Games with Many Players and Abstract Economies Permitting Differentiated Commodities, Clubs, and Public Goods

In a seminal paper relating economic and game theoretic structures, Shapley and Shubik (1969) demonstrate that a game is a market game -- that is, a game derived from a finite-dimensional private goods exchange economy where all participants have continuous, concave utility functions. In this paper, to accommodate models of economies with public goods, clubs, indivisibilities, and other deviations from the classic model of Shapley and Shubik, we demonstrate an equivalence between homogeneous market games with many players, possibly all with different characteristics, and abstract economies permitting differentiated commodities, clubs, public goods, coalition production, unbounded short sales and other deviations from standard economic models. By a homogeneous market game we mean a game derived from a market where all individuals have the same concave and continuous utility function. We also demonstrate that the condition of small group effectiveness -- that small groups of players can realize almost all gains to collective activities -- is equivalent to the condition of asymptotic negligibility -- that small groups of players cannot have significant impacts on average payoff to large groups of players.