Gaussian learning rate.
Finally, it is also easy to verify that it satisfies the Gaussian triangle condition:
ρ̂t (i, j) + ρ̂t (j, k) =
√ (µi − µj + µj − µk )
√
t
= ρ̂t (i, k).
2
Proof of Theorem 1 — Sufficiency
Suppose ρ0 is a continuous, monotonic random choice process that starts uninformed, ultimately learns, has a Gaussian learning rate and satisfies the Gaussian
triangle condition.
Suppose ρ0t (i∗ , j ∗ ) > 1/2 for some i∗ , j ∗ , t∗ .
Step 1: Time change
Let f : T → T be given by
f (s) =
ρ̂0s (i∗ , j ∗ )
ρ̂0t∗ (i∗ , j ∗ )
2
,
∀s ∈ T
and note that it is well-defined since ρ̂0t∗ (i∗ , j ∗ ) 6= 0. Since ρ0 is continuous and
monotonic, f is continuous and strictly increasing. So f is a time change. In
particular we have f (0) = 0 and f (t∗ ) = 1. We use this time change to define the
random choice process ρ by ρt (i, a) = ρ0f −1 (t) (i, a) for each alternative i, menu a
and time t.
Step 2: Time change sets learning rate
For any pair of alternatives k, l ∈ A we have
ρ̂t (k, l) = ρ̂0f −1 (t) (k, l)
ρ̂0f −1 (t) (i∗ , j ∗ )
=
=
=
hence
√
√
ρ̂0t∗ (i∗ , j ∗ )
!
× ρ̂t∗ (k, l)
t × ρ̂0f −1 (1) (k, l)
t × ρ̂1 (k, l)
√
√
√
s × ρ̂t (k, l) = stρ̂1 (k, l) = t × ρ̂s (k, l)
34
f (t)
�
ρ̂�s (i∗ , j ∗ )
ρ̂�t (i∗ , j ∗ )
�2
1
s
t*
t
Figure 8: Time change.
Step 3: Gaussian triangle condition
Note ρ satisfies Gaussian triangle condition, just like ρ0 . The Gaussian triangle
condition is a restriction on each random choice rule ρt , and therefore not affected
by the time change.
Step 4: construct utility signal structure
Take an alternative i∗ ∈ A and, for each j ∈ A define µ : A → R by
µ(j) :=
√
2 ρ̂1 (j, i∗ ),
for all j ∈ A
Since we use the convention ρt (i, i) = 1/2 for all i ∈ A, this implies µ(i∗ ) = 0.
Define γ : A × A → [−1, 1] so that (µ, γ) is simple: let γ(i, i) = 1 for all i ∈ A
and γ(i, j) = 0 for all i 6= j.
Step 5: (µ, γ) is a representation for ρ
By the Gaussian learning rate and the time change, for each t > 0 and j ∈ A we
have
√ √
√
√
t µ(j) = 2 t ρ̂1 (j, i∗ ) = 2 ρ̂t (j, i∗ )
35
By Gaussian triangle condition, for every i, j ∈ A and t > 0
ρt (i, j) = Φ (ρ̂t (i, j))
= Φ (ρ̂t (i, i∗ ) + ρ̂t (i∗ , j))
= Φ (ρ̂t (i, i∗ ) − ρ̂t (j, i∗ ))
√ (µ(i) − µ(j))
√
t
=Φ
2
which together with the necessity part shows that ρ is a Bayesian learning process
with a simple utility signal structure.
Proof of Proposition 2
Since i j we have µi > µj so E[mi (t)−mj (t)] → µi −µj > 0. Since Var[m(t)] →
0 when t goes to infinity, the probability that mi (t) > mj (t) goes to one.
