The stochastic choice problem: A game-theoretic approach

JOURNAL

OF MATHEMATICAL

PSYCHOLOGY

36,

547-554 (1992)

The Stochastic Choice Problem: A Game-Theoretic Approach Dov MONDERER Faculty

of Industrial Technion-Israel

Engineering Institute

and Management,

of Technology

The stochastic choice problem is formulated as a problem in the theory of values for cooperative games. The new formulation yields an independent proof of a conjecture of Block and Marschak (1960), which has been already proved (to be true) by Falmagne (1978). The game-theoretic approach gives a new insight and provides new tools to deal with the related binary stochastic choice problem, which is still open. 0 1992 Academic press, IIIC.

1. INTBODUCTI~N Let N be a finite set of n alternatives. Consider a choice experiment, where an individual is presented with various subsets of N, each of which is presented to him a large number of times. Given such a subset, he is asked to choose the most preferred alternative in it. Typically, individuals do not stick to a fixed ranking over N in their choices. Such a phenomenon may be explained by a stochastic model, where the individual’s choices are governed by a probability distribution over the set of n! rankings on N, and not necessarily by a fixed deterministic ranking. Suppose the choice experiment results in a function j, where for i E T c N, b( i, T) denotes the frequency of choosing the alternative i out of T. In order to be able to test our stochastic model hypothesis, we want to find a simple mechanism that can determine whether fl can be rationalized by a probability distribution over the set of rankings on N (such p will be called consistent). Actually, for a given N we need many much mechanisms, one for each class of observed subsets T. The stochastic choice problem is the one in which T ranges over all subsets of N. The binary stochastic choice problem is the one in which T ranges over all 2-element subsets of N. The term “simple mechanism” has yet to be defined. In this paper we take the commonly used definition: An explicit finite set of linear conditions on j that are necessary and sufficient for the consistency of b. The existence of such a finite set of linear conditions can be easily proved. The main difficulty is, therefore, stating them explicitly. Address all reprint request to Dov Monderer at the Faculty of Industrial Engineering Management, Technion-Israel Institute of Technology, Technion City, Haifa 32OOO,Israel.

and

547 0022-2496192 $5.00 Copyright I<‘ 1992 by Academic Press. Inc. All rights 01 reproduction in any lorm reserved

DOV MONDERER

548

Block and Marschak (1960) presented a set of necessary conditions for the stochastic choice problem (known today as the Block-Marschak conditions) and conjectured that the conditions are also sufficient. Their conjecture was proved to be true by Falmagne (1978). Barbera and Pattanaik (who were unaware of Falmagne’s proof) reproved it in 1986 (see Barbera & Pattanaik (1986) with the editorial note). In this paper we give a game-theoretic formulation to the stochastic choice problem: To every choice experiment, whose result is a function /I as described above, we associate a solution function for cooperative games. We prove that p is rationalized by a probability distribution iff the associated solution function is a quasivalue (see Weber (1988) and Section 3 of this paper for the definitions). We then use the characterization theorem of Weber for quasivalues in order to prove Falmagne’s result. We want to emphasize that in this paper we discuss just the technical relationships between the stochastic choice problem and value theory. The potential conceptual relations are yet to be explored. Unlike the stochastic choice problem, the binary stochastic choice problem is still open (for n > 5). For recent results as well as many previous references the reader is refered to Fishburn (1988), Fishburn and Falmagne (1989), Gilboa (1989), Cohen and Falmagne (1990), and Koppen (1990). In the second part of this paper we provide a game-theoretic formulation for the binary stochastic choice problem. For this problem, though, the new formulation has not yielded a solution. However, it exposes a new potential way to deal with this stubborn problem, as can be seen in Gilboa and Monderer (1990), where new results were obtained using the game-theoretic approach. 2. THE STOCHASTIC CHOICE PROBLEM Let N= { 1, 2, .... n} be the set of alternatives. Let W be the set of all one-to-one functions 8: N-r N. Interpret every 0 E W as a ranking on N; i is prefered to j according to f3 iff O(i) > O(j). Let B = b(i, T)i, Tc N be a vector of numbers. We say that /I is consistent, if there exists a probability distribution Pr on W such that p(i, T) is the probability that i is the most prefered element in T. That is, for all i E T c N. B(i, T) = We) (2.1) c {0:6J(i)~O(j)Vj~~{i})

Obvious necessary conditions P(i, T) 20

for the consistency of /I are for all T # @ and for all i E T,

(2.2)

and i& B(i, T) = 1

for all

T# 0.

