A characterization, existence proof and dimension bounds for the kernel of a game

Abstract

The kernel of a cooperative game is a subset of the bargaining set M1(i). It is sensitive to symmetry relations and their generalizations, which may exist in the characteristic function. The present paper offers an interesting representation formula for the kernel. This formula is applied to deriving properties of the kernel as well as practical methods for its computation. In particular, we provide an algebraic proof to the theorem stating that for each coalition structure in a cooperative game there exists a payoff in the kernel (and therefore also in the bargaining set M1(i)). (All other known proofs of this theorem are based on the Brouwer fixed-point theorem.) We also prove that the maximal dimension of the kernel of an n-person game is and this bound is sharp. Two players in a game are called symmetric, if the game remains invariant when these players exchange roles. One generalizes this concept by defining a player k to be more desirable than a player l, if player k always contributes not less than player l by joining coalitions which contain none of these players. It turns out that the payoffs in the kernel always preserve the order determined by the desirability relations. This fact may simplify the representation formula significantly. © 1966 by Pacific Journal of Mathematics.