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={C. {\`A}lvarez and J. Gabarr{\'o} and M. Serna},
  journal={Electron. Colloquium Comput. Complex.},
  year={2005}
}
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… Expand
Polynomial Space Suffices for Deciding Nash Equilibria Properties for Extensive Games with Large Trees,
TLDR
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. Expand
Pure Nash equilibria: hard and easy games
TLDR
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. Expand
The Computational Complexity of Nash Equilibria in Concisely Represented Games
TLDR
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. Expand
Symmetric games with piecewise linear utilities
TLDR
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. Expand
On the complexity of constrained Nash equilibria in graphical games
TLDR
A formal framework for specifying these kinds of requirement is introduced and investigated in the context of graphical games, where a player p may directly be interested in some of the other players only, called the neighbors of p, and the complexity of deciding the existence and of computing constrained equilibria is investigated. Expand
Complexity of mixed equilibria in Boolean games
TLDR
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. Expand
Rational Generating Functions and Integer Programming Games
TLDR
This work presents efficient algorithms for enumerating all pure Nash equilibria, and other computations of interest, such as the pure price of anarchy and pure threat point, when the dimension and number of “convex” linear pieces in the payoff functions are fixed. Expand
Integrated Project Member of the FET Proactive Initiative Complex Systems DELIS-TR-404 Equilibria problems on Games: Complexity versus succintnes
We study the computational complexity of problems involving equilibria in strategic games and in perfect information extensive games when the number of players is large. We consider, among others,Expand
Weighted Boolean Formula Games
TLDR
This work introduces a new class of succinct games, called weighted boolean formula games, which make a natural mutuality assumption on the payoffs of distinct players, and proves that each weighted, linear-affine (network) congestion game with player-specific coefficients and constants is polynomial, sound Nash-Harasanyi-Selten homomorphic to a weighted Boolean formula game. Expand
The complexity of game isomorphism
TLDR
The question of whether two multiplayer strategic games are equivalent and the computational complexity of deciding such a property is addressed and two notions of isomorphisms, strong and weak, are introduced. Expand
...
1
2
...

References

SHOWING 1-10 OF 24 REFERENCES
Pure Nash equilibria: hard and easy games
TLDR
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. Expand
The computational complexity of nash equilibria in concisely represented games
TLDR
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. Expand
On the NP-completeness of finding an optimal strategy in games with common payoffs
TLDR
It is shown that the problem of determining whether there exists a joint strategy where each player has an expected payoff of at least r is NP-complete as a function of the number of nodes in the extensive-form representation of the game. Expand
Complexity Results about Nash Equilibria
TLDR
A single reduction demonstrates NP- hardness of determining whether Nash equilibria with certain natural properties exist, and demonstrates the NP-hardness of counting NashEquilibria (or connected sets of Nash Equilibria). Expand
The complexity of two-person zero-sum games in extensive form
Abstract This paper investigates the complexity of finding max-min strategies for finite two-person zero-sum games in the extensive form. The problem of determining whether a player with imperfectExpand
The Complexity of Games on Highly Regular Graphs
TLDR
Algorithms and complexity results are presented for the problem of finding equilibria in games with extremely succinct description that are defined on highly regular graphs such as the d-dimensional grid; it is argued that such games are of interest in the modelling of large systems of interacting agents. Expand
Nash and correlated equilibria: Some complexity considerations
This paper deals with the complexity of computing Nash and correlated equilibria for a finite game in normal form. We examine the problems of checking the existence of equilibria satisfying a certainExpand
Nash Equilibria in Discrete Routing Games with Convex Latency Functions
TLDR
This work presents a complete characterization of the instances for which a fully mixed Nash equilibrium exists, and proves that (in case of its existence) it is unique. Expand
The complexity of pure Nash equilibria
TLDR
This work focuses on congestion games, and shows that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLS-complete in general. Expand
The structure and complexity of Nash equilibria for a selfish routing game
TLDR
This work provides a comprehensive collection of efficient algorithms, hardness results (both as NP-hardness and #P-completeness results), and structural results for these algorithmic problems related to the computation of Nash equilibria for the selfish routing game the authors consider. Expand
...
1
2
3
...