(2.3)

549

STOCHASTIC CHOICE PROBLEM

Another set of necessary conditions was given by Block and Marschak For each S # N and for each i 4 S define a’(S)= where s = (SI. The Block-Marschak

n-l 1 (-l)‘-” *=s conditions

a’(S) > 0

C ( =,$pyW are conditions for all

P(k Tu {i>,

>

,

(1960):

(2.4)

(2.2), (2.3), and

S c N and i $ S.

(2.5)

The necessity of (2.5) is clear as one can easily verify that if (2.1) holds, then a’(S) is precisely the probability that i is prefered to Jo S and that k is prefered to i for all k E (N\S)\{i}. In th e next section we develop the game-theoretic tools needed for the proof of the sufficiency of the Block-Marschak conditions.

3. GAMES AND QUASIVALUES Let N= (1, 2, .... n}. Members of N will be called players and subsets of N will be called coalitions. A game on N is a function u : 2N -+ R with u(0) = 0, where 2N is the set of coalitions and R is the set of real numbers. u(S) will be called the worth of the coalition S, it is interpreted as the amount that the players in S can obtain for themselves if they form a coalition. The space of all games will be denoted by G. G is a linear space of dimension 2” - 1. We now introduce a linear base for G: For each T # 0 define if if

TsS TGS.

It is well known (see, e.g., Owen (1982)) that { W, : T# a} is a linear base for G. The game W, represents a situation where a decision can be carried out iff each of the members of T votes for it. A game u is additive if u(S u T) = u(S) + u(T) for all S n T = 0. The space of all additive games will be denoted by A. Every a E A is uniquely. characterized by the worth it associates to the l-element coalitions, because a(S) = 1 ai

V ScN,

icS

where ai=a({i}) for all ieN. A solution is a function $ : G + A. If t++(u)= a and ai = a( { i}), then ai may be interpreted as the index of power of Player i in the game u. Many other interpretations are available in the literature of game theory (see, e.g., Owen (1982) and Roth (1988)).

550

DOVMONDERER

Let II/ be a solution. II/ is linear (or $ satisfies the linearity axiom) if $(CUI + bu) = c+(v) + /Iti for all u, u E G and for all a, j E R. $ is efficient (or $ satisfies the efficiency axiom) if $(u)(N) = u(N) for all v E G. II/ is a projection (or + satisfies the projection axiom) if $(a) = a for all a E A. For the next axiom we need more notation: Recall that W is the set of all one-toone functions 6’ : N -+ N. Each t? is interpreted as a change of names for the players. For each 8 E W and v E G define a game Bv by (eu)(s) = 4NS))

for all

S s N,

where e(S) = {O(i): iE S}. + is symmetric (or II/ satisfies the symmetry axiom) if

+(eo)(i)= tiu(ei) for all UEG, ieN, and 8Eg. A game u is monotonic if u(S) < v(T) whenever S c T. II/ is positive (or $ satisfies the positivity axiom) if $(u) is monotonic whenever v is monotonic. A value is a solution that satisfies the axioms of linearity, efficiency, symmetry, and positivity and is also a projection. A quasioalueis a solution that satisfies the axioms of a value except, possibly, the symmetry axiom. For further discussions on the various types of solutions the reader is referred to Owen (1982) and Roth (1988). We now define a special type of solution, due to Weber (1988). For each 8 E B and i E N denote Qf= {jEN:

Let Pr be a probability distribution each l-element set ( i},

ej<&}.

over 6%‘.Define the solution tip, as follows: for (3.1)

A solution Ic/ is a random order oalue if there exists a probability distribution Pr on W such that 1,5= tip,. The following theorem relates the theory of quasivalues to that of social choice: THEOREM

3.1

(Weber 1988).

Let + be a solution. Then II/ is a quasivalue iff it

is a random order value.

