Smooth Nash Equilibria: Algorithms and Complexity

• URL: http://arxiv.org/abs/2309.12226v2
• Date: Sat, 20 Jul 2024 05:39:44 GMT
• Title: Smooth Nash Equilibria: Algorithms and Complexity
• Authors: Constantinos Daskalakis, Noah Golowich, Nika Haghtalab, Abhishek Shetty,
• Abstract summary: 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.
• Score: 38.08108978808664
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A fundamental shortcoming of the concept of Nash equilibrium is its computational intractability: approximating Nash equilibria in normal-form games is PPAD-hard. In this paper, inspired by the ideas of smoothed analysis, we introduce a relaxed variant of Nash equilibrium called $\sigma$-smooth Nash equilibrium, for a smoothness parameter $\sigma$. 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, which is a distribution that does not put too much mass (as parametrized by $\sigma$) on any fixed action. We distinguish two variants of $\sigma$-smooth Nash equilibria: strong $\sigma$-smooth Nash equilibria, in which players are required to play $\sigma$-smooth strategies under equilibrium play, and weak $\sigma$-smooth Nash equilibria, where there is no such requirement. We show that both weak and strong $\sigma$-smooth Nash equilibria have superior computational properties to Nash equilibria: when $\sigma$ as well as an approximation parameter $\epsilon$ and the number of players are all constants, there is a constant-time randomized algorithm to find a weak $\epsilon$-approximate $\sigma$-smooth Nash equilibrium in normal-form games. In the same parameter regime, there is a polynomial-time deterministic algorithm to find a strong $\epsilon$-approximate $\sigma$-smooth Nash equilibrium in a normal-form game. These results stand in contrast to the optimal algorithm for computing $\epsilon$-approximate Nash equilibria, which cannot run in faster than quasipolynomial-time. We complement our upper bounds by showing that when either $\sigma$ or $\epsilon$ is an inverse polynomial, finding a weak $\epsilon$-approximate $\sigma$-smooth Nash equilibria becomes computationally intractable.

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)
• Differentiable Arbitrating in Zero-sum Markov Games [59.62061049680365]
  We study how to perturb the reward in a zero-sum Markov game with two players to induce a desirable Nash equilibrium, namely arbitrating. The lower level requires solving the Nash equilibrium under a given reward function, which makes the overall problem challenging to optimize in an end-to-end way. We propose a backpropagation scheme that differentiates through the Nash equilibrium, which provides the gradient feedback for the upper level.
  arXiv  Detail & Related papers  (2023-02-20T16:05:04Z)
• 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)
• Approximate Nash Equilibrium Learning for n-Player Markov Games in Dynamic Pricing [0.0]
  We investigate Nash equilibrium learning in a competitive Markov Game (MG) environment. We develop a new model-free method to find approximate Nash equilibria. We demonstrate that an approximate Nash equilibrium can be learned, particularly in the dynamic pricing domain.
  arXiv  Detail & Related papers  (2022-07-13T19:27:07Z)
• Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics [28.815822236291392]
  We show that no game dynamics can converge to $epsilon$-Nash equilibria. We also prove a stronger result for $epsilon$-approximate Nash equilibria.
  arXiv  Detail & Related papers  (2022-03-26T18:27:40Z)
• Multi-Leader Congestion Games with an Adversary [0.5914780964919123]
  We study a multi-leader single-follower congestion game where multiple users (leaders) choose one resource out of a set of resources. We show that pure Nash equilibria may fail to exist and therefore, we consider approximate equilibria instead. We show how to compute efficiently a best approximate equilibrium, that is, with smallest possible $alpha$ among all $alpha$-approximate equilibria of the given instance.
  arXiv  Detail & Related papers  (2021-12-14T14:47:43Z)
• 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)
• Near-Optimal Reinforcement Learning with Self-Play [50.29853537456737]
  We focus on self-play algorithms which learn the optimal policy by playing against itself without any direct supervision. We propose an optimistic variant of the emphNash Q-learning algorithm with sample complexity $tildemathcalO(SAB)$, and a new emphNash V-learning algorithm with sample complexity $tildemathcalO(S(A+B))$.
  arXiv  Detail & Related papers  (2020-06-22T05:00:13Z)

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.