The Core of a Cooperative Game without Side Payments^)

The core of an M-person game, though used already by von Neumann and Morgenstern [15], was first explicitly defined by Gillies [5]. Gillies's definition is restricted to cooperative games with side payments and unrestrictedly transferable utilities(2), but the basic idea is very simple and natural, and appears in many approaches to game theory. We consider a certain set of "outcomes" to a game, and define a relation of "dominance" (usually not transitive) on this set. The core is then defined to be the subset of outcomes maximal with respect to the dominance relation; in other words, the subset of outcomes from which there is no tendency to move away—the equilibrium states. To turn this intuitive description of the core notion into a mathematical definition, we need precise characterizations of (a) the kind of game-theoretic situation to which we are referring (cooperative game, noncooperative game, etc.); (b) what we mean by "outcome"; and (c) what we mean by "dominance." Different ways of interpreting these three elements yield different applications of the generalized "core" notion, many of them well-known in game theory. Gillies's core, Luce's ^-stability [lO], Nash's equilibrium points [12], Nash's solution to the bargaining problem [l3](3), and the idea of Pareto optimality—to mention only some of the applications—can all be obtained in this way. Here we shall be concerned exclusively with cooperative games without side payments(4). Our procedure will be to generalize von Neumann's fundamental notion of characteristic function to this case, and on the basis of this generalization to define the core in a way that generalizes and parallels the core in the classical theory—i.e., Gillies's core. The generalization of the characteristic function is of interest for its own sake also; for example, a theory of "solutions" has been developed that generalizes and parallels the classical theory of solutions and is based on the characteristic function [3; 16].