A selective review of recent characterizations of stochastic choice models using distribution and functional equation techniques

Mathematical

Social Sciences

23 (1992) 5-29

North-Holland

A selective review of recent characterizations of stochastic choice models using distribution and functional equation techniques A.A.J. Marley Deparfment H3A

of Psychology,

IBI.

McGill

University,

1205 Avenue

Docteur

Penfield,

Montreal,

Quebec

Canada

Communjcated by W. Batchelder Received 10 August 1990 Revised 27 June

1991

There are two major classes of stochastic choice function models assume that the choice probabilities of the values models

of an appropriate

assume

propriate

vectors

and

functional

associated

Key words: models.

classes of such stochastic theory

with such characterizations.

to generalized

extreme value models, Stochastic

choice

over the potential

can be represented

scale values over the potential

of broad equation

vector

probabilities

of real-valued

recent characterizations tion

random

that the choice

models that have received extensive study: random can be represented by some deterministic function

models;

techniques

choice models,

is on random

with some comment characterization

options;

choice options.

in this domain,

The emphasis

choice

constant

by some deterministic

and

This paper

illustrating states

and constant

functional

of ap-

summarizes

the use of distribu-

various function

on recent work on generalized theorems;

function

function

equations;

open

problems

models related normal random

models. utility

1. Introduction A task that has received considerable theoretical and empirical study in psychology, economics, and statistics is the selection of the single ‘best’ option from some available set of options. Typical situations involve the selection of: the loudest of some presented set of pure tones; the preferred mode of transportation amongst, say, automobile, bus, and subway; the cause of death from some set of potential causes. Particularly in the psychology and economics literature, it is often the case that such choices (decisions) are stochastic, i.e. ‘repeated’ decisions in the ‘same’ situation will not necessarily lead to identical choices. Such stochastic choice is conceptualized as either being due to variability in the decisions of a given individual and/or variability in the decisions across individuals. Variability within an individual is the case of major interest in psychology and variability across individuals the case of major interest in economics; however, both cases occur in both fields, 016554896/92/$05.00

0

1992-Elsevier

Science

Publishers

B.V. All rights

reserved

6

A. A. J. Ma&y

/ Stochastic

choice models

and my focus in this paper is on models for the choices of a single individual. Given the known variability in repeated choices, many of the models that have been developed for dealing with such preference or choice data are probabilistic, and the purpose of this review is to summarize recent work on certain of these probabilistic (or stochastic) models of choice and preference. There have traditionally been two major classes of such stochastic models for choice and preference (Lute and Suppes, 1965; Colonius, 1984; Hilton, 1985): random function models assume that the choice probabilities can be represented by some deterministic function of the values of an appropriate random vector over the potential choice options; constant function models assume that the choice probabilities can be represented by some deterministic function of appropriate vectors of real-valued scale values over the potential choice options. I use the word function for the random and constant models, rather than the traditional word utility, to emphasize the possibly multidimensional nature of the representations of the choice options. Also I make no attempt in this paper to give adequate general definitions of the two classes of models-in the case of random function because I do not need such generality; and in the case of constant function because, as I later discuss in detail, neither I (nor anyone else in my opinion) has an adequate formal definition of that class, even though we may have an intuitive idea (captured by my earlier verbal definition) of what the definition looks like. Basically, constant function has something to do with the way scale values (or vectors of scale values) are assigned to alternatives, and the consistency of those measures across subsets of options. Clearly, any random function model where the random vector is determined by its compound moments (Kagan, Linnik and Rao, 1973) is a constant function model, albeit one possibly requiring an infinite sequence of real-valued scale values for its specification. Also, there are interesting constant function models that are not random function models-for instance, the generalized Rotondo model (Rotondo, 1986; Marley, 1991a; Section 3 of this paper). Finally, many common probabilistic choice models are both constant and random function (utility) models, the most familiar being random utility models based on multivariate dependent probit (or normal) distributions and those based on multivariate dependent logit (or extreme value) distributions; Lute (1977a, b) gives excellent summaries of theoretical and empirical results on these two classes of models at that date. Significant additional work has been done on the probit and logit models since Lute wrote his reviews, and a major purpose of the present paper is to summarize recent theoretical work on generalized logit models and related constant function models. It would be useful to have a similar review of recent excellent work on the probit model, e.g. that of Hausman and Wise (1978), Iverson (1979, 1987), and Iverson and Sheu (1992), and even an integrative approach that gave the probit and logit models as special cases of a more general class of models. (I later make some remarks on this latter problem.) A major portion of the theoretical and empirical work on stochastic choice models has concerned choice probabilities over sets of choice options where no addi-

A.A. J. Marley

tional

structure

(e.g. dimensional)

/ Stochastic

is assumed

7

choice models

for the options.

In this review,

I also

discuss models that jointly consider choice probabilities and choice reaction times; I also summarize results when the choice options have an explicit multidimensional structure. Since this article is intended as a review, I do not include any proofs (except two short ones as an appendix for illustrative purposes), but refer to the original articles for this material. The major mathematical technique that has been used in developing these theoretical results is functional equations (Aczel, 1966, 1987; Aczel and Dhombres, 1989). In the case of random function models, usually some plausible condition is stated regarding the properties of the underlying random vector, which then leads to a functional equation whose solution is sought, i.e. one develops a characterization theorem for the underlying distributions (Galambos and Kotz, 1978; Galambos, 1981, 1982, 1987). In the case of constant function models, usually some plausible condition on the choice probabilities immediately leads to a functional equation, the solution of which is then sought. In both cases, difficulties often arise with respect to possible empirical tests because of the need to add ‘technical’ (mathematical) assumptions (e.g. solvability, continuity) to the major substantive assumptions of interest. I avoid such issues as much as possible in this review (see Falmagne, 1981, and Marley, 1991a,b, for discussions of these issues). The remainder of the review has the following organization. I begin by discussing models without concern for the ‘dimensional’ structure of the options chosen. First, I present results on random function models for choice probabilities and reaction times, followed by results on constant function models for choice probabilities. I omit reaction time from the constant function models since very little such work has been done; a major exception being Lute (1960). (Of course, since numerous random function models are also constant function models, some of the results on reaction time for the random models could be redescribed in terms of constant models, but then the motivating assumptions would be ‘hidden’.) Next I introduce explicit dimensional structure on the objects of choice, discuss plausible relations between the associated unidimensional and multidimensional choice probabilities,’ and summarize the resulting models. The final technical section briefly applies these results to models of ranking. The presentation is discursive, in the form of presenting a problem, discussing known solutions and any remaining related open problems, and pointing to relevant theorems and techniques in the mathematical literature, especially that on functional equations. I do not explicitly point to uses of distribution and functional equation techniques in developing the results since the techniques are pervasive, and transparent, throughout the text. ’ Phrases such as ‘unidimensional choice probabilities’ (respectively, ‘multidimensional choice probabilities’) should be interpreted as meaning ‘choice probabilities over unidimensional options’ (respectively, ‘choice probabilities over multidimensional options’), i.e. it is the options, not the probabilities, that are uni- or multidimensional. fusion.

