Values of games with infinitely many players

This chapter studies the theory of value of games with infinitely many players.Games with infinitely many players are models of interactions with many players. Often most of the players are individually insignificant, and are effective in the game only via coalitions. At the same time there may exist big players who retain the power to wield single-handed influence. The interactions are modeled as cooperative games with a continuum of players. In general, the continuum consists of a non-atomic part (the "ocean"), along with (at most countably many) atoms. The continuum provides a convenient framework for mathematical analysis, and approximates the results for large finite games well. Also, it enables a unified view of games with finite, countable, or oceanic player-sets, or indeed any mixture of these.The value is defined as a map from a space of cooperative games to payoffs that satisfies the classical value axioms: additivity (linearity), efficiency, symmetry and positivity. The chapter introduces many spaces for which there exists a unique value, as well as other spaces on which there is a value.A game with infinitely many players can be considered as a limit of finite games with a large number of players. The chapter studies limiting values which are defined by means of the limits of the Shapley value of finite games that approximate the given game with infinitely many players.Various formulas for the value which express the value as an average of marginal contribution are studied. These value formulas capture the idea that the value of a player is his expected marginal contribution to a perfect sample of size t of the set of all players where the size t is uniformly distributed on [0,1]. In the case of smooth games the value formula is a diagonal formula: an integral of marginal contributions which are expressed as partial derivatives and where the integral is over all perfect samples of the set of players. The domain of the formula is further extended by changing the order of integration and derivation and the introduction of a well-crafted infinitesimal perturbation of the perfect samples of the set of players provides a value formula that is applicable to many additional games with essential nondifferentiabilities.