We can finally formulate the sufficiency result: THEOREM A. Let B = (/?(i, T))i, T be a vector satisfying the Block-Marschak conditions (2.2), (2.3), and (2.5). Then /I is consistent. That is, there exists a probability distribution Pr on W such that (2.1) holds.

551

STOCHASTIC CHOICE PROBLEM

Proof:

We need the following lemma:

LEMMA 3.1. Let /I = (fi(i, T))i, T be a vector of real numbers. Let a= (ai(S))i,, be the vector defined by (2.4). Then

B(4T)=

c {S:

ai

for all

iE TGN.

(3.2)

T\(i)cSEN\{i)}

Proof of Lemma 3.1. It suffkes to prove that the claim holds for a linear base for the space of all vectors /I = (j?(i, T))i, T. Let {pi, + TEN, iE T} be such a linear base, where

Pi, di, T) = i

if i=i,and otherwise.

3

T=T,

Note that by (2.4), for /I = fl,, T,, a’(S) = 0 if i# i, or SG/ TO\(&), and a’O(S)=(-l)$-‘-“P(io, TO) if SC TO\{&,}, where to= /TOI and s= lS1. The result follows now from an obvious direct computation utilizing the fact that for ma1

We now return to the proof of Theorem A. Define a solution II/ as follows: @7((i))=

1

(v(Su {i))-u(S))d(S).

srN\(i)

(3.3)

We claim that it suffices to show that t+Qis a quasivalue. Indeed, if tj is a quasivalue then by Theorem 3.1 there exists a probability distribution Pr on 3 such that * = *PT. By (3.3) for every ie TG N,

J/W,(i)=

C {S:

ai( S).

T\{i}ESEN\(i)]

Therefore, by Lemma 3.1 ICIW,(i) = Hi, 0. On the other hand by (3.1) $W,(i)=$,,W,(i)=Pr(i>j

where i >j, denotes the event that i is preferred to j. Therefore (2.1) holds. 480/36/4-7

vj~T\{i}),

552

DOVMONDERER

So, all we have to show is that Ic/ is a quasivalue. $ is obviously linear. It is positive because of (2.5). We proceed to prove that + is a projection. Let UEA be given. Then

@4{W=4{i>)

1 4s). SEN\{i]

By Lemma 3.1 we have

and by (2.3) fi(i, {i})= 1. Therefore @=a. To prove that ti is efficient, it s&ices to show that II/W,(N) = W,(N) for every Tf a. Indeed, we have already shown that t,bW,(i) = fl(i, T) for i E T. For i $ T, Hence, $ W,(i) = 0. Therefore by W,(Su {i)) - W,(S) = 0 f or all S c N\(i). (2.5 ), $W,(N)=

i

$W,(i)=

1 $W,(i)=

i=l

As W,(N)

icT

= 1 the result follows.

1 b(i, T)=l. UET

1

4. THE BINARY STOCHASTIC CHOICE PROBLEM AND QUASIVALUES Recall that the binary stochastic choice problem is defined as follows: Given a vector /I = (/?(i,j))i+i (where fl(i,j) is short for B(i, {i,j})) find a finite number of necessary and sufficient conditions on /I for the existence of a probability distribution Pr on 9 such that

B(i,j) = Pr(i>j) Obvious necessary conditions

Vi#j.

(4.1)

are and

B(U)20

B(U) + W, 4 = 1.

(4.2)

We now proceed to formulate a problem in the theory of quasivalues, which is equivalent to the binary stochastic choice problem. Let Q2 be the linear space of games spanned by the additive games and all games Wii,il, i # j. A typical game u in Q2 has the form U=a+

C

: i,j ) where a E A, and 61i,j1 are real numbers.

6{i.j}

w{i.j},

(4.3)

553

STOCHASTIC CHOICE PROBLEM

Let p = (p(i,j)),+i

satisfy (4.2). Define + : Q2 + A as follows:

*(a+c

6{i,j}

W{i,j}

Glcxi+ 1

(i.i)

LEMMA

4.1.

C

(4.4)

s{i,j}P(i9j)*

j#i

I) is a quasivalue on Q2.

