Existence of Equilibria in Infinite Games

这个奖资助博弈论的研究。博弈论推理对经济学中的大量问题至关重要，从有效拍卖的设计（例如出售无线电频谱许可证）到健康保险市场的分析。分析这类问题的一个基本工具是纳什1950年提出的均衡概念，现在简称为纳什均衡。纳什的想法已被证明非常有用，涵盖了很多领域。然而，在玩家行为的微小变化可能对自己或他人的收益产生巨大影响的情况下（如拍卖），将其应用于游戏却很困难。更确切地说，在不连续博弈中，纳什均衡的存在并不总是得到保证。第一个项目是PI先前工作的延续，即识别不连续博弈，尽管存在不连续，但仍承认纳什均衡。这已经为纳什1950年的存在性定理所未涵盖的博弈分析开辟了大量的类别。该项目寻求的纯策略均衡的新存在性结果的一个重要方面是它是序数的，这是该领域大多数先前工作无法满足的性质。序数意味着新结果独立于用于定义游戏中玩家收益的任意数值尺度。因此，新的结果不仅将扩大我们能够分析的战略博弈的类别，它也将为我们理解纳什均衡的存在性问题提供更深入的基础。第一个项目关注的是战略形式的游戏和不连续性，而第二个项目关注的是广泛形式的游戏，其中信息和动态被明确地建模。自Kreps和Wilson（1982）引入有限扩展形式博弈的顺序均衡理念以来，已经过去了30年。然而，对于具有无限行动集和任意信息结构的游戏，我们仍然没有一个有效的顺序平衡定义。尽管许多动态经济模型都是用无限的行动空间来制定的。分析这类游戏的研究人员必须使用与有限游戏的定义不一致的临时解决方案。该提案的第二个项目旨在为这种无限行动扩展形式博弈提供缺失的顺序均衡定义。我们的目标是定义一个顺序均衡的概念，即(i)无论博弈是否是连续的（在收益或信息中），都允许存在结果；（ii）这样的定义与Kreps和Wilson对有限博弈的定义自然地一致；（iii）在一类典型例子中产生“自然均衡”。最终目标是提供一个广泛适用、有用和明智的定义。由于社会科学和计算机科学的许多领域都使用博弈论作为建模工具，因此该研究将具有重大的跨学科影响。

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.