The phrasing

used is clearly more compact

and should not lead to con-

8

A. A. J. Marley / Stochastic choice models

2. Basic notation theory)

and results for random function

models

(based on distribution

In this and the next several sections, I assume that there is a finite set R of potential choice options available and that on any given choice opportunity some subset Xc R with 1x122 is presented to a person, who has to select one element of X as ‘best’ according to some (usually specified) criterion. The assumption of finite R considerably simplifies the presentation; Resnick and Roy (1992) extend various of the following results to the case of infinite R, plus they reference relevant earlier work on that case. I assume that choices are stochastic (i.e. the same choice is not necessarily made on repeated presentations of the same set X) and that both the time of choice and the option chosen can be observed. The following notation covers these points. R = {rlr . . . . r,} is a finite set of potential choice options; Xc R with /Xl 2 2 is the currently available choice set; T(X) is a random variable denoting the time at which a choice is made; and C(X) is a random variable denoting the element chosen. For t20, Px(x;t)=Pr[C(X)=xn T(x)>t] is

the

probability

that

option

x

is

chosen

from

X

after

time

t,

and

(Px (x: t) : x E XL R} , or (R, P) for short, is called a joint structure of choice probabilities and reaction times. A joint structure is complete if it is defined for all subsets of R. In this paper, unless otherwise noted, all joint structures are assumed to be complete. The first class of models that I discuss generalizes the classic random utility family to include reaction times. The underlying assumption is that, associated with each option XEX, there is a non-negative random variable t(x) which denotes the time of occurrence of some ‘event’ associated with that option; the element is chosen whose ‘event’ occurs first. I also assume that simultaneous events do not occur.’ Remembering the notation R = {rl, . . . , r,}, we thus have Definition 1. A (complete) joint structure of choice probabilities and reaction times (R, P) is a ‘horse race’ random utility model if there is a &n-negative random vector (t(r,), . . . . t(r,)) such that for all XEX~

Px(x;t)=Pr[t
R with 1x122, for all yEX-{x}].

(I)

Note that with

P(x:X)=Pr[t(x)
’ The

distribution

pp. 136-140) simultaneous

associated

with

for all yEX-

a random

variable

{x}].

is absolufely

confinuous

(Feller,

provided it has a density and no singular component. This condition prevents occurrence of two or more ‘events’ in the ‘horse race’ random utility model (Definition

1966, the 1).

A.A.J.

Marley

{P(x: X): x E X c R} constitutes

/ Stochastic

9

choice models

a system of choice probabilities or a choice struc-

ture (R,P) with P(x:X)=Pr[t(x)=min{t(y),y~X}], i.e. (R, P) satisfies a random utility model (Lute and Suppes, ing the usual max). We are led immediately solution. Problem 1. Given a {Px(x ; t): x E X } on Px(x; t) are necessary utility representation ture {Px(x: ~):xEXC

to the first characterization

1965, with min replac-

problem

and its (partial)

joint structure of choice probabilities and reaction times a fixed set X, what conditions on the (survival) functions and sufficient for the existence of a ‘horse race’ random for that joint structure; similarly, for a complete joint struc-

R}?

Theorem 1 (Marley and Colonius, 1991). (a) For a set of choice probabilities and reaction times {Px(x: t): XE X}

on a fixed set X with 1XI 2 2, there exist independent random variables tX (x), XE X, with unique distributions, such that equation (1) holds provided Px(x ; t), XE X, are positive and absolutely continuous for aN t 2 0. (b) A complete joint structure of choice probabilities and reaction times {Px(x ; t), XE X C R} can be represented by an independent ‘horse race’ random utility model, with unique distributions, if the conditions of (a) hold and

c

YEX

Px(Y;s)

1

-‘d&W

is independent of XC R for all ~10. Proof.

See the appendix.

Related

results

appear

in Bundesen

(1991) and

Vorberg (1991). The condition for the proof of Theorem l(a) to be valid. A more general sufficient condition is that there is a fixed a>0 such that Px (x ; t) > 0 for all x E X if and only if (Y2 t 10, which is perhaps more empirically plausible. Each condition leads to a set of random variables that are nonzero on a common interval of the non-negative reals. Weaker conditions, which lead to random variables that are not all nonzero on the same interval, can be used, but they require more complex proofs. Marley and Colonius (1991) relate the ‘horse race’ random utility model, and the results of Theorem 1, to the competing risks literature. This is done by reinterpreting the ‘event’ associated with each XE X as a cause (e.g. of failure or death), and thus the selection of XE Xthen corresponds to the ‘cause’ associated with x being the first to occur when X contains the set of possible ‘causes’. The formula in Theorem l(b)

Px (x ; t) > 0 for all x E X and all t I 0 is sufficient

A.A.J.

