多边纳什均衡在游戏族中的存在性
Existence of Multilateral Nash equilibria for families of games
AI总结：
本文提出多边纳什均衡和游戏族的概念，证明在游戏族中存在多边均衡，并利用Kneser图的克里夫覆盖数作为关键工具。
摘要：
本文介绍了博弈论中两个全新的概念：多边纳什均衡和游戏族。从非合作博弈开始，我们展示了这些概念如何无缝整合并自然扩展经典理论，同时使我们能够证明一个强大的（多边）纳什均衡存在性结果，仅需对游戏做出最小的假设。传统上，纳什均衡是一个全局策略，无论哪个玩家单方面偏离均衡，都会减少自己的利润。对于k-边纳什均衡，我们现在要求无论哪k个玩家集体改变策略，所有偏离玩家的利润都会减少。通过这种方式，我们获得了一种均衡的过滤，其中高边均衡更少见。此外，我们推导了多边纳什均衡的存在标准，并展示了它如何反映更高边均衡的稀有性。此外，我们证明了一些经典游戏具有更高边纳什均衡，每种情况都从新的视角揭示了这些游戏的结构。游戏族是参数化的非合作博弈集合，其中参数影响游戏的每一个方面。通常，我们假设这种依赖是连续的，从而引入了新的结构。这样，我们可以避免逐个分析游戏，而是将整个族视为整体。这使参数在我们的理论中扮演了核心角色，并将我们的注意力从寻找特殊策略转移到寻找具有首选策略的特殊游戏。我们的主要结果证明了在游戏族中存在多边均衡，保持了对单个游戏的最小假设。令人惊讶的是，克里夫覆盖数在Kneser图中起到了核心作用。
英文摘要：
This paper introduces two fundamentally new concepts to game theory: multilateral Nash equilibria and families of games. Starting with non-cooperative games, we show how these notions together seamlessly integrate into and naturally extend the classical theory, and simultaneously enable us to prove a powerful (multilateral) Nash equilibrium existence result with minimal assumptions on the game. Classically, a Nash equilibrium is a global strategy such that whichever player unilaterally deviates from the equilibrium, also reduces his own profit. For a k-lateral Nash equilibrium we now require that whichever group of k players collectively changes their strategies, also reduces all of the deviating players' profits. In this way, we obtain a filtration of equilibria, where the higher-lateral equilibria are less frequent. Furthermore, we derive an existence criterion for multilateral Nash equilibria and demonstrate how it reflects the increasing rarity of higher-lateral equilibria. Additionally, we show that some classical games have higher-lateral Nash equilibria, which in every case reveal the structure of these games from a new point of view. A family of games is a parameterized collection of non-cooperative games, where the parameter affects every aspect of the game. Typically, we assume that this dependence is continuous, thereby introducing a new structure. That way, we can avoid analyzing the games one at a time, and instead treat the family as a whole. This allows the parameter to take a central role in our theory, and shifts our attention from seeking a special strategy to searching for a special game with preferred strategies. Our main result proves the existence of a multilateral equilibrium in a family of games, maintaining minimalistic assumptions on the games individually. Surprisingly, the clique covering number of the Kneser graph makes a central appearance.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