Proof. It is easily verified that II/ is well defined, linear, efficient, and a projection. We still have to show that $ is positive. Suppose that the game u given in (4.3) is monotonic. Let in N. We will show that Ii/u(i) 2 0. For each S c N\{ i}, u( S u ( i} ) - u(S) 2 0. Therefore

Ui+ 1 d(i,j)BO*

(4.5 )

jsS

Denote S,= 61i,jj for all j~N\(i}.

Then (4.5) implies that 1 6,zj> -a, is N\(i)

(4.6 )

for every 2 = (z~)~.~,(~) which is a vertex of the unit cube D of RN\fi). Hence (4.6) holds for every z in the unit cube, and in particular it is true for z = (B(i,j))j,,\iil. Therefore rl/u(i) >, 0. As $0 E A, then $u(i) 2 0 for all i E N implies the monotonicity of *u. [ It is now clear that the binary stochastic choice problem following problem:

is equivalent

to the

Problem A. Find a finite number of conditions on a quasivalue $ : Q, + A that are necessary and sufficient to ensure that it can be extended to a quasivalue on the space of all games.

The literature about extending solutions that are defined on linear subspaces is not rich. Hart and Mas-Cole11 (1989), and Neyman (1989) implicitly deal with extensions of a value to a value, and Monderer (1988) deals with extensions of values and semivalues to semivalues (where a semivalue is a solution that satisfies all the axioms of the value except, possibly, the efficiency axiom). However, in view of the equivalence theorem formulated in this section, it seems to us that a deep investigation about the extension property of quasivalues on subspaces may shed some light on the binary stochastic choice problem and related topics.

5. CONCLUSIONS.

In this paper we established technical relationships between stochastic choice problems and game theory (value theory). Roughly speaking, to each experiment’s result /?, we associated a solution

DOV

554

MONDERER

function for cooperative games. We then proved that /3 is consistent with a stochastic choice model iff the associated solution is reasonable. The above result enabled us to use the existing game-theoretic tools in order to provide an independent proof for Falmagne’s result on the stochastic choice problem. It also enabled us to point out a new potential way to deal with the open binary stochastic choice problem.

REFERENCES

BARBERA, S., & PATTANAIK, P. K. (1986). Falmagne and the rationalizability of stochastic choices in terms of random orderings. Econometrica, 54, 707-715. BLOCK, H. D., & MARXHAK, J. (1960). Random orderings and stochastic theories of response. In I. Olkin et al. (Eds.), Confributions to probability and statistic (pp. 97-132). Stanford: Stanford Univ. Press. COHEN, M., & FALMAGNE, J.-CL. (1990). Random utility representation of choice probability: A new class of necessary conditions. Journal of Mathematical Psychology, 34( 1), 88-94. FALMAGNE, J.-CL. (1978). A representation theorem for finite random scale. Journal of Mathematical Psychology,

18, 52-72.

FISHBURN,P. (1988). Binary probabilities induced by rankings. [Mimeograph] FISHBURN,P., & FALMAGNE, J.-CL. (1989). Binary choice probabilities and rankings. Economic Lettres, 31, 113-l 17. GILBOA, I. (1989). A necessary but insufficient condition for the stochastic binary choice problem. Journal

of Mathematical

Psychology,

34, 371-392.

GILBOA, I., & MONDERER, D. (1990). A game-theoretic approach to the binary stochastic choice problem. [Mimeograph] HART, S., & MAS-COLELL, A. (1989). Potential, value, and consistency. Economerrica, 57(3), 589-614. KOPPEN, M. (1990). Random utility representations of binary choice probabilities. [Mimeograph] MONDERER, D. (1988). Values and semivalues on subspaces of finite games. International Journal of Game Theory, 17(4), 301-310. NEYMAN, A. (1989). Uniqueness of the Shapley value. Games and Economic Behavior, l(l), 116-118. OWEN, G. (1982). Game Theory. Orlando, FL: Academic Press. ROTH, A. E. (1988). The Sharpley Value. New York: Cambridge Univ. Press. WEBER, R. J. (1988). Probabilistic values for games. In A. E. Roth (Ed.), The Sharpley Value (pp. 101-119). New York: Cambridge Univ. Press. RECEIVED:

August 25, 1989