# 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…

## 34 Citations

Settling the Complexity of Computing Approximate Two-Player Nash Equilibria

- Computer Science2016 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 ScienceSTOC
- 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 ScienceElectron. 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

- EconomicsAAMAS
- 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, EconomicsAlgorithmica
- 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 ScienceElectron. 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

- EconomicsHardness 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 ScienceAAMAS
- 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.

## References

SHOWING 1-10 OF 65 REFERENCES

Settling the complexity of computing two-player Nash equilibria

- EconomicsJACM
- 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…

Approximating the best Nash Equilibrium in no(log n)-time breaks the Exponential Time Hypothesis

- EconomicsElectron. 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 the complexity of approximating a Nash equilibrium

- Computer ScienceSODA '11
- 2011

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.…

How hard is it to approximate the best Nash equilibrium?

- EconomicsSODA
- 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…

On oblivious PTAS's for nash equilibrium

- Mathematics, Computer ScienceSTOC '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.

Computing Approximate Nash Equilibria in Polymatrix Games

- Economics, Computer ScienceWINE
- 2014

The main result is that an (0.5 + δ)-Nash equilibrium of an n-player polymatrix game can be computed in time polynomial in the input size and \(\frac{1}{\delta}\).

Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games

- Computer ScienceITCS
- 2017

It is proved that, for some constant $\epsilon>0$, any randomized oracle algorithm that computes an $\epSilon$-ANE in a binary-action, $n$-player game must make $2^{\Omega(n/\log n)}$ payoff queries.

Playing large games using simple strategies

- EconomicsEC '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.