On the set of solvable 𝑛-person games

Abstract

The set Gn of all n-person games 1 may be considered a convex subset of a euclidean space; Gn has dimension 2 n — n — 2. Although it has not been proved that all games possess a solution, large classes of solvable games have been discovered. In [2], Shapley defined a certain class of solvable games—the quota games—and showed that from the point of view of dimension, the set Qn of all ^-person quota games constitutes a considerable part of Gn. For n odd (the more favorable case), the dimension of Qn differs from that of Gn by (n — l)(n — 2)/2. Shapley also showed that by suitable extensions, it is possible to increase slightly the dimensionality of the set of games known to have a solution. Here we introduce a modification of the notion of quota called a partial quota, and demonstrate the existence of partial quota solutions for all partial quota games (games possessing a partial quota). We then show that the set of all partial quota games has dimension equal to that of Gn. Thus the probability is positive that an ^-person game picked at random has a solution. A similar result holds for zero-sum games: the set of zero-sum partial quota w-person games has dimension equal to that of all zero-sum w-person games. Notation will be as in [2], except that (0, 1) normalization will be used throughout. We will show that a sufficient condition for a game v to possess a partial quota solution is that there is at least one triple {i, j , k} of players such that