10

Marley

/ Stochastic

choice models

then becomes the negative of the cause-specific (or crude) hazard rate associated with x, i.e. the instantaneous rate of ‘failure’ from ‘cause’ x. Thus the requirement that this quantity be independent of the set X corresponds to independent ‘causes’ of ‘failure’. From this perspective, Theorem l(a) says that the observable effects, i.e. ‘failures’, of any given set of (possibly dependent) ‘causes’ can be modelled by an independent set of ‘causes’. This is seen as a problem in the competing risks literature, and attempts are made to avoid it by assuming that the distributions associated with the ‘real’ causes belong to some specified parametric family. Of course, similar problems would arise in psychological applications of Theorem l(a), but fortunately (when the data are from an experiment) we can vary the set X and hence have the additional constraints of Theorem l(b) available to test for independence. Nonetheless, noting that the above result only characterizes independent ‘horse race’ random utility models, we are led to: Open Problem 1. Characterize (complete) joint structures {Px(x; t): XE XC R} of choice probabilities and reaction times that have a (possibly) dependent ‘horse race’ random utility representation. Before discussing partial solutions to this problem, I summarize the known parallel results and open problems for random utility models (i.e. on choice probabilities only). Falmagne (1978) and Cohen (1980) characterize (possibly dependent) random utility models on complete structures (i.e. choice probabilities are given on all subsets of some set); to my knowledge, no one has characterized independent random utility models in this case; nor is a characterization known for structures of choice probabilities that are not complete. For instance, Marley (1990) summarizes recent work on the latter problem when only the binary choice probabilities are given; there is continuing recent interest in this binary case, including papers by Fishburn (1992), Heyer and Niederee (1992), Gilboa and Monderer (I 992), Monderer (1991), and Suck (1992). Fishburn presents an integrative summary of results and techniques on this problem. In the context of these results, Theorem 1 (above) and the discussion (below) are thus somewhat surprising in that for ‘horse race’ the independent is not (below). I now discuss joint structures x, is independent

random utility models for choice probabilities and reaction times, case is understood (Theorem l), yet the (possibly) dependent case partial solutions to Open Problem 1. Results to date only apply to of choice probabilities and reaction times where the option chosen, of the time of choice, t, i.e. where

Px(x;

t)=Pr[C(X)=x]

Pr[T(X)>t].

I will show next that a very large class of ‘horse race’ random utility models, based on extreme value distributions, satisfies equation (2). First, however, I indicate that together the above two senses of independence-the independent ‘horse

A.A. J. Marley

/ Stochasiic

choice models

I1

race’ random utility model and equation (2)-characterize Lute’s (1959) choice model. Remember, Lute’s choice model holds for a system of choice probabilities {P(x: X): x EXC R} provided P(x: X)#O,

there

is a ratio

scale u on R such that,

provided

1, P(x: X) =

v(x)

cyc,y U(Y).

For simplicity in stating the next result, I assume that all the choice probabilities are nonzero. The result can be generalized when this is not the case by adding a connectivity and a transitivity condition (Lute, 1959, Theorem 4, p. 25). and Colonius, 1991). Consider a (complete) independent ‘horse race’ random utility model (R, P) where for each x E X G R with 1x12 2, Px(x ; t) is positive and absolutely continuous for all t > 0. If the option chosen is independent of the time of choice, then the choice probabilities satisfy Lute’s choice axiom. Theorem 2 (Marley

Related (but not identical) characterizations of the choice model are given by Robertson and Strauss (1981, Theorem 2), who refer to Strauss (1979, Theorem 4) for proof, and by Bundesen (1991) and Vorberg (1991). Given the known limitations of Lute’s choice model when applied to preference (Tversky, 1972a,b), if we wish equation (2) to hold we must study dependent ‘horse race’ random utility models. Thus I now present a class of dependent models that includes all the ‘classical’ choice models based on extreme value distributions, including McFadden’s (1978) generalized extreme value (GEV) class and the transition probabilities of Tversky’s (1972a,b) elimination by aspects (EBA) model, with Lute’s choice model a special case of each. These models all satisfy the following definition with q = -log, which gives Weibull univariate marginals, i.e. for t,>O, i=l , . . ..n. PR(tj)=exp-AR(i). t/‘, with A,(i)>0 and usually with AR(i)=p(i). With R={r ,,..., r,), t;zO, i=l,..., n let PR(f,, .‘.1 t,)=Pr(t(r,)>t,,...,t(r,)>t,), where t is a random

vector

as in Definition

1. Such a PR is a survival function.

2. For a finite set R, a survival function PH is a generalized stable survival function if there is a strictly monotonic decreasing function q and a constant pu>O such that for all o>O, ti>O, i= l,..., n, Definition

. . . . at,)=apqP,(t,,...,t,).

rlMatl, Letting

G&f,, . . . , tn)=vPR(f,,...,fn), this yields G&tr,

. . . . a,)=aPGR(tr,...,

i.e. G, is (generalized)

homogeneous

t,), of degree p.

12

A.A.J.

Marley

/ Stochastic

choicemodels

In order to state the next theorem in its currently most generally known form, I call a survival function PR a strictly monotone transform of a generalized stable survival function QK if there is a strictly monotonic increasing function y with y(O)=O, y(o3)=00 such that for t,>O, i=l,..., n,

PR(t,, *,-, t,> = QK(Y@I), --., dt,,>), i.e. if t (respectively, s) is the random vector associated with PR (respectively, QR), then s;=y(t,) for each i= l,..., n. Note that I should really talk of ‘strictly monotone transforms of the random variables generating the generalized stable survival function’ rather than the more convenient ‘strictly monotone transforms of the survival function’. We then have 1989a; Resnick and Roy, 1992). Any ‘horse race’ random utility model that is generated by a strictly monotone transform of a generalized stable survivalfunction is such that the option chosen is independent of the time of choice. Theorem 3 (Marley,

I stated and proved this result for generalized stable survival functions on finite sets (i.e. without the possibility of a strictly monotone transform), and also demonstrated that the choice probabilities generated by such survival functions are invariant under strictly monotone transforms-my proof is easily adapted to show that strictly monotone transforms of a generalized stable survival function also satisfy independence of the option chosen and the time of choice. Resnick and Roy extend these results to infinite choice sets. I can now formulate Open Problem 2. Is the converse regularity conditions)?

