On the Nucleolus of a Characteristic Function Game

Abstract

Introduction. The nucleolus is a solution concept for games in characteristic function form. It was introduced by D. Schmeidler in [1]. This paper is concerned with some of the properties of the nucleolus. Some of the results are new (Theorems 2, 4, 5) whereas others are known (Theorems l and 3), but the proofs given here are, in a sense, simpler than those in [1]. A characteristic function game is a pair (N, v) consisting of a set N = {1, 2, , n} of n players, and a characteristic function v, which maps each subset S of N, called a coalition, to a number v(S). In addition, it is assumed that v(N) _ 0, and that v(S) = 0 for all one-person coalitions, as well as for the empty coalition. A payoff vector is an n-tuple x = (x1, , xn) such that xi > 0 for all i = 1, , n, and x(N) = v(N), where for each coalition S, x(S) denotes Zi,eS Xi. X is the set of all payoff vectors. Of course, X depends on the particular game v. For any x in X, let q(x) be a vector in E2n, the components of which are the numbers v(S) x(S), arranged in order of decreasing magnitude, where S runs over all the coalitions of N. We shall say that x is at least as acceptable as y (with respect to v), and write x > y, if q(x) is not greater than q(y), in the lexicographical order on E2n. If q(x) is smaller than q(y), we shall say that x is more acceptable than y and write x >y. The nucleolus of the game is the set of those points in X which are "most acceptable," that is, {x E X :x > y for all y E X}. In [1], Schmeidler proved the basic result that every game possesses a nonempty nucleolus. He further showed that the nucleolus actually consists of just one point, and that-viewed as a point function of the game-it is continuous. Our aim is to give new proofs of the uniqueness and of the continuity of the nucleolus. To this end, we will have to introduce a few definitions and to prove an auxiliary theorem (Theorem 2), which may have some interest in its own right. DEFINITION. Let bo, b1, ... , bp be a sequence of sets whose elements are coalitions of N. This sequence is a coalition array whenever: (i) every coalition of N is contained in exactly one of the sets b1, bp, (ii) bo contains only one-element coalitions. For every game v and payoff vector x, let b1(x, v) be the set of those S c N for which max {v(S) x(S): S c N} is attained. Similarly, b2(x, v) is the set of those S c N where max {v(S) x(S) :S ? b1(x, v)} is attained, and so on. Finally, let bo(x) = {{i} : xi = 0}. It is obvious that bo(x), b1(x, v), . . , bp(x, v) is a coalition array. We shall say that it is the array that belongs to (v, x). DEFINITION 2. A coalition array bo, , bp has property I if for all k= 1,2,...,pandanyyinE n, (1) y(S) _ 0forall Sebo, (2) y(S) ? O for all Se b1 U U bk, and (3) y(N) = 0