# Pure Nash Equilibria in Games with a Large Number of Actions

@article{lvarez2005PureNE, title={Pure Nash Equilibria in Games with a Large Number of Actions}, author={Carme {\`A}lvarez and Joaquim Gabarr{\'o} and Maria J. Serna}, journal={Electron. Colloquium Comput. Complex.}, year={2005}, volume={TR05} }

We study the computational complexity of deciding the existence of a Pure Nash Equilibrium in multi-player strategic games. We address two fundamental questions: how can we represent a game? and how can we represent a game with polynomial pay-off functions? Our results show that the computational complexity of deciding the existence of a pure Nash equilibrium in a strategic game depends on two parameters: the number of players and the size of the sets of strategies. In particular we show that…

## 26 Citations

### Polynomial Space Suffices for Deciding Nash Equilibria Properties for Extensive Games with Large Trees,

- EconomicsISAAC
- 2005

This paper proposes three ways of representing a game with different degrees of succinctness for the components of the game and shows that when the number of moves of each player is large and the player function and the utilities are represented succinctly the considered problems are PSPACE-complete.

### Pure Nash equilibria: hard and easy games

- EconomicsTARK '03
- 2003

It is shown that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether agame has a strong Nash equilibrium is ΣP2-complete, and practically relevant restrictions that lower the complexity are studied.

### The Computational Complexity of Nash Equilibria in Concisely Represented Games

- EconomicsTOCT
- 2012

Two models of concisely represented games are studied: circuit games, where the payoffs are computed by a given boolean circuit, and graphGames, where each agent’s payoff is a function of only the strategies played by its neighbors in a given graph.

### On the Complexity of Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games

- EconomicsMath. Oper. Res.
- 2008

It is proved that it actually is strongly NP-hard to determine whether a given weighted network congestion game has a pure-strategy Nash equilibrium, regardless of whether flow is unsplittable (has to be routed on a single path for each player) or not.

### Pairwise-Interaction Games

- EconomicsICALP
- 2011

It is shown that pairwise-interaction games form a proper subclass of the usual graphical games, and a new defective graph colouring problem called Nash colouring is defined, which is of independent interest, and its decision version is NP-complete.

### Symmetric games with piecewise linear utilities

- EconomicsBQGT
- 2010

This work gives polynomial-time algorithms to count the number of PSNE (thus deciding if such an equilibrium exists) and to find a sample PSNE, when one exists, and focuses on a natural representation of utility as piecewise-linear functions, and shows that such a representation has nice computational properties.

### Computing pure strategy nash equilibria in compact symmetric games

- Computer ScienceEC '10
- 2010

This work analyzes the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games with a fixed number of actions, and shows that in the general case, where utility functions are represented as arbitrary circuits, the problem of deciding the existence of PSNE is NP-complete.

### On the complexity of constrained Nash equilibria in graphical games

- EconomicsTheor. Comput. Sci.
- 2009

### Complexity of mixed equilibria in Boolean games

- EconomicsArXiv
- 2017

The present work focuses on the complexity of algorithmic problems dealing with mixed strategies in Boolean games and shows that the problem of determining whether a two-player game has an equilibrium satisfying a given payoff constraint is NEXP-complete.

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