of Theorem

3 true or false (under

reasonable

Robertson and Strauss (1981) give a partial answer, namely that the converse is true if the survival function PR (or some strictly monotone transform of it) belongs to the generalized Thurstone class: for t;z 0, i = 1,. . . , n,

P&l, . . . . t,)=P(u(x,)t,,...,u(x,)t,), where u(x,)LO and P is a survival function that is independent of R. (Their results are stated in terms of quantities ,LL(x,)- f,, which can take on real values, but the proofs can equally well be written in the above multiplicative form on the nonnegative reals.) Unfortunately, Robertson and Strauss’s proof depends heavily on the fact that the generalized Thurstone restriction allows one acecss to PR(tl, . . . , t,,) for all distinct t,, i= 1, . . . , n, which does not appear possible from equation (2) in general. reaction 1991).

Note that one can ‘trivialize’ Open Problem 2 by assuming that rank order times (of Definition 2) are observable (see Bundesen, 1991, and Vorberg,

A.A.J.

Marley

/ Stochastic

choice models

13

Additional interesting properties of ‘horse race’ random utility models generated by generalized stable survival functions are given in Marley (1989a). For instance, in special cases, the choice probabilities generated by such models satisfy Yellott’s (1977) k-copies condition: for each set X={x,,...,x,} c R with q22, let X,= lx,,, . . ..X.k, . . . . x41, . . . . xqk}, where for each i, x0, j = 1, . . . , k, are k identical copies of Xi. Then the choice structure (R,P) (with the corresponding satisfies the k-copies condition provided that for every positive

k-fold structures) integer k and each

X,EX, P(x;:X)=

f P(x;] j=l

:x/o.

Yellott shows that any independent (generalized) Thurstone random utility model that satisfies the k-copies condition on all subsets of a set R with IRI z 3 generates a structure of choice probabilities (R,P) that satisfies Lute’s choice model. It follows from the discussion in Marley (1989a) that any structure (R,P) generated by a generalized stable survival function with q = -log also satisfies the k-copies condition. However, as discussed in Marley’s papers, technical difficulties (owing to the lack of independence and/or absolute continuity of the survival function) prevented the proof of a suitable converse, i.e. that any ‘horse race’ random utility model that satisfies the k-copies condition is generated by a generalized stable survival function with q = -log. I now summarize some ideas on how the k-copies condition might lead to a rapprochement between probit (normal) and logit (extreme value) random function models; these ideas are quite preliminary, but their inclusion appears warranted in the context of my presenting open problems. Consider the following reinterpretation of the k-copies condition in the context of random utility models. For each option, the person takes k independent samples of the associated random variables and notes the largest value (note well: here and in the remainder of this aside I assume the standard maximum based random utility model). The person then compares these largest values across options, and selects as ‘best’ the option that has the overall largest value, i.e. a max-max process. Then a variant of the k-copies condition says that the resulting choice probabilities are independent of the sample size k. Now consider an alternative process where instead of basing judgments on the largest of the k samples the person bases judgments on the sum of the k samples, i.e. chooses as ‘best’ the option that has the largest sum over the samples. (This process could be made observable by actually making k copies of each choice option available to the person, who is then asked to choose the ‘best’ of those k-element sets on the assumption that he or she will receive all k copies of the selected option.) One can then ask: Which (independent) random utility model has choice probabilities that are independent of the sample size under this latter decision rule? It is not the independent probit (normal) model (see below), but note that we have broadened the perspective by considering a different decision rule (sum rather than

14

A.A. J. Marley

max). One is then immediately

/ Stochastic

choice models

led to think of considering

the general

class of rules

based on generalized means: (j,

,tsx;)lyp,

where tj(Xi) is thejth of the k samples of the random variable associated with option xi and p is a real number. Clearly, p = 1 gives the sum rule, p = CXJgives the max rule, and other values of p give other rules. One can then explore what distributions (if any) give invariance of the choice probabilities for these various combination rules. Returning to the sum rule, this does not give choice probabilities that are invariant under sample size for the normal distribution (see below). However, one may nonetheless be able to characterize the independent probit (normal) model from this rule. I only present the idea for binary choices; to solve the problem it may be necessary to go to larger set sizes. Let a (respectively, b) be an arbitrary choice option with associated random utility that is independent normal with mean u(a) (respectively, u(b)) and variance a2(a) (respectively, a2(b)). Then (e.g. Iverson, 1979)

=@

L

u(a) - u(b)

(a2(a) + 02(b))l”

1’

where @ is the cumulative normal distribution. Now consider the special case a2(a) =02(b)=02. Let a(k) (respectively, b(k)) denote a set containing k identical copies of a (respectively, 6), and assume that the person’s choice between a(k) and b(k) is based on the sum of k independent samples of the random variables associated with a,b, i.e. a(k) is chosen over b(k) if the associated sum for a is greater than that for b. Then clearly the random variable for a(k) (respectively, b(k)) is normal, with mean ku(a) (respectively, ku(b)) and variance ka2. Thus

p(aW, b(k)) = @ which clearly depends model is not invariant parison of a, b),

on k, in support of my statement above that the normal under the sum rule. Nonetheless, with k= 1 (i.e. the com-

p(a,b)=@

[ u(a~~(b)]

@-‘p(a(k),

b(k)) = k”2@-‘p(a,

9

and so

which implies

(but is not implied

b),

by) that for options

a, 6, c, d,

A.A. J. Marley

@-‘aGw,

b(k))

/ Stochastic

15

choice models

@- ‘P(U, 6)

@-‘p(c(k),d(k)) = @-‘p(c,d) ’ which in turn

implies

(but is not implied

P@(k), W))

by)

>P(C(k), d(k)) ++P(Q, 6) >iD(c, 4.

