# From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory

@article{Papadimitriou2018FromNE, title={From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory}, author={Christos H. Papadimitriou and Georgios Piliouras}, journal={Entropy}, year={2018}, volume={20} }

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…

## 18 Citations

Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics

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It is shown that any Nash equilibrium which is not strict (in that every player has a unique best response) cannot be stable and attracting under the dynamics of FTRL, and shows unequivocally that only strict Nash equilibria can emerge as stable limit points thereof.

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A stress test for classic routing games with arbitrarily many strategies, polynomial cost functions, non-atomic as well as atomic routing games and heteregenous users shows that every system has a carrying capacity, above which it becomes unstable.

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- Computer ScienceArXiv
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It is proved here that the well-studied evolutionary learning algorithm of replicator dynamics (RD) seamlessly minimizes the strongest possible form of Φ-regret in generic 2 × 2 games, without any modiﬁcation of the underlying algorithm itself.

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- Computer ScienceCOLT
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It is proved here that the well-studied evolutionary learning algorithm of replicator dynamics (RD) seamlessly minimizes the strongest possible form of Φ-regret in generic 2 × 2 games, without any modification of the underlying algorithm itself.

Fast and Furious Learning in Zero-Sum Games: Vanishing Regret with Non-Vanishing Step Sizes

- Computer ScienceNeurIPS
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We show for the first time, to our knowledge, that it is possible to reconcile in online learning in zero-sum games two seemingly contradictory objectives: vanishing time-average regret and…

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New tools to understand and control the dynamics in n-player differentiable games are developed and basic experiments show SGA is competitive with recently proposed algorithms for finding stable fixed points in GANs -- while at the same time being applicable to, and having guarantees in, much more general cases.

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This paper proposes to view the outcomes of the agents’ dynamics as inducing a “meta-game” between the users, and proposes a general framework to model and analyze these strategic interactions between users of learning agents for general games and analyze the equilibria induced on the users in three classes of games.

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