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

@article{Rubinstein2016SettlingTC,
title={Settling the Complexity of Computing Approximate Two-Player Nash Equilibria},
journal={2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)},
year={2016},
pages={258-265}
}
• A. Rubinstein
• Published 14 June 2016
• Computer Science
• 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
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 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…
66 Citations
Smoothed Complexity of 2-player Nash Equilibria
• Economics
2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
• 2020
We prove that computing a Nash equilibrium of a two-player ($n\times n$) game with payoffs in [−1, 1] is PPAD-hard (under randomized reductions) even in the smoothed analysis setting, smoothing with
Simple Approximate Equilibria in Games with Many Players
• Economics, Computer Science
EC
• 2017
This work considers ε-equilibria notions for a constant value of ε in n-player m-action games and proves its equivalence to the well-known Beck-Fiala conjecture from discrepancy theory, the first result that introduces a connection between game theory and discrepancy theory.
An Improved Quasi-Polynomial Algorithm for Approximate Well-Supported Nash Equilibria
• Computer Science
AAAI
• 2019
A much better quasi-polynomial time algorithm that computes a (1/2 + e)-well-supported Nash equilibrium in time nO(log logn1/e/e2), for any e > 0, is proposed.
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.
Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games
• Economics, Mathematics
SAGT
• 2021
This work develops a query-efficient reduction from more general games to Lipschitz games, and provides an exponential lower bound on the deterministic query complexity of finding -approximate correlated equilibria of n-player, m-action, λ-Lipschitzer games for strong values of , motivating the consideration of explicitly randomized algorithms in the above results.
Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries
• Economics, Computer Science
ACM Trans. Economics and Comput.
• 2016
It is shown that randomized algorithms require Ω(k2) payoff queries in order to find an ϵ-Nash equilibrium with ϵ < 1/4k, even in zero-one constant-sum games, which rules out query-efficient randomized algorithms for finding exact Nash equilibria.
Playing Anonymous Games using Simple Strategies
• Computer Science, Mathematics
SODA
• 2017
The approach 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.
A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games
• Computer Science
ArXiv
• 2022
A new refinement of the Tsaknakis-Spirakis algorithm is proposed, resulting in a polynomial-time algorithm that computes a ( 1 3 +δ)-Nash equilibrium, for any constant δ > 0.3393 + δ.
Distributed Methods for Computing Approximate Equilibria
• Computer Science, Economics
WINE
• 2016
A new, distributed method to compute approximate Nash equilibria in bimatrix games by solving a single LP that combines the two players' payoffs, which can be applied to give an improved bound in the limited communication setting and to obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication.

## References

SHOWING 1-10 OF 96 REFERENCES
Settling the Complexity of Computing Approximate Two-Player Nash Equilibria
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
A note on approximate Nash equilibria
• Economics
Theor. Comput. Sci.
• 2009
Settling the complexity of computing two-player Nash equilibria
• Economics
JACM
• 2009
We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by
On the complexity of approximating a Nash equilibrium
We show that computing a relative---that is, multiplicative as opposed to additive---approximate Nash equilibrium in two-player games is PPAD-complete, even for constant values of the approximation.
Approximating the best Nash Equilibrium in no(log n)-time breaks the Exponential Time Hypothesis
• Economics
Electron. Colloquium Comput. Complex.
• 2014
A reduction from the PCP machinery to finding Nash equilibrium via free games, the framework introduced in the recent work by Aaronson, Impagliazzo and Moshkovitz is introduced, and the lower bound matches the quasi-polynomial time algorithm by Lipton, Markakis and Mehta for solving the problem.
On oblivious PTAS's for nash equilibrium
• Mathematics, Computer Science
STOC '09
• 2009
It is proved that any oblivious PTAS for anonymous games with two strategies and three player types must have 1/εα in the exponent of the running time for some α ≥ 1/3, rendering the algorithm in [Daskalakis 2008] (which works with any bounded number of player types) essentially optimal within oblivious algorithms.
How hard is it to approximate the best Nash equilibrium?
• Economics
SODA
• 2009
The quest for a PTAS for Nash equilibrium in a two-player game seeks to circumvent the PPAD-completeness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a
Inapproximability of NP-Complete Variants of Nash Equilibrium
• Economics
APPROX-RANDOM
• 2011
It is shown that optimal Nash equilibrium is just one of several known NP-hard problems related to Nash equilibrium, all of which have approximate variants which are as hard as finding a planted clique, and for general Bayesian games the problem is NP- hard.
Playing large games using simple strategies
• Economics
EC '03
• 2003
The existence of ε-Nash equilibrium strategies with support logarithmic in the number of pure strategies is proved and it is proved that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support.