From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory.
From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory.
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DOI：
10.3390/e20100782
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发表时间：
2018-10-12
期刊：
影响因子：
--
通讯作者：
Piliouras G
中科院分区：
文献类型：
--
作者：
Papadimitriou C;Piliouras G
1950年，纳什（Nash）提出了一个自然的等效解决方案概念。 ，在他时代的拓扑结构中，最复杂的结果 - 实际上，最近的算法工作确定NASH等效在本文中与固定点相等。 NASH不可用的动态系统的拓扑既始于游戏，而是在混合策略上定义的动态。在动态系统理论中，NASH等同于链条重复集，一旦我们专注于这个解决方案概念（“游戏的结果”的概念，每个游戏都像潜在的游戏一样换句话说（加权）潜在的游戏，新概念与动态的固定点/平衡相吻合。动力学满足运动的特定信息，我们讨论了许多新型计算以及该链条复发概念提出的结构性的组合问题。
In 1950, Nash proposed a natural equilibrium solution concept for games hence called Nash equilibrium, and proved that all finite games have at least one. The proof is through a simple yet ingenious application of Brouwer’s (or, in another version Kakutani’s) fixed point theorem, the most sophisticated result in his era’s topology—in fact, recent algorithmic work has established that Nash equilibria are computationally equivalent to fixed points. In this paper, we propose a new class of universal non-equilibrium solution concepts arising from an important theorem in the topology of dynamical systems that was unavailable to Nash. This approach starts with both a game and a learning dynamics, defined over mixed strategies. The Nash equilibria are fixpoints of the dynamics, but the system behavior is captured by an object far more general than the Nash equilibrium that is known in dynamical systems theory as chain recurrent set. Informally, once we focus on this solution concept—this notion of “the outcome of the game”—every game behaves like a potential game with the dynamics converging to these states. In other words, unlike Nash equilibria, this solution concept is algorithmic in the sense that it has a constructive proof of existence. We characterize this solution for simple benchmark games under replicator dynamics, arguably the best known evolutionary dynamics in game theory. For (weighted) potential games, the new concept coincides with the fixpoints/equilibria of the dynamics. However, in (variants of) zero-sum games with fully mixed (i.e., interior) Nash equilibria, it covers the whole state space, as the dynamics satisfy specific information theoretic constants of motion. We discuss numerous novel computational, as well as structural, combinatorial questions raised by this chain recurrence conception of games.
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