Proof of Proposition 7
In general, for random utility models that specify Gaussian error terms, calculating choice probabilities from the parameters of the model involves the numerical
computation of multivariate integrals. Here we show that for the starting time
of a random choice process, we can obtain expressions for ternary choices in
terms of elementary (trigonometric) functions. We interpret choices at time zero
as the limit of the distribution of choices at time t when t is taken arbitrarily
small. We will obtain expressions that show that a Bayesian learning model and
a naı̈ve learning probit model sharing the same utility signal structure (µ, γ) will
in general exhibit different choice behavior for times close to zero. The two will
coincide only in the knife-edge case in which the common utility signal structure
is simple. In this knife-edge case, both choice processes start with a uniform
distribution over the available alternatives.
Let (µ, γ) be a utility signal structure and let b = {1, 2, 3}. First consider the
naı̈ve learning probit model. The utility signal X corresponding to b has three
dimensions. For every time t > 0 we have X(t) ∼ N (tµ, tΛΛ0 ). Let L1 be the
2 × 3 matrix given by
L1 =
"
−1 1 0
−1 0 1
36
#
so that L1 X(t) = (X2 (t) − X1 (t), X3 (t) − X1 (t)). Then
ρt (1, b) = P{X1 (t) > X2 (t) and X1 (t) > X3 (t)} = P{L1 X(t) ≤ 0}.
The vector L1 X(t) is jointly Gaussian with mean tL1 µ and covariance tL1 ΛΛ0 L01 .
√
So L1 X(t) has the same distribution as tL1 µ+ tM B where B is a bi-dimensional
standard Gaussian random vector and M M 0 = L1 ΛΛ0 L01 has full rank. Take M
as the Cholesky factorization
 p

2(1 − γ12 )
0

M =  1+γ23 −γ12 −γ13 q
−γ12 −γ13 )2
√
2(1 − γ12 ) − (1+γ23
2(1−γ12 )
2(1−γ12 )
then we can write for each t > 0,
ρt (1, b) = P{L1 X(t) ≤ 0}
√
= P{tL1 µ + tM B ≤ 0}
√
= P{M B ≤ − tL1 µ}
and taking t → 0 we obtain
ρ0 (1, b) = P{M B ≤ 0}
Now M B ≤ 0 if and only if

p

0 ≥ B1 2(1 − γ12 )


and
q


 0 ≥ 1+γ
√23 −γ12 −γ13 B1 + 2(1 − γ12 ) −
2(1−γ12 )
(1+γ23 −γ12 −γ13 )2
B2
2(1−γ12 )
if and only if


B ≤0

 1
and


 B2 ≤ −B1 √
(1+γ23 −γ12 −γ13 )
2(1−γ12 )2(1−γ13 )−(1+γ23 −γ12 −γ13 )2
which describes a cone in R2 and, due to the circular symmetry of the standard
Gaussian distribution, we have
ρ0 (1, b) =
1
1
+
arctan
4 2π
(1 + γ23 − γ12 − γ13 )
p
2(1 − γ12 )2(1 − γ13 ) − (1 + γ23 − γ12 − γ13 )2
37
!
.
with analogous expressions for ρ0 (2, b) and ρ0 (3, b).
Now consider the Bayesian learning model with the same utility signal structure and prior parameters (m0 , s20 ). Let L1 be defined as before so that L1 m(t) =
(m2 (t) − m1 (t), m3 (t) − m1 (t)). Then the Bayesian learner chooses alternative
one at time t > 0 with probability
ρt (1, b) = P{m1 (t) > m2 (t) and m1 (t) > m3 (t)} = P{L1 m(t) ≤ 0}
The vector L1 m(t) is jointly Gaussian with mean L1 s(t) [s(0)−1 m(0) + t(ΛΛ0 )−1 µ]
and covariance tL1 s(t)ΛΛ0 s(t)0 L01 .
The two-dimensional random vector L1 m(t) has the same distribution as
√
L1 s(t)s(0)−1 m(0) + tL1 s(t)(ΛΛ0 )−1 µ + tM (t)B
where B is a bi-dimensional standard Gaussian and M (t)M (t)0 = L1 s(t)ΛΛ0 s(t)0 L01 .
