The Complexity of Games on Highly Regular Graphs (original) (raw)

Equilibria problems on games: Complexity versus succinctness

Journal of Computer and System Sciences, 2011

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, the problems of deciding the existence of a Pure Nash Equilibrium in strategic games or deciding the existence of a pure Nash or a subgame perfect Nash equilibrium with a given payoff in finite perfect information extensive games. We address the fundamental question of how can we represent a game with a large number of players? We propose three ways of representing a game with different degrees of succinctness for the components of the game. For perfect information extensive games we show that when the number of moves of each player is large and the input game is represented succinctly these problems are PSPACE-complete. In contraposition, when the game is described explicitly by means of its associated tree all these problems are decidable in polynomial time. For strategic games we show that the complexity of deciding the existence of a Pure Nash equilibrium depends on the succinctness of the game representation and then on the size of the action sets. In particular we show that it is NP-complete when the number of players is large and the number of actions for each player is constant, while the problem is Σ p 2-complete when the number of players is a constant and the size of the action sets is exponential.

On the Complexity of Nash Equilibria of Action-Graph Games

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2009

We consider the problem of computing Nash Equilibria of action-graph games (AGGs). AGGs, introduced by Bhat and Leyton-Brown, is a succinct representation of games that encapsulates both 'local'dependencies as in graphical games, and partial indifference to other agents' identities as in anonymous games, which occur in many natural settings. This is achieved by specifying a graph on the set of actions, so that the payoff of an agent for selecting a strategy depends only on the number of agents playing each of the neighboring strategies in the action graph. We present a Polynomial Time Approximation Scheme for computing mixed Nash equilibria of AGGs with constant treewidth and a constant number of agent types (and an arbitrary number of strategies), together with hardness results for the cases when either the treewidth or the number of agent types is unconstrained. In particular, we show that even if the action graph is a tree, but the number of agent-types is unconstrained, it is NP-complete to decide the existence of a pure-strategy Nash equilibrium and PPAD-complete to compute a mixed Nash equilibrium (even an approximate one); similarly for symmetric AGGs (all agents belong to a single type), if we allow arbitrary treewidth. These hardness results suggest that, in some sense, our PTAS is as strong of a positive result as one can expect.

The computational complexity of Nash equilibria in concisely represented games

2006

Abstract Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows.

Nash and correlated equilibria: Some complexity considerations

Games and Economic Behavior, 1989

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 certain condition, such as "Given a game G and a number r, is there a Nash (correlated) equilibrium of G in which all players obtain an expected payoff of at least r?" or "Is there a unique Nash (correlated) equilibrium in G?" etc. We show that such problems are typically "hard" (NP-hard) for Nash equilibria but "easy" (polynomial) for correlated equilibria.

Settling the complexity of computing two-player Nash equilibria

Journal of The ACM, 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 Papadimitriou in 1991. Our result, building upon the work of Daskalakis et al. [2006a] on the complexity of four-player Nash equilibria, settles a long standing open problem in algorithmic game theory.

Pure Nash equilibria: complete characterization of hard and easy graphical games

2010

We consider the computational complexity of pure Nash equilibria in graphical games. It is known that the problem is NP-complete in general, but tractable (ie, in P) for special classes of graphs such as those with bounded treewidth. It is then natural to ask: is it possible to characterize all tractable classes of graphs for this problem? In this work, we provide such a characterization for the case of bounded in-degree graphs, thereby resolving the gap between existing hardness and tractability results. In particular, we analyze the ...

New results on the complexity of uniformly mixed Nash equilibria

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2005

We are interested in the complexity of finding Nash equilibria with one uniformly mixed strategy (that is, equilibria in which at least one played strategy is a uniform probability function). We show that, even in imitation bimatrix games, where one player has a positive payoff if he plays the same pure strategy as the opponent, deciding the existence of such an equilibrium is an NP-complete problem. This main result follows from the NP-completeness of graph-theoretical problems strictly related to this class of equilibria.

Computing nash equilibria gets harder: New results show hardness even for parameterized complexity

2009

In this paper we show that some decision problems regarding the computation of Nash equilibria are to be considered particularly hard. Most decision problems regarding Nash equilibria have been shown to be NP-complete. While some NP-complete problems can find an alternative to tractability with the tools of Parameterized Complexity Theory, it is also the case that some classes of problems do not seem to have fixed-parameter tractable algorithms. We show that k-Uniform Nash and k-Minimal Nash support are W [2]-hard. Given a game G=(A,B) and a nonnegative integer k, the k-Uniform Nash problem asks whether G has a uniform Nash equilibrium of size k. The k-Minimal Nash support asks whether G has Nash equilibrium such that the support of each player's Nash strategy has size equal to or less than k. First, we show that k-Uniform Nash (with k as the parameter) is W [2]-hard even when we have 2 players, or fewer than 4 different integer values in the matrices. Second, we illustrate that even in zerosum games k-Minimal Nash support is W [2]-hard (a sample Nash equilibrium in a zero-sum 2-player game can be found in polynomial time (von Stengel 2002)). Thus, it must be the case that other more general decision problems are also W [2]-hard. Therefore, the possible parameters for fixed parameter tractability in those decision problems regarding Nash equilibria seem elusive.

The complexity of computing a Nash equilibrium

Communications of the ACM, 2009

How long does it take until economic agents converge to an equilibrium? By studying the complexity of the problem of computing a mixed Nash equilibrium in a game, we provide evidence that there are games in which convergence to such an equilibrium takes prohibitively long. Traditionally, computational problems fall into two classes: those that have a polynomial-time algorithm and those that are NP-hard. However, the concept of NP-hardness cannot be applied to the rare problems where "every instance has a solution"---for example, in the case of games Nash's theorem asserts that every game has a mixed equilibrium (now known as the Nash equilibrium, in honor of that result). We show that finding a Nash equilibrium is complete for a class of problems called PPAD, containing several other known hard problems; all problems in PPAD share the same style of proof that every instance has a solution.