One can then apply techniques similar to those of Iverson and Sheu (1992) to various such conditions to attempt to find a useful characterization of the probit model, and/or other models with stable or semi-stable distributions (Iverson and Sheu, 1992), and/or other models motivated by the above conditions. In summary, by replacing the ‘maximum’ rule in Yellott’s k-copies condition by a generalized mean, we are led to a more general characterization problem which may have the logit (Lute’s choice) model and the probit (normal) model as solutions in special cases, with other distributions (models) in other cases.

3. Additional functional

notation equations)

and

results

for

constant

function

models

(based

on

As indicated in the Introduction, I do not have a satisfactory definition of the class of constant function models. Keep this in mind whilst reading the following and I will refer back to the definitional problem at various points. For r;ER and XC R, let

A large class of stochastic

notation

where the -

P(x:X)=

means

choice models

has the form:

for R = {r,, . . . , r,,},

that

FIX : XX (rl 1,.. . , xx 0-J

c,,xFty:x~(rlX...,xx(rn)

for some specific function F. For instance, the following example, which belongs to McFadden’s (1978) generalized extreme value (GEV) class, has the above form (and satisfies a random utility model that can be generated by a generalized stable survival functionDefinition 2). Examplel. necessarily

LetR={l,..., n} denote the master disjoint subsets of R with $,

R,= {l,...,n},

set. Let B, c R, k = 1, . . . , m, be not

16

A.A. J. Marley

and let a,>O, Osakr i= l,..., n, such that

/ Sfochasiic

choice models

1, k= 1, . . . . n. Also, we have non-negative

scale values

u(i),

(with u(0) = 0). Returning to the general representation, it is desirable to characterize system of choice probabilities is of the form: for XEX~ R = {rl, . . . . r,},

when a

for some function F. One might think that such a characterization has been given by simple scalability (Suppes, Krantz, Lute and Tversky, 1989), which yields representations of the form:

where Fj,, is a function that might depend on 1x1 and w is a real-valued function. However, it is just the possible dependence of F on 1x1 that I consider unsatisfactory for my idea of what would be meant by a ‘consistent’ representation. For instance, suppose that R = {x, y, z} and that there exists a non-negative function w on R such that for arbitrary a, b E R,

P(a: {a,b})=

w(a) w(a) + w(a)

and W:

I~Y,z>)=

w2(a) w2(x) + w2( y) + w2(2) *

Then (R, P) satisfies simple scalability but is not ‘consistent’ in the strong sense that I wish since the ‘same’ combination function F is not applied to the scale w on all subsets

of R. Thus,

we have the vaguely

stated

Open Problem 3. Give a sensible definition of a ‘consistent’ structure babilities (R, P) and characterize such consistent structures.

of choice pro-

Remember that Falmagne (1978) has solved the parallel problem for random utility (function) models. It has long been known that such random utility models are determined by the values of the probabilities of the rank orders on the set R, which gives a ‘handle’ on the problem. I do not currently see how to get a similar ‘handle’ on Open Problem 3. Narens (1991) gives a qualitative characterization of the choice model, which is a special case of the above representation; however, Naren’s approach is so heavily dependent on the form of the choice model-as a (generalized) conditional additive probability measure-that his techniques do not obviously generalize to other representations.

A.A. J. Marley / Stochastic choice models

17

I now present two further examples of the ‘consistent’ class, which are interesting in their own right but which also lead naturally into the discussion of models defined on multidimensional choice options. Example 2. Generalized

Rotondo

P(x:X)-

n

(Rotondo,

1986; Marley,

1991a). For X~XC

R,

V(X,y)“8(x’,

YEX-_(XI

where q on Rx R and 6’ on R are non-negative

functions.

This model generalizes Lute’s choice model and can handle various of the ‘classical’ counterexamples to that model (Debreu, 1960). Its conceptual basis is that for each x, y E R there is a measure q of the binary advantage of x over y such that the probability of selecting a particular element x from a subset X is proportional to the (geometric mean) advantage of x vis-a-vis all the other elements of X. The major reasons that this model is context dependent (in contrast to Lute’s choice model which satisfies the independence of irrelevant alternatives condition) are that the exponent e(X) can depend on the current choice set X and that the function q can reflect dependencies between the utilities of pairs of alternatives. Example

3. Ideal based (e.g. Bossuyt,

1990). For XEX~

R,

P(x : X) - r/(x,x*), where q is a non-negative function on R x R and X* is a referent, ment of X or some ‘ideal’ element that may not be in X.

e.g. the ‘best’ ele-

The conceptual basis here is similar to that in Example 2, except that in this model the element x is only compared for binary advantage to the relevant referent for the available choice set. Neither Example 2 nor 3 is (in general) a random utility model, although each is a (binary) random advantage mode/ (Marley, 1991a), a generalization of the random utility class. Problem

2. Characterize

the generalized

Rotondo

model.

To keep technicalities to a minimum in the presentation of the solution to this problem, I restrict attention to nonzero choice systems (R,P), i.e. for each XEX~ R, P(x:X)#O. Now, given a nonzero choice structure (R, P) and x,yeX~R, let P(x:X)

g(x,Y)=

P(y:X)’

i.e. Lf;(x, y) is the likelihood

or odds ratio for choosing

x versus y in context

X.

18

A.A.J.

Marley / Stochastic choice models

4 (Marley, 1991a). Given a nonzero choice system (R, P) and a nonnegative function b’ on R, the following three conditions are equivalent. For all x,y~SflT with S,TCR andx,yEXCR, Theorem

(3) There exists a non-negative function P(x:X)-

q on R x R such that

n Y/(x,y)“O@? YEXm{XI

Open Problem 4. Characterize the generalized Rotondo model without the function 0 is known. (See Marley, 1991a, for partial solutions.)

presuming

Would one expect e(X) in the above model to depend on individual elements of X? This is unclear, but it is certainly the case that one might wish to look at cases other than B(X) = 1x1 -1 (which is the value Rotondo selected). For instance, will this model ever satisfy the k-copies condition whereby choice probabilities are suitably invariant when the choice set is expanded by k copies of each choice element? Yes, it will if e(x) = JXJ(assuming q(x,x) = 1 for all distinct copies of xE X), but it will not if e(X) = /Xl - 1 (Marley, 1991a). In the next section I use the generalized Rotondo model to motivate the study of choice structures on multidimensional options.

