A characterization theorem for random utility variables

JOURNAL

OF MATHEMATICAL

23, 184-189 (1981)

PSYCHOLOGY

Note A Characterization

Theorem

for Random

Utility Variables

C. A. ROBERTSON AND D. J. STRAUSS University

of California,

Riverside

In this note we investigate the condition that the distribution of the maximum of a set of random variables does not depend on which variable attains the maximum. This problem arises in random utility theory. When the random variables are independent, the property implies that all the marginal distributions must be Double Exponential (with distribution function exp(-e-‘) in standard form). When dependence is allowed the property characterizes a much broader class consisting of arbitrary functions of arbitrary homogeneous functions of the variables emX’, a result stated without proof in D. J. Strauss (Journal of Mathematical Psychology, 1979, 20, 35-52). These are the distributions such that the maximum has the same distribution (apart from a location shift) as the marginals, provided the marginals are the same.

1. INTRODUCTION Let U, , U, ,..., U, be continuous random variables whose marginal distributions are the same except for location shifts. Let Y be the maximum of these variables. We examine the conditions under which the distribution of Y is independent of which Ui attains the maximum. The problem can arise in random utility theory, where it is supposed that for each of n choice alternatives there is a random variable Ui such that the probability of i being chosen from a subset A is P(U, = maxjEA Uj); it is assumed that P(Ui = Uj) = 0 for all i f j. In practice, attention has been focussed on Thurstone type models (Thurstone, 1927), where Ui = m, +X,, the m’s being constants corresponding to the utilities and the X’s independent with a common distribution. We shall assume throughout that all our distribution functions have nonvanishing derivatives (on IR or IF?“, as appropriate).

2. A CHARACTERIZATION

Let

THEOREM

F(x,, x2 ,..., x, ) be the joint distribution of the X’s, and let Y be as before. Let Z be the maximizing value of i; that is, max,,ij Y = m, + X,. We assume that Z is unique with probability 1. Recall that a function 184 0022-2496/E l/O20 184-06%02.00/O Copyright All rights

0 1981 by Academic Press, Inc. of reproduction in any form reserved.

THEOREMS

FOR

RANDOM

UTILITY

VARIABLES

185

H(y, , y2 ,..., y,) is homogeneous of degree d iff H(ky, , ky, ,..., ky,) = kWY,t Y2 ,...9 y,). The following results were stated without proof as Theorem 5 of Strauss (1979), in which they are attributed to Robertson and Strauss (1978); this work was an earlier draft version of the present paper. THEOREM 1. (i) Y is independent of I, for all (m, ,..., m,), I@ F(x, ,..., x,,) = homogeneous function and 4 is an @(H(eCX1,..., em’ n)}, where H is an arbitrary arbitrary scalar function (subject to F being a proper distribution function).

(ii) If the {Xi} h ave a common marginal distribution then Y has this same distribution (apart from a location shift) lr F = #{H(e-“I,..., eeXn)} as before, provided that H(0 ,..., 0, x, 0 ,..., 0) is the same whichever argument is non zero. Proof of (i)

andI=i)/P(I=i)

P(Y
Fi(Y - ml,..., Y -m,) = I’?‘, Fi(t - m, ,..., t - m,) dt *

Here, and subsequently, we use subscripts such as i, j on functions to denote partial differentiation with respect to the ith, jth arguments. Thus the condition for Y to be independent of I is that Fi(X - m,,

x

- m2,..., x - m,) = CFj(x - m, , x - m2 ,..., x - m,),

where C may depend on the m, but not on x. Differentiation ?’ Fik(x-m,,x-m

2,-.,x

- m,) = C 2

with respect to x yields

Fjk(x - m, ,. x - m2 ,..., x - m,)

k=l

k=l

and elimination

of C between these equations gives

Fj 2

F, = Fi i

k=l

Fjk ; for convenience we have dropped

k=l

the argument of the functions, which is clearly (x - m,, x - m2,..., x - m,) for all. We now transform to variables yi = exp(-xi + m,); the subscripts here do not of course denote differentiation. It follows that a/3x, = -yi(3/ayi), and our condition becomes Fjyiyj

Fi + Ck

YkF

lk) =Fiyjyi

(Fj+TykFjk)v

reducing to Fj Ck ykF, = Fi Ck y,Fjk, where again we may suppress the arguments; differentiations are now with respect to the y variables.

186

ROBERTSON

Defining a function G to be Ck ykFk condition becomes

AND

STRAUSS

and noting that Gi = Fi + zk y,Fik,

our

Fj Gi = Fi Gj for all pairs (i, j).

But this asserts that all the Jacobians a(F, G)/a( yi, yj) vanish, and hence we find that our condition is satisfied if and only if F and G are functionally related. Hence Ck y,F, = d(F), for an arbitrary scalar function #. We now show that by a suitable transformation of F we may reduce this equation to the simpler form 1 y, Hk = H.