Notice the contrast with the naı̈ve learner model, where M did not depend on t.
We can take M (t) to be the Cholesky factorization
"
#
M11 (t)
0
M (t) =
M21 (t) M22 (t)
given by
M11 (t) =
s
C1 (t)
−1 −
3s4 t2
−
s6 t3
+
2
γ12
+
2
γ13
2 + s2 t −3 + γ 2 + γ 2 + γ 2
− 2γ12 γ13 γ23 + γ23
12
13
23
2
where
C1 (t) = s4 −2s8 t4 (−1 + γ12 ) − 2s6 t3 −4 + 2γ12 (1 + γ12 ) + (γ13 − γ23 )2
2
2
2
−4s2 t(2 + γ12 ) −1 + γ12
+ γ13
− 2γ12 γ13 γ23 + γ23
2
2
2
−s4 t2 −12 + 7γ13
+ 2γ12 γ12 (5 + γ12 ) + γ13
− 6(1 + 2γ12 )γ13 γ23 + (7 + 2γ12 )γ23
2
2
2
− −1 + γ12
+ γ13
− 2γ12 γ13 γ23 + γ23
2 + 2γ12 − (γ13 + γ23 )2
and the expressions for M21 (t) and M22 (t) are similarly cumbersome and omitted.
Now we can write, for each t > 0,
ρt (1, b) = P{L1 m(t) ≤ 0}
√
= P{L1 s(t)s(0)−1 m(0) + tL1 s(t)(ΛΛ0 )−1 µ + tM (t)B ≤ 0}
√
= P{ tM (t)B ≤ −L1 s(t)s(0)−1 m(0) − tL1 s(t)(ΛΛ0 )−1 µ}
√
1
= P{M (t)B ≤ − √ L1 s(t)s(0)−1 m(0) − tL1 s(t)(ΛΛ0 )−1 µ}
t
38
Lemma 10. In the limit as t goes to zero,
1
lim √ L1 s(t)s(0)−1 m(0) = 0,
t→0+
t
s
2
2
− 2γ13 γ23 − γ23
2 + 2γ12 − γ13
lim M11 (t) = s2
> 0,
2
2
2
t→0+
1 + 2γ12 γ13 γ23 − γ12
− γ13
− γ23
and
(1 + γ12 )(1 + γ13 ) − γ23 (1 + γ12 + γ13 + γ23 )
M21 (t)
.
=p
2
2
2
t→0+ M22 (t)
)
− γ23
− γ13
(3 + 2γ12 + 2γ13 + 2γ23 )(1 + 2γ12 γ13 γ23 − γ12
lim
Proof. Long and cumbersome, omitted.
Using Lemma 10 we have
ρ0 (1, b) = lim ρt (1, b)
t→0+
√
1
−1
0 −1
= lim P M (t)B ≤ − √ L1 s(t)s(0) m(0) − tL1 s(t)(ΛΛ ) µ
t→0+
t
M21 (t)
=P B1 lim M11 (t) ≤ 0 and B2 ≤ −B1 lim
t→0+
t→0+ M22 (t)
n
o
(1+γ12 )(1+γ13 )−γ23 (1+γ12 +γ13 +γ23 )
√
=P B1 ≤ 0 and B2 ≤ −B1 (3+2γ +2γ +2γ )(1+2γ γ γ −γ 2 −γ 2 −γ 2 )
12
13
23
12 13 23
12
13
23
and by the circular symmetry of the standard Gaussian distribution we obtain
1
1
(1+γ12 )(1+γ13 )−γ23 (1+γ12 +γ13 +γ23 )
ρ0 (1, b) = +
arctan √(3+2γ +2γ +2γ )(1+2γ γ γ −γ 2 −γ 2 −γ 2 )
12
13
23
12 13 23
12
13
23
4 2π
with analogous expressions for ρ0 (2, b) and ρ0 (3, b).
