Weighted values and the core

Abstract

The main result of this paper is that the set of all weighted Shapley values of a cooperative game contains the core of the game. That there is such a general relationship between core and values is somewhat surprising in light of the difference in concept behind these solutions. Indeed cooperative game theory tells us very little about the relations between core and values. Such relations are known to exist for convex games and for market games with a continuum of players. In such games the Shapley value is always in the core. Convex games have, in a sense, large cores, which 'explains' why they contain the Shapley value. In the case of market games with a continuum of players, it is the homogeneity of the games and the diagonal property of the Shapley value that guarantee this fact. More relations can be found when we consider core-like and value-like solutions. In a recent paper Owen (1990) shows that for spatial voting games the Copeland winner outcome, which is a near core solution concept, is an analogue of the Shapley value. A result concerning a relation between the core and value-like solutions for general games was noted by Weber (1988) who showed that the set of all random order values of a game contains the core. Our result generalizes Weber's since weighted values constitute a subset (dimensionally, a very small one) of random order values. Weighted Shapley values (weighted values for short) were defined by Shapley (1953 a, b) alongside the standard Shapley value and were extensively discussed in the literature (e.g., Owen (1972), Kalai, and Samet (1987), and Hart and Mas-Colell (1989)). For these values weights are assigned to the players. The value is then determined in one of two equivalent ways. In the random order approach the weights are used to determine a probability distribution over orders of the players and the value is the expected contributions of the players according to this probability distribution. In the algebraic approach the value of a unanimity game is determined first, by allocating one unit among the players of the carrying coalition according to their