Existence of Equilibria in Infinite Games

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This award funds research in game theory.Game theoretic reasoning is critical to a vast array of problems in economics, from the design of efficient auctions (e.g. to sell radio spectrum licenses), to the analysis of markets for health insurance. A fundamental tool for the analysis of such problems is Nash's 1950 concept of equilibrium, now called simply Nash equilibrium. Nash's idea has proven immensely useful and covers a lot of ground. However, applying it in contexts (such as auctions) where small changes in a player's behavior can have dramatic effects on his and/or on others' payoffs has been difficult. More precisely, the existence of a Nash equilibrium cannot always be guaranteed in discontinuous games. The first project is a continuation of the PI's previous work to identify discontinuous games that, despite the presence of discontinuities, admit Nash equilibria. This has already opened up large classes of games for analysis that were not previously covered by Nash's 1950 existence theorem. An important aspect of the new existence result for pure strategy equilibria that sought by the project is that it is ordinal, a property that most of the previous work in the area fails to satisfy. Ordinality means that the new result is independent of the arbitrary numerical scale that is used to define the players' payoffs in the game. Thus, the new result will not only expand the class of strategic games that we are able to analyze, it will provide a deeper foundation to our understanding of the question of existence of Nash equilibrium as well. While the first project focuses on strategic form games and discontinuities, the second project concerns games in extensive form, where information and dynamics are explicitly modeled. It has been about 30 years since Kreps and Wilson (1982) introduced the idea of a sequential equilibrium for finite extensive form games. Yet, we still do not have a working definition of sequential equilibrium for games with infinite action sets and arbitrary information structures. This is despite the fact that many dynamic economic models are formulated with infinite action spaces. Researchers analyzing such games must resort to using ad hoc solutions that do not correspond to the definition used for finite games. The second project of this proposal seeks to provide the missing definition of sequential equilibrium for such infinite-action extensive form games. The goal is to define a notion of sequential equilibrium that (i) permits an existence result whether or not the game is continuous -- in payoffs or in information -- and (ii) such that the definition coincides in a natural way with Kreps and Wilson's definition for finite games and (iii) yields the "natural equilibria" in a class of canonical examples. The ultimate goal is to provide a definition that is widely applicable, useful, and sensible.Because many areas of social science and computer science use game theory as a modeling tool, the research will have a significant interdisciplinary impact.