Proof of Corollary 8
Immediate from Proposition 7.
Proof of Theorem 3
When the menu of available alternatives is {1, 2}, both 1 and 2 are equally likely
to be chosen at the start of the random choice process, i.e., in the limit as t → 0+.
By Propostion 7, when the menu includes alternative 3, the probability that
alternative 1 is chosen at the start of the random choice process is given by
1
1
+
arctan
4 2π
(1 + γ12 )(1 + γ13 ) − γ23 (1 + γ12 + γ13 + γ23 )
p
2
2
2
(3 + 2γ12 + 2γ13 + 2γ23 )(1 + 2γ12 γ13 γ23 − γ12
− γ13
− γ23
)
39
!
and the probability for alternative 2 is given by
(1 + γ12 )(1 + γ23 ) − γ13 (1 + γ12 + γ13 + γ23 )
1
1
+
arctan
4 2π
p
2
2
2
)
− γ23
− γ13
(3 + 2γ12 + 2γ13 + 2γ23 )(1 + 2γ12 γ13 γ23 − γ12
!
since the function arctan is strictly increasing, the probability of alternative 1 is
larger than the probability of alternative 2 if and only if
(1+γ12 )(1+γ13 )−γ23 (1+γ12 +γ13 +γ23 ) > (1+γ12 )(1+γ23 )−γ13 (1+γ12 +γ13 +γ23 )
which holds if and only if
(γ13 − γ23 )(2 + 2γ12 + γ13 + γ23 ) > 0
which holds since all γij are positive and since the pair (1, 3) is more similar than
the pair (2, 3) we have γ13 > γ23 . This shows that for t close to zero, by continuity,
introducing alternative 3 hurts alternative 2 more than it hurts alternative 1.
The second part of the Theorem follows from the fact that, as t goes to
infinity, ρt converges to a standard probit random choice rule with E[m(t)] → µ
and Var[m(t)] → ΛΛ0 .
Proof of Corollary 4
Fix utility signal parameters γ12 , γ13 , γ23 ∈ [−1, 1] such that γ13 > γ12 and γ13 >
γ23 . Since γij form a positive definite matrix for every finite menu, we have the
following positive determinant
2
2
2
1 + 2γ12 γ13 γ23 − γ12
− γ13
− γ23
>0
therefore
γ12 γ23 −
q
q
2
2
2
2
(1 − γ12
)(1 − γ23
) < γ13 < γ12 γ23 + (1 − γ12
)(1 − γ23
)
which provides bounds for γ13 for fixed values of γ12 and γ23 . The next Lemma
shows that, when γ13 is the largest of the three parameters, the upper bound is
strictly positive.
Lemma 11. Let γ be utility signal parameters with γ13 > γ12 and γ13 > γ23 .
p
2
2
Then γ12 γ23 + (1 − γ12
)(1 − γ23
) > 0.
40
Proof. If γ12 and γ23 are negative then γ12 γ23 is positive and we are done. Suppose
p
2
2
) > γ13 > γ12 ≥ 0 and we are done. The
)(1 − γ23
γ12 ≥ 0, then γ12 γ23 + (1 − γ12
same applies if γ23 ≥ 0.
By Propostion 7, to prove (i) it suffices to show that, as γ13 approaches the
p
2
2
) from the left, we have
limit γ12 γ23 + (1 − γ12
)(1 − γ23
(1 + γ12 )(1 + γ13 ) − γ23 (1 + γ12 + γ13 + γ23 )
p
→ +∞
2
2
2
)
− γ23
− γ13
(3 + 2γ12 + 2γ13 + 2γ23 )(1 + 2γ12 γ13 γ23 − γ12
(6)
so it is sufficient to show that the numerator above converges to a strictly positive
number, while the denominator is positive and converges to zero.
