Query Complexity of Approximate Nash Equilibria

We study the query complexity of approximate notions of Nash equilibrium in games with a large number of players $n$. Our main result states that for $n$-player binary-action games and for constant $\varepsilon$, the query complexity of an $\varepsilon$-well-supported Nash equilibrium is exponential in $n$. One of the consequences of this result is an exponential lower bound on the rate of convergence of adaptive dynamics to approxiamte Nash equilibrium

Axiomatic Approach to Solutions of Games

We consider solutions of normal form games that are invariant under strategic equivalence. We consider additional properties that can be expected (or be desired) from a solution of a game, and we observe the following: - Even the weakest notion of individual rationality restricts the set of solutions to be equilibria. This observation holds for all types of solutions: in pure-strategies, in mixed strategies, and in correlated strategies where the corresponding notions of equilibria are pure-Nash, Nash and coarse-correlated. An action profile is (strict) simultaneous maximizer if it simultaneously globally (strictly) maximizes the payoffs of all players. - If we require that a simultaneous maximizer (if it exists) will be a solution, then the solution contains the set of pure Nash equilibria. - There is no solution for which a strict simultaneous maximizer (if it exists) is the unique solution

Query Complexity of Correlated Equilibrium

We study lower bounds on the query complexity of determining correlated equilibrium. In particular, we consider a query model in which an n-player game is specified via a black box that returns players' utilities at pure action profiles. In this model we establish that in order to compute a correlated equilibrium any deterministic algorithm must query the black box an exponential (in n) number of times.Comment: Added reference

Graphical potential games

We study the class of potential games that are also graphical games with respect to a given graph $G$ of connections between the players. We show that, up to strategic equivalence, this class of games can be identified with the set of Markov random fields on $G$. From this characterization, and from the Hammersley-Clifford theorem, it follows that the potentials of such games can be decomposed to local potentials. We use this decomposition to strongly bound the number of strategy changes of a single player along a better response path. This result extends to generalized graphical potential games, which are played on infinite graphs.Comment: Accepted to the Journal of Economic Theor

Approximate Nash Equilibria via Sampling

We prove that in a normal form n-player game with m actions for each player, there exists an approximate Nash equilibrium where each player randomizes uniformly among a set of O(log(m) + log(n)) pure strategies. This result induces an $N^{\log \log N}$ algorithm for computing an approximate Nash equilibrium in games where the number of actions is polynomial in the number of players (m=poly(n)), where $N=nm^n$ is the size of the game (the input size). In addition, we establish an inverse connection between the entropy of Nash equilibria in the game, and the time it takes to find such an approximate Nash equilibrium using the random sampling algorithm