---
title: "Complexity Results about Nash Equilibria"
authors:
  - "Vincent Conitzer"
  - "Tuomas Sandholm"
doi: ""
url: "https://ijcai.org/proceedings/2003/111"
pdf: "https://ijcai.org/proceedings/2003/0111.pdf"
venue: "Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence"
year: 2003
pages: "765-770"
track: ""
keywords:
---

# Complexity Results about Nash Equilibria

Vincent Conitzer, Tuomas Sandholm

*Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence*, pages 765-770.

## Abstract

Noncooperative game theory provides a normative framework for analyzing strategic interactions. However, for the toolbox to be operational, the solutions it defines will have to be computed. In this paper, we provide a single reduction that 1) demonstrates NP-hardness of determining whether Nash equilibria with certain natural properties exist, and 2) demonstrates the #P-hardness of counting Nash equilibria (or connected sets of Nash equilibria). We also show that 3) determining whether a purestrategy Bayes-Nash equilibrium exists is NPhard, and that 4) determining whether a purestrategy Nash equilibrium exists in a stochastic (Markov) game is PSPAC£-hard even ifthe game is invisible (this remains NP-hard if the game is finite). All of our hardness results hold even if there are only two players and the game is symmetric.