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

• URL: http://arxiv.org/abs/2203.14129v1
• Date: Sat, 26 Mar 2022 18:27:40 GMT
• Title: Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics
• Authors: Jason Milionis, Christos Papadimitriou, Georgios Piliouras, Kelly Spendlove
• Abstract summary: We show that no game dynamics can converge to $epsilon$-Nash equilibria. We also prove a stronger result for $epsilon$-approximate Nash equilibria.
• Score: 28.815822236291392
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players' behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this becomes a problem in the theory of dynamical systems. We apply this theory, and in particular the concepts of chain recurrence, attractors, and Conley index, to prove a general impossibility result: there exist games for which any dynamics is bound to have starting points that do not end up at a Nash equilibrium. We also prove a stronger result for $\epsilon$-approximate Nash equilibria: there are games such that no game dynamics can converge (in an appropriate sense) to $\epsilon$-Nash equilibria, and in fact the set of such games has positive measure. Further numerical results demonstrate that this holds for any $\epsilon$ between zero and $0.09$. Our results establish that, although the notions of Nash equilibria (and its computation-inspired approximations) are universally applicable in all games, they are also fundamentally incomplete as predictors of long term behavior, regardless of the choice of dynamics.

Related papers

• Games played by Exponential Weights Algorithms [0.0]
  We consider a repeated interaction in discrete time, where each player uses an exponential weights algorithm characterized by an initial mixed action and a fixed learning rate. We show that whenever a strict Nash equilibrium exists, the probability to play a strict Nash equilibrium at the next stage converges almost surely to 0 or 1.
  arXiv  Detail & Related papers  (2024-07-09T08:49:51Z)
• On Tractable $Φ$-Equilibria in Non-Concave Games [53.212133025684224]
  Non-concave games introduce significant game-theoretic and optimization challenges. We show that when $Phi$ is finite, there exists an efficient uncoupled learning algorithm that converges to the corresponding $Phi$-equilibria. We also show that Online Gradient Descent can efficiently approximate $Phi$-equilibria in non-trivial regimes.
  arXiv  Detail & Related papers  (2024-03-13T01:51:30Z)
• Smooth Nash Equilibria: Algorithms and Complexity [38.08108978808664]
  A fundamental shortcoming of the concept of Nash equilibrium is its computational intractability. In a $sigma$-smooth Nash equilibrium, players only need to achieve utility at least as high as their best deviation to a $sigma$-smooth strategy. We show that both weak and strong $sigma$-smooth Nash equilibria have superior computational properties to Nash equilibria.
  arXiv  Detail & Related papers  (2023-09-21T16:22:07Z)
• Abstracting Imperfect Information Away from Two-Player Zero-Sum Games [85.27865680662973]
  Nayyar et al. (2013) showed that imperfect information can be abstracted away from common-payoff games by having players publicly announce their policies as they play. This work shows that certain regularized equilibria do not possess the aforementioned non-correspondence problem. Because these regularized equilibria can be made arbitrarily close to Nash equilibria, our result opens the door to a new perspective to solving two-player zero-sum games.
  arXiv  Detail & Related papers  (2023-01-22T16:54:06Z)
• Global Nash Equilibrium in Non-convex Multi-player Game: Theory and Algorithms [66.8634598612777]
  We show that Nash equilibrium (NE) is acceptable to all players in a multi-player game. We also show that no one can benefit unilaterally from the general theory step by step.
  arXiv  Detail & Related papers  (2023-01-19T11:36:50Z)
• Learning Correlated Equilibria in Mean-Field Games [62.14589406821103]
  We develop the concepts of Mean-Field correlated and coarse-correlated equilibria. We show that they can be efficiently learnt in emphall games, without requiring any additional assumption on the structure of the game.
  arXiv  Detail & Related papers  (2022-08-22T08:31:46Z)
• Towards convergence to Nash equilibria in two-team zero-sum games [17.4461045395989]
  Two-team zero-sum games are defined as multi-player games where players are split into two competing sets of agents. We focus on the solution concept of Nash equilibria (NE) We show that computing NE for this class of games is $textithard$ for the complexity class $mathrm$.
  arXiv  Detail & Related papers  (2021-11-07T21:15:35Z)
• Learning to Compute Approximate Nash Equilibrium for Normal-form Games [15.321036952379488]
  We propose a general meta learning approach to computing approximate Nash equilibrium for finite $n$-player normal-form games. Unlike existing solutions that approximate or learn a Nash equilibrium from scratch for each of the games, our meta solver directly constructs a mapping from a game utility matrix to a joint strategy profile.
  arXiv  Detail & Related papers  (2021-08-17T07:06:46Z)
• No-regret learning and mixed Nash equilibria: They do not mix [64.37511607254115]
  We study the dynamics of "follow-the-regularized-leader" (FTRL) We show that any Nash equilibrium which is not strict cannot be stable and attracting under FTRL. This result has significant implications for predicting the outcome of a learning process.
  arXiv  Detail & Related papers  (2020-10-19T13:49:06Z)
• Chaos, Extremism and Optimism: Volume Analysis of Learning in Games [55.24050445142637]
  We present volume analyses of Multiplicative Weights Updates (MWU) and Optimistic Multiplicative Weights Updates (OMWU) in zero-sum as well as coordination games. We show that OMWU contracts volume, providing an alternative understanding for its known convergent behavior. We also prove a no-free-lunch type of theorem, in the sense that when examining coordination games the roles are reversed: OMWU expands volume exponentially fast, whereas MWU contracts.
  arXiv  Detail & Related papers  (2020-05-28T13:47:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.