# Settling the Complexity of Computing Approximate Two-Player Nash Equilibria

@inproceedings{Rubinstein2016SettlingTC,
title={Settling the Complexity of Computing Approximate Two-Player Nash Equilibria},
author={Aviad Rubinstein},
booktitle={FOCS},
year={2016}
}
We prove that there exists a constant e > 0 such that, assuming the Exponential Time Hypothesis for PPAD, computing an e-approximate Nash equilibrium in a two-player (n × n) game requires quasi-polynomial time, nlog1-o(1) n. This matches (up to the o(1) term) the algorithm of Lipton, Markakis, and Mehta [54]. Our proof relies on a variety of techniques from the study of probabilistically checkable proofs (PCP), this is the first time that such ideas are used for a reduction between problems…
Settling the Complexity of Computing Approximate Two-Player Nash Equilibria
• A. Rubinstein
• Computer Science
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
• 2016
We prove that there exists a constant ε > 0 such that, assuming the Exponential Time Hypothesis for PPAD, computing an ε-approximate Nash equilibrium in a two-player (n × n) game requires
Explorer Playing Anonymous Games using Simple Strategies
• Computer Science, Mathematics
• 2016
This work exploits the connection between Nash equilibria in anonymous games and Poisson multinomial distributions and proves a new probabilistic lemma establishing the following: Two PMDs, with large variance in each direction, whose first few moments are approximately matching are close in total variation distance.
Sum-of-squares meets Nash: lower bounds for finding any equilibrium
• Computer Science
STOC
• 2018
This work proposes a framework of roundings for the sum-of-squares algorithm (and convex relaxations in general) applicable to finding approximate/exact equilbria in two player bimatrix games and strengthens the classical unconditional lower bound against enumerative algorithms for finding approximate equilibria due to Daskalakis-Papadimitriou and the classical hardness of computing equilibira due to Gilbow-Zemel.
Bounds for the Communication Complexity of Two-Player Approximate Correlated Equilibria
• Computer Science
Electron. Colloquium Comput. Complex.
• 2017
This paper provides a communication protocol that outputs a ε-approximate correlated equilibrium after exchanging Õ(nε−2) bits, saving over the naive protocol which requires O(n)-bits, and shows that the dependence on the number of players is unavoidable.
Approximate Nash Equilibria of Imitation Games: Algorithms and Complexity
• Economics
AAMAS
• 2020
It is shown that much like the general case, for any c > 0, computing a 1 nc -approximate NE of imitation games remains PPADhard, where n is the number of moves available to the players and a polynomial-time algorithm is designed to find ε-approximates NE for any given constant ε > 0 (PTAS).
Distributed Methods for Computing Approximate Equilibria
• Computer Science, Economics
Algorithmica
• 2018
A new, distributed method to compute approximate Nash equilibria in bimatrix games that first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then computes an approximate Nash equilibrium using only limited communication between the players.
Sum-of-Squares meets Nash: Optimal Lower Bounds for Finding any Equilibrium
• Computer Science
Electron. Colloquium Comput. Complex.
• 2018
This work presents an algorithmic model based on the sum-of-squares (SoS) hierarchy that allows escaping this inherent limitation of integrality gaps of computing equilibria and shows a lower bound that matches these upper bound up to constant factors in the exponent.
Bounds for the Communication Complexity of Approximate Correlated Equilibria
• Computer Science
• 2017
This paper provides a communication protocol that outputs a ε-approximate correlated equilibrium for multiplayer multi-action games after exchanging Õ(mn4ε−4) bits, saving over the naive O(mn)-bits protocol when the number of players is large.
Hardness of Approximation Between P and NP
• A. Rubinstein
• Economics
Hardness of Approximation Between P and NP
• 2017
This book provides strong evidence that even finding an approximate Nash equilibrium is intractable, and proves several intractability theorems for different settings (two-player games and many- player games) and models (computational complexity, query complexity, and communication complexity).
Zero-Sum Game Techniques for Approximate Nash Equilibria
• Economics, Computer Science
AAMAS
• 2017
It is proved that computing Nash equilibria in bimatrix games is PPAD-complete even if both of the payoff matrices are symmetric, which motivates the interest in computing additive approximate NashEquilibria efficiently for bim atrix games with symmetric payoffmatrices.

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