4. Relations

between univariate

and multivariate

representations

In this section X is a fixed finite set and we are only concerned with the choice probabilities on that set X. I now expand the notation for a system of choice probabilities (X, P) to explicitly include the dimensional representation of the choice options (probabilities). So let i= 1, . . . , m be the dimensions, Pj(x: X) the probability of selecting x as ‘best’ on dimension i, and P(x: X) the probability of selecting x as ‘best’ overall. For notational completeness, I should now write (X, P, P,, . . . , P,) for such a system of multidimensional and unidimensional choice probabilities. However, for notational simplicity, I continue to write (X, P) with the understanding that P,, . . . , P,,, are also defined, and I refer to such (X, P), (X, Q), etc. on a fixed set X as a class of choice probabilities on the set X. I now investigate plausible relations between the component choice probabilities Pi, i= 1, . . . . m, and the overall choice probabilities P using a special case of the generalized Rotondo model to motivate one of the main representations that I finally characterize. This special case assumes that the univariate and multivariate choice probabilities all satisfy Rotondo’s model, with the implied ‘natural’ relation be-

19

A.A. J. Marley / Stochastic choice models

tween the unidimensional for dimensions

and the multidimensional

advantage

measures.

That is,

i = 1, . . . , m,

P&x:X)-

rnx)

rlik_YP’~?

and n

P(x:X)-

q(_&Y) 14x1 -n,

YE,+_(X)

where czj20.

rlx, Y) = ii Vi(X, Y)“‘, i=l

Note that these relations P(x:X)-

imply

that

fi P;(x:Xp, i=l

which is a general relation between the component choice probabilities Pi, i= 1, . . . , m, and the overall choice probabilities P, i.e. it does not involve the original generalized Rotondo model that led to it. Note that the special case of (Y;= 1, i=l , . . . , m, can be motivated by the following process. When instructed to select the ‘best’ option overall, the person selects the ‘best’ option on each dimension; if the same option is chosen (probabilistically) as ‘best’ on every dimension, that option is selected as the overall ‘best’ option; otherwise, the person resamples over all the dimensions until such an option is selected. I now present the solution of Problem 3. Characterize the above relation (with general pi, i= 1, . . . . m, that may depend on the set X) between univariate and multivariate choice probabilities. The solution assume

nonzero

is based on the following

assumptions.

For simplicity

I continue

to

choice systems.

Assumption Ll (likelihood independence property). such that for all nonzero systems of choice probabilities

There exists a function F, (X, P) and for each x, y E X,

L; (~9Y) = Fx Wf;’(x, YX.. . , Lp;m(x, Y)I . Clearly, the interpretation of this condition is that it does not matter, in calculating likelihood ratios, whether one first calculates them on the individual dimensions and then combines these ratios over dimensions, or simply calculates likelihood ratios of the choice probabilities on the multidimensional set. This is a plausible independence condition, similar in motivation to that which gives Lute’s choice model on the component choice probabilities. The condition effectively constrains the solution to the earlier product form when

20

A.A.J.

Marley

/ Stochastic

models

choice

IX/ ~2 (below), but when 1x1 =2 there is a broad class of solutions. In fact when 1x1 =2 the above condition reduces to the following (Marley, 1991b). For i=l,..., m, let pi(x,y)=Pj(x: {x;y}) and let a;=p;(X,y)/pi(r,X). Then, with Ll that F standing for Ftx,YJ for notational simplicity, it follows from Assumption

This reciprocal property (Aczel, tions) solutions of the form:

1984) has (under

reasonable

mathematical

condi-

F@b...,a,)=exp where wk and W1 are odd functions (a function g is odd if g(-t) = -g(t)). Turning to the case 1x1~3, I need some regularity conditions (see Marley, 1991b, for a discussion of the reasonableness of these conditions). Assumption L2. For any m-dimensional positive real vectors (rl, . . . , rm), (s,, . . . , sm), provided 1x1 2 3 it is possible to select a system of nonzero choice probabilities (X,P) and x, y,z~X such that for i= 1, . . . . m,

L;( y,z) = s,.

L$(X,Y)= rj,

Assumption L3 (dominance principle). For structures of nonzero choice abilities (X, Pj), (X, Qi), i= 1, . . . , m, and (X, P), (X, Q), and for x, y E X, if L!j(x, y)SL$(x,

y),

prob-

for every i= 1, . . . . m,

then J$ (x, Y) 5 La (x9Y). The general Theorem

solution

5 (Marley,

of Problem

1991b).

3 is then given by:

If a class of nonzero

finite

choice

set X with /XI L 3 satisfies Assumptions Ll-L3, constants wx(i), i=l,...,m, such thatforxEX, P(x:X)-

probabilities

(X,P)

on a

then there exist non-negative

fi P;(x:x)w+ i=l

Proof.

See the appendix.