To verify that the numerator in (6) converges to a strictly positive number,
p
2
2
first suppose γ23 ≤ 0. Then as γ13 → γ12 γ23 + (1 − γ12
)(1 − γ23
), the numerator
tends to
q
2
γ12
)(1
2
γ23
)
(1 + γ12 ) 1 + γ12 γ23 + (1 −
−
q
2
2
− γ23 1 + γ12 + γ12 γ23 + (1 − γ12 )(1 − γ23 ) + γ23
q
2
2
2
= (1 + γ12 ) 1 − γ23 + γ12 γ23 + (1 − γ12 )(1 − γ23 )
q
2
2
− γ23 (1 + γ12 ) − γ23 (1 − γ12
)(1 − γ23
)
p
2
2
2
> 0, γ12 γ23 + (1 − γ12
)(1 − γ23
)>
which is strictly positive since 1+γ12 > 0, 1−γ23
0 by the previous Lemma, and since we are assuming γ23 ≤ 0, the terms −γ23 (1 + γ12 )
p
2
2
and −γ23 (1 − γ12
)(1 − γ23
) are also positive.
Now suppose instead that γ23 > 0. Since the pair (1, 3) is more similar than
p
2
2
(2, 3) we have γ12 γ23 + (1 − γ12
)(1 − γ23
) > γ23 which holds if and only if
41
√
√
1 + γ12 / 2. The numerator in (6) converges to
q
2
2
2
(1 + γ12 ) 1 − γ23 + γ12 γ23 + (1 − γ12 )(1 − γ23 )
q
2
2
)
− γ23 (1 + γ12 ) − γ23 (1 − γ12
)(1 − γ23
q
2
2
2
> (1 + γ12 ) 1 − γ23
+ γ23 − γ23 − γ23 (1 − γ12
)
)(1 − γ23
√
q
1 + γ12
2
2
2
√
> (1 + γ12 ) 1 − γ23
−
)
)(1 − γ23
(1 − γ12
2
q
p
1
2
2
= (1 + γ12 ) 1 − γ23
− √ (1 + γ12 ) 1 − γ12 1 − γ23
2
q
√
q
1
−
γ
12
2
2
√
= (1 + γ12 ) 1 − γ23
−
1 − γ23
2
p
2
> 0 and the expression in parenthesis is strictly
where (1 + γ12 ) > 0, 1 − γ23
√
√
2
positive since 0 < γ23 < 1 + γ12 / 2 implies γ23
< (1 + γ12 )/2 which implies
r
√
q
1 + γ12
1 − γ12
2
√
1 − γ23 > 1 −
=
.
2
2
γ23 <
Hence the numerator in (6) converges to a strictly positive number.
2
Now consider the denominator in (6). The determinant (1 + 2γ12 γ13 γ23 − γ12
−
2
2
γ13
− γ23
) is positive and tends to zero as γ13 goes to the upper bound γ12 γ23 +
p
2
2
)(1 − γ23
(1 − γ12
) from below. Moreover, it is easy to check that the bounded
2
−
set of utility signal parameters {(γ12 , γ13 , γ23 ) ∈ [−1, 1]3 : (1 + 2γ12 γ13 γ23 − γ12
2
2
) > 0} is entirely contained in the half space {(γ12 , γ13 , γ23 ) ∈ [−1, 1]3 :
− γ23
γ13
(3 + 2γ12 + 2γ13 + 2γ23 ) ≥ 0}. Therefore the expression (3 + 2γ12 + 2γ13 + 2γ23 ) in
the denominator is positive and bounded, so the entire denominator is positive
and goes to zero as desired.
To prove (ii), note that as t → ∞ we have tVar[m(t)] → ΛΛ0 = (γij ) and
E[m(t)] → µ. Since 1 ∼ 2 ∼ 3 we have µ1 = µ2 = µ3 and therefore ρt (i, {1, 2, 3})
tends to ρ̈0 (i, {1, 2, 3}) given in Proposition 7 as t → ∞. In other words, in this
knife-edge case in which the µ parameters of all three alternatives are identical,
the Bayesian Probit Process behaves, in the limit as t goes to infinity, exactly as
the naı̈ve probit behaves at time zero. The result then follows from Propostion 7.