Indeed, if we take F = v(H), satisfies

it is easy to see that provided the scalar function w

our two partial differential equations are equivalent, and hence that for arbitrary $ we can find a suitable w. Now Eq. (1) is simply Euler’s equation for homogeneous functions of order 1; and since all solutions of this equation are homogeneous functions, we may summarize by saying that the only possible solutions satisfying the independence condition are distribution functions which are arbitrary functions of arbitrary homogeneous functions of variables eCxi. We now verify that F(x, , xz ,..., x,,) = v{H(eWX1, epx2,..., CXn)}, for arbitrary w (subject, of course, to the condition that F is a distribution and homogeneous H, does in fact satisfy the condition. We find F,(x - m,, x - m, ,..., x - m,) = -e-X+miHi(e-Xt

mt, emxtmz ,..., e-“+““)

function)

dH,

Now if H is homogeneous of order a, its partial derivatives Hi are homogeneous of order a - 1 (since differentiation of C y/H] = aH with respect to y, leads at once to G yiHrj = (a - 1) H,), and so Fi(x - m,, x - m, ,..., x - m,) = -e-x’“‘e-‘“-“XH,(eml,

dv em*,..., em,) dH

whence Fi(x - m,, x - m, ,..., x - m,) emiHl(eml, em2,..., em”) Fj(x - m, , x - m, ,..., x - m,) = e”‘JHj(eml, em=,...,em.) ’

which is independent of x, as required.

THEOREMS

FOR

RANDOM

UTILITY

187

VARIABLES

Proof of (ii)

If Y has the same distribution

as each of the Xi (apart from location shifts) then

F(x - m, ,...I x - m,) = F&x - c(m)),

where c(m, ,..., m,) is independent of x and F, is the common marginal distribution. Thus Fi = F~(x - c(m)) c,(m), and Fi

If,

I;;

Fi dx = JE, Fj dX ’

(where subscripts i, j denote partial differentials), which states that the distribution of the maximum does not depend on the i attaining the maximum. This leads again to the family F(x, ,..., x,,) = y{H(e-“I ,..., eexn)). For the converse, we must show that if w 1,..., xJ = ${H(e-‘I ,..., eexn )} then indeed, for appropriate choice of c(m), we have F(x - m, ,..., x - m,) = F(x - c(m), co,..., co)

In fact, the left member of (2) is t,u{e-*“H(e”‘,..., vW(e-

(2)

em”)}; the right member is

(x-c(m)), 0,..., 0)} = ay{e-“Xe”C’m’H(1, 0 ,..., 0)}

and so (2) is satisfied provided that c(m) = $ ln(H(e”‘,...,

emn)/H(1, 0 ,..., O)}.

1

Strauss (1979) gives examples of the class of functions F = #[HI. As well as the double exponential distribution, it includes distributions such as F(x,, x2, x3) = exp[-{(e-““I

+ ePLLX2)‘Ia + emx3}].

This distribution is of interest as a natural model for situations of Debreu type, where two choice alternatives are similar and the third is dissimilar to the others. The parameter a is related to the correlation I between X, and X,; in fact r = 1 - l/a*. Distributions of the type F(x, ,..., x,J = exp{-C e-axi}“u, which are still in the family F = y(H), have the property that

P xi t mi = ,FF”,“n(Xj t mj) = $$-, (

i

(3)

where vi = emj. When a = 1 we obtain the usual choice probability formula (obeying Lute’s Choice Axiom) based on the independent double exponential Thurstone model: thus we have a natural extension to the case of correlated utility variables.

188

ROBERTSON AND STRAUSS

3. THURSTONE

MODELS

When the variables are independent and identically distributed apart from location shifts, we have the usual Thurstone model. Let F be the common distribution function of the X’s, let Y be max i ,,i&Xi + mi) and Z be the maximizing i, as before. The analogue of Theorem I is THEOREM 2. (i) Y is independent of Z for all (m, , m2 ,..., m,) iff F is the double exponential distribution; (ii) for all (m,, m,,..., m,), Y has the same distribution function as the Xs apart from a location shift 13 F is double exponential.

The proof is given in Strauss (1979) as part of Theorem 4; it is there attributed to Robertson and Strauss (1978), an earlier draft version of this paper. We therefore omit it here. Instead we offer an informal argument which may give insight into why, for double exponential variables, Y and Z should be independent. (i)

For each integer i (1 < i < n), suppose we have an independent random double exponential variables. Let sample wi, 9 wi* 9***9Wiki of standard { W,}. Then it is known from Fisher and Tippett (1928) that Zi has zi = maxlgjgk, the distribution of Wi, + In ki. (ii) Given m, ,..., m,, suppose we can find a real a and positive integers k i ,..., k, such that exp(a + mi) = ki

1
(4)

In general, of course, this will not be possible exactly, but we can approximate (4) as closely as we wish with suitable large a and ki’s. (iii) Suppose that (4) does hold. Let X, ,..., X, be standard double exponentials and let Y= max{X, + mi}. Then Y + Cl = max {Xi + a + mi) = max {Xi + In ki}. According to (i), this has the distribution

of

Now the k, + -.- + k, variables W, are i.i.d., and clearly the distribution of their maximum is unaffected if we condition on that maximum being attained in a particular subset. I

THEOREMS FORRANDOM

UTILITY VARIABLES

189

REFERENCES FISHER, R. A., & TIPPETT, L. H. C. Limiting forms of the frequency distribution of the largest or the smallest of a sample. Proceedings of the Cambridge Philosophical,Society. 1928, 24, 180-I 90. ROBERTSON,C. A., & STRAUSS, D. J. A characterization of an extreme value distribution. Manuscript, Department of Statistics, University of California, Riverside, 1978. STRAUSS. D. J. Some results on random utility models. Journal of Mathematical Psychology, 1979, 20, 35-52.

THURSTONE, L. L. A law of comparative judgment. Psychological RECEIVED:

May 22, 1980

Review,

1927, 34, 213-286.