I call the combination rule of Theorem 5 weighted geometric aggregation (Marley, 1991b). Note that wx(i) is independent of X if the function Fx in Assumption Ll is independent of X. I noted earlier that if all the unidimensional choice probabilities satisfy the

Marley / Stochastic choice models

A.A.J.

generalized theorem,

Rotondo

model

then the overall

and the combination

choice probabilities

rule is as given

21

by the above

also satisfy the generalized

Rotondo

model provided wx(i) is independent of X, i.e. the generalized Rotondo model is in that case ‘closed’ under the combination rule of Theorem 5. We are thus led quite naturally to Open Problem 5. Characterize the most general weighted geometric aggregation.

choice model that is ‘closed’ under

The functional equation approach of Appendix B of Marley (1991a) can probably be generalized to solve this problem. Also, a solution to Open Problem 3 would probably be helpful in formulating and solving the general version of Open Problem 5. I continue the discussion of Theorem 5 with some comments regarding its application to Lute’s choice model, which is a special case of the generalized Rotondo model (Marley, 1991a). In particular, when the choice model holds on each dimension, i.e. there are ratio scales u,, i = 1, . . . , m, such that for each XE Xc R, P;(x : X) - u;(x), and the conditions

of Theorem

P(x : X) - ;

5 also hold,

then

u;(xy~(?

r=l Thus, when wx(i) is independent dent of X), we obtain:

of X (i.e. when F, in Assumption

Ll is indepen-

P(x:X)-o(x), where u(x) = fi u;(x)W(i), i=l

i.e. the choice model also holds on (R,P). Thus, weighted geometric aggregation is one ‘correct’ way to combine choice probabilities satisfying Lute’s choice model. Unfortunately, the ‘natural’ way of combining unidimensional random utility representations of the choice model to give multidimensional representations also satisfying the choice model does not yield representations that satisfy weighted geometric aggregation. For instance, the following is the general independent random utility representation of the choice model (Lute and Suppes, 1965; Yellott, 1977) based on choosing the option with the curent minimum value: there exist strictly decreasing functions g, , i = 1, . . . , m, on the non-negative reals with g,(O) = 1, g;(m) = 0, such that on dimension i there are independent random variables t;(x), XER, with Pr(t,(x) > t) =gj(t)u’(x).

22

A.A.J.

Marley

Now if the random variables XER, t(x)= min t;(x),

/ Stochastic

representing

choice models

the overall

choices

have the form:

for

15ism

and g;=g

for all i= l,...,m, Pr(t(x)

then

> t) = g(t)U(X),

where u(x)=

t Uj(X). i=l

Thus, under these conditions, the overall choice probabilities also satisfy the choice model, but the overall scale IJ is a sum (equivalently, arithmetic mean), not a geometric mean as required by weighted geometric aggregation. One can weaken the above conditions by allowing the random variables t; to be dependent on i (and each other). There are two distinct ways to formulate such dependence. One can assume there is a function G such that for each XE R, t(x) = G [tl (x), . . . , t,(x)1

(so G has to be such that t(x) is a random hl(t)=Pr(t,(x)
XER,

variable),

i=l,...,

or one can let

m,

and F(t) and assume

= Pr(t(x)l

t),

that there is a function

H such that

hx(t) = H(h;(t), . . . , h;(t)) (where now H has to be such that hX is a cumulative distribution function, i.e. it is related to an n-copula-see Schweizer and Sklar, 1983, and Marshall and Olkin, 1988). Nonetheless, it is not obvious whether or not such multidimensional ‘horse race’ random utility representations will ever give Lute’s choice model representations on the dimensions and overall that also satisfy weighted geometric aggregation. In fact, Alsina (1989) shows in a similar situation that aggregation of random variables and of cumulative distributions (as just presented) are in general incompatible; it is quite possible that each of these is in turn incompatible with aggregation at the level of choice probabilities (of the kind given in weighted geometric aggregation). See Marley (1991a,b,c) for further discussion of these issues, and Marshall and Olkin (1988) for general representations of multivariate distributions via functional combination, i.e. H above, of associated univariate marginals. Theorem 5 gives a weighted geometric mean relationship between the component and the overall choice probabilities. Are there any other ‘natural’ combination rules? In what follows I summarize the characterization of arithmetic mean combination rules (under which the choice model is not invariant).

A.A. J. Marley / Stochastic choice models

Assumption

Ml (simple marginalization

that for all systems

property).

of choice probabilities

P(x:X)=F,[x,P,(x:X)

23

There exists a function

F, such

(X, P) and for each XEX,

,..., P,(x:X)].

In this case, when 1x1 = 2 we again obtain with a;=p,(x,y),

a functional

equation.

With Fdenoting

Fix,,) it has the form: F(x,a,,

. . . . a,)+F(y,

l-a, ,..., l-a,)=

1.

This functional equation has a broad class of solutions (summarized below-1 have not seen this particular function in the literature). When 1x1~ 3, with regularity conditions similar to those discussed for the likelihood independence principle, the solutions satisfying the simple marginalization property are of the form (Marley, 1991b): for each XEX,

where the weights wx(i) E [-1, l] and y/(. :X) is a fixed probability measure for each X. The solutions when 1x1 =2 generalize this form by having a function q in front of the whole right-hand sum, and a transform function r,ui in front of P;(x : X) for each i with q(a) + ~(1 - a) = w;(a) + I,v;(1 - a) = 1.

5. Ranking

and subset selection

The above techniques and results can be easily extended to cases where the person must rank order a set of options according to some criteria, or choose some ‘acceptable’ subset of the available options (Marley, 1991~). A particularly interesting application that I have studied is in the area of probabilistic social choice. I will not develop that, and ,other, applications in detail here since they follow quite immediately from the theorems and representations already presented; in fact, given the similarity of these problems-for choice, subset selection, and ranking-it is probably worthwhile in the future to develop a single general notation with each of the above then a special case. Here I simply state a ranking version of the generalized Rotondo model and mention two open problems associated with that model. Consider a finite set R and ‘dimensions’ (voters) i, i= 1, . . . , m. For each Xc R distribution & over the rank with IX/ 22 and i, i= 1, .._, m, there is a probability orders of X, and there exists (or is to be developed) an overall probability distribution r,, Xc R, over the rank orders of X. Let Q = er . . . Q, be an arbitrary rank order of Xc R (suppressing notation for the dependence on X for simplicity). Note the following set of conditions and result.

24

A.A. J. Marley /

Ranking version of the generalized r;l (@I-

II

Stochastic choice models

Rotondo

model on the component

rank orders:

ri(@h@J)“‘x’*

I sh
where 0 is a non-negative

function

with e(X) = 1 when

1x1 = 2, and

Consistency:

where

ry is a non-negative

These representations

Ranking

function.

imply

version of the generalized r,Y(@I-

n

Rotondo

model on the overall rank orders:

r(@h@j)B’X’,

ISh
where r(@h@j)-

fi r=l

ri(@h@j)v/(i).