42
Proof of Theorem 5
Let ε > 0 and recall m(t) is the (random) vector of posterior mean beliefs at time
t, which has a joint normal distribution. The probability that alternative 1 is
chosen at time t is equal to the probability that m1 (t) > m2 (t) and m1 (t) > m3 (t).
This happens if and only if the bi-dimensional vector (m2 (t) − m1 (t), m3 (t) −
m1 (t)) has negative coordinates. This vector has a joint normal distribution
with mean given by
E[m2 (t) − m1 (t)] =s2 t µ3 (1 + s2 t + γ12 )(γ13 − γ23 )
2
+ µ1 (−1 − s2 t)(1 + s2 t + γ12 ) + γ13 γ23 + γ23
and
+µ2 1 + γ12 + s2 t(2 + s2 t + γ12 ) − γ13 (γ13 + γ23 ) /
2
2
2
1 + s2 t 1 + s2 t 2 + s2 t − γ12
− γ13
− γ23
+ 2γ12 γ13 γ23
E[m3 (t) − m1 (t)] =s2 t µ2 (1 + s2 t + γ13 )(γ12 − γ23 )
2
+ µ1 (−1 − s2 t)(1 + s2 t + γ13 ) + γ12 γ23 + γ23
+µ3 1 + γ13 + s2 t(2 + s2 t + γ13 ) − γ12 (γ12 + γ23 ) /
2
2
2
1 + s2 t 1 + s2 t 2 + s2 t − γ12
− γ13
− γ23
+ 2γ12 γ13 γ23
The denominator in both expressions can be written as
2
2
2
1 + s2 t 1 + s2 t 2 + s2 t − γ12
− γ13
− γ23
+ 2γ12 γ13 γ23 =
2
2
2
s2 t s2 t 3 + s2 t + 3 − γ12
− γ13
− γ23
+
2
2
2
1 + 2γ12 γ13 γ23 − γ12
− γ13
− γ23
2
2
2
− γ23
which is clearly positive since s, t > 0, γij2 < 1 and 1 + 2γ12 γ13 γ23 − γ12
− γ13
is the positive determinant of the covariance matrix of X(t).
In both numerators, the expression multiplying the coefficient µ3 is positive.
In the first case, note that 1 + γ12 + s2 > 0 and that γ13 > γ23 since the pair
(1, 3) is more similar than the pair (2, 3). In the second case, the expression
multiplying µ3 can be written as
2
[1 − γ12 γ23 ] + 2s2 t + s4 t2 + γ13 (1 + s2 t) − γ12
where each expression in brackets is positive. Therefore for fixed t both coordinates of the mean vector (E[m2 (t) − m1 (t)], E[m3 (t) − m1 (t)]) can be made
arbitrarily negative by taking µ3 negative and sufficiently large in absolute value.
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The covariance matrix Var[m(t)] = ts(t)(ΛΛ0 )−1 s(t)0 does not depend on µ.
Since µ1 = µ2 we have both ρt (1, {1, 2, 3}) and ρt (2, {1, 2, 3}) converging to 1/2
when t goes to infinity. Note that, while increasing the absolute value of the
negative parameter µ3 does not change Var[m(t)] for any t, it decreases both
E[m2 (t) − m1 (t)] and E[m3 (t) − m1 (t)] for every t > 0 and therefore increases
ρt (1, {1, 2, 3}) for every t > 0. Moreover, for fixed t > 0, ρt (1, {1, 2, 3}) can be
made arbitrarily close to 1 by taking µ3 sufficiently negative. This guarantees
that we can have the attraction effect starting arbitrarily early in the random
choice process. Moreover, since E[m2 (t) − m1 (t)] above converges to zero from
below as t goes to infinity, ρt (1, {1, 2, 3}) will converge to 1/2 from above, while
ρt (2, {1, 2, 3}) will converge to 1/2 from below.
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