Connectivity and equilibrium in random games

We study how the structure of the interaction graph of a game affects the existence of pure Nash equilibria. In particular, for a fixed interaction graph, we are interested in whether there are pure Nash equilibria arising when random utility tables are assigned to the players. We provide conditions for the structure of the graph under which equilibria are likely to exist and complementary conditions which make the existence of equilibria highly unlikely. Our results have immediate implications for many deterministic graphs and generalize known results for random games on the complete graph. In particular, our results imply that the probability that bounded degree graphs have pure Nash equilibria is exponentially small in the size of the graph and yield a simple algorithm that finds small nonexistence certificates for a large family of graphs. Then we show that in any strongly connected graph of n vertices with expansion $(1+\Omega(1))\log_2(n)$ the distribution of the number of equilibria approaches the Poisson distribution with parameter 1, asymptotically as $n \to +\infty$.Comment: Published in at http://dx.doi.org/10.1214/10-AAP715 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Pure Nash Equilibria: Hard and Easy Games

We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (the actions of) a logarithmically small number of other players. The dependency structure of a game G can be expressed by a graph DG(G) or by a hypergraph H(G). By relating Nash equilibrium problems to constraint satisfaction problems (CSPs), we show that if G has small neighborhood and if H(G) has bounded hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class NC2 of highly parallelizable problems

A Complete Solver for Constraint Games

Game Theory studies situations in which multiple agents having conflicting objectives have to reach a collective decision. The question of a compact representation language for agents utility function is of crucial importance since the classical representation of a $n$-players game is given by a $n$-dimensional matrix of exponential size for each player. In this paper we use the framework of Constraint Games in which CSP are used to represent utilities. Constraint Programming --including global constraints-- allows to easily give a compact and elegant model to many useful games. Constraint Games come in two flavors: Constraint Satisfaction Games and Constraint Optimization Games, the first one using satisfaction to define boolean utilities. In addition to multimatrix games, it is also possible to model more complex games where hard constraints forbid certain situations. In this paper we study complete search techniques and show that our solver using the compact representation of Constraint Games is faster than the classical game solver Gambit by one to two orders of magnitude.Comment: 17 page

A distributed asynchronous solver for Nash Equilibria in hypergraphical games

Hypergraphical games provides a compact model of a network of self-interested agents, each involved in simultaneous subgames with its neighbors. The overall aim is for the agents in the network to reach a Nash Equilibrium, in which no agent has an incentive to change their response, but without revealing all their private information. Asymmetric Distributed constraint satisfaction (ADisCSP) has been proposed as a solution to this search problem. In this paper, we propose a new model of hypergraphical games as an ADisCSP based on a new global constraint, and a new asynchronous algorithm for solving ADisCSP that is able to find a Nash Equilibrium. We show empirically that we significantly reduce both message passing and computation time, achieving an order of magnitude improvement in messaging and in non-concurrent computation time on dense problems compared to state-of-the art algorithms