Thus we have a ranking model that is ‘closed’ under the above consistency condition. [One could extend this representation by allowing w(i) to depend on X (as in previous results).] We then have (paralleling Open Problem 5)

Open Problem 6. Characterize the most under the above consistency condition. And

general

ranking

model

that

is ‘closed’

finally

Open Problem 7. Characterize ranking.

the above generalized

Rotondo

model on choice and

The given representation of the ranking probabilities generalizes that in Marley’s (1968) reversible ranking model, but the given representation of the choice probabilities is not a generalization of the parallel representation in that model. In fact, there are numerous other possible relations between choice and ranking probabilities, as reviewed in Critchlow, Fligner and Verducci (1991) and discussed in Marley (1991~). Few of these have as yet been formalized via characterization theorems.

A.A. J. Marley / Stochastic choice models

25

6. Summary I have presented a selected review of recent characterizations of stochastic choice models. It is clear that, although significant progress has been made in characterizing and generalizing various such models, many open problems remain. For instance, the present paper does not discuss in detail recent results concerning probit (normal) models and their generalizations, nor has any serious attempt been made to present and/or characterize a more general class of models that includes the probit (normal) and logit (extreme value) models as special cases. No constant function models were presented for reaction time-this is partly because most of the models presented satisfy independence of the option chosen and the time of choice, thus separating the representation of the choice probabilities from that of the reaction times, but also partly because we do not yet have an adequate definition of the constant function class on choice probabilities and thus even less so on reaction times. Marley (1992) extends many of the results in this paper to identification tasks (Nosofsky, 1985; Ennis and Ashby, 1991), where now the structures of interest involve the conditional probabilities P(i 1j) of making response i when stimulus j is presented. Such generalizations, especially of Theorem 5, yield models similar to those discussed by Nosofsky (1990), although some of his models do not satisfy that theorem (or the alternative additive version). Thus, further work is warranted to characterize the models that he and others (e.g. Ennis and Ashby, 1991) have studied; some of those models appear to possibly have forms compatible with the multidimensional random utility models mentioned in passing in Section 4. In summary, I have illustrated the fruitfulness of using characterization theorems in the development and integration of stochastic choice models, and the importance of functional equations in such research. Acknowledgments This work was supported in part by the Natural Science and Engineering Research Council of Canada. The paper is based on an Invited Address at the Twenty First Annual Meeting of the Society for Mathematical Psychology, Northwestern University, July 1988, and a Colloquium at the Irvine Research Unit in Mathematical Behavioral Sciences, January 1990. I thank Hans Colonius, Geoff Iverson, and John Rotondo for comments on earlier drafts. I am also grateful to W. Batchelder, the communicator, and the anonymous referees, for their constructive comments.

Appendix:

Selected proofs

Proof of Theorem 1. (The following proof is adapted from Berman, 1963, and Marley and Colonius, 1991.) We wish to find, for given Xc R with /XI 22, (nonnegative) random variables tX(x), XE X, such that: for XE X and for t L 0,

26

A.A. J. Marley / Stochastic choice models

PX(x;t)=Pr[t
tX exists, let G:(t)

(Al)

= Pr [tX(x) > t], and note that

c

Pr[TW)>tl=

for all YEX-{x}].

n G;(t),

b(v;t)=

(A3

YEX

YEX

each term being the probability that T(X) exceeds t. Inserting the right-hand side of (A2) in the right-hand side of (Al), where the right-hand side of (Al) equals

‘m n ! I YE,-@) %

G,x(d d[l- G,x(dl,

we obtain: Px(x;

t)= -

=-

!‘mIyFxPx(~; $1dllog G,XNl.

t Conversion

with (unique,

I

to a differential

equation

dPx(x;$=

Px(Y;~

proper G:(t)

dG,x(s)

R%)l-

‘, ,Fx G;(s).

c I YEX

or improper)

= exp il

yields:

1

W&(4),

(A3)

solution

‘1 y~xf’xbvI]-’ [

dP,(x;dj

.

(A4)

(Remember, we are assuming that Px (x ; t), x E X, are positive and absolutely continuous for all tr0.) Now suppose {H1?, XEX) is a second solution. Then it also satisfies (A3) for all S, and therefore d{logG,X(s)}=d{logH,X(s)J, i.e. log G,X (s) = log H,“(s) + C, so G,X(s)=AH,X(s). But we require that for s = 0, G:(s) = H,” (s) = 1, so A = 1. We can now define (proper or improper) random variables non-negative reals with, for t 20,

tX(x),

XEX,

on the

Pr [tX (x) > t] = G,“(t), which gives tX as the desired ‘horse race’ random utility {Px(x; t), XE X}. Clearly, to have a common random utility we require that tX(x) be independent of X {Px(x; 0, XEXLR}, that the right-hand expression of (A4) be independent of X, so

representation of representation of for all Xc R, i.e. in particular

27

A. A. J. Marley / Stochastic choice models

c

Px(Y;s)

YEX is independent

1

-‘dP,(x;s). q

of X for all ~10.

Proof of Theorem 5. (The following proof is adapted from Genest, Weerahandis and Zidek, 1984, and Marley, 1991b.) Assuming /X/ 23, let Y and s be positive real vectors and x,y,z~ R having the properties in Assumption L2, i.e. Q(x,y)=r;, Also,

L$(y,z)=s;. of L$,

from the definition

L;(w)L~(Y,z)=L~(x,z), i.e. using Assumption

Ll with the above,

F,(r)F,(s)

= F,(r.

we have

s).

Now F, is defined for all positive real vectors (Assumption L2), and is monotonic increasing (Assumption L3), so we can use a multivariate extension of Aczel (1966, Theorem 3, p.41) to conclude that F,(r)

= fi

r,?(?

i=l

with w,(i)rO,

m. The result is immediate.

i=l,...,

0

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