This project focuses on fundamental mathematical questions in game theory and its applications to economics. In particular, the investigation centers on value theory for games with a continuum of players. Theory of value for such games was developed by Aumann and Shapley in the period between 1964 and 1974. It has provided a mathematical model of perfect competition in productive economies and in some exchange economies. The existence and uniqueness of values for various classes of games, characterized by mathematical properties such as atomic or non-atomic probability measures, finite number of generators, or finite number of lattice operations from a finite set of measures, is of current vivid interest. Neyman and Dubey intend to prove that, for exact market games, any continuous value is necessarily an element of the core, and to extend this investigation to more general market games. Complementary to this axiomatic approach is the asymptotic one, in which the model involves the approximations of a game with a continuum of players by finite games. The asymptotic value is defined as a limit of the Shapley values of such finite games. One of the central problems is the existence of the asymptotic value. Neyman has recently settled a long standing problem by proving that all games defined by a monotone continuous function and a probabilty measure with a non-atomic part and possibly countably many atoms have asymptotic value. He and Dubey now propose to study the existence of asymptotic values for games with either positive or signed vector measures. Another area of proposed study is the axiomatic theory of allocations and payoffs that would encompass economies with core allocations, value allocations and Walrasian allocations. One goal is to explain the much remarked fact their solutions become equivalent on the domain of smooth, perfectly competitive economies. Another goal is to pinpoint the differences that emerge between them on bigger domains. The approach taken in this research is as follows: given a domain of economies and a solution, one lists properties that categorically determine the solution there. If these properties are shared by another solution on that domain, one deduces that they coincide there. On the other hand, if a minor variant of the properties categorizes the other solution, then this pinpoints the difference between them. The investigators intend to analyze the economies both with and without perfect competition. Mathematical game theory of the type described here is a large research activity bridging economic theory and pure and applied mathematics. I relies on mathematical tools such as measure theory, multivalued maps, saddle point theorems and stochastic processes. The results of this and other investigations are aimed at interpreting the mechanisms governing various economic models . They create a conceptual framework for understanding of real economic systems.
