AWESOME: A General Multiagent Learning Algorithm that Converges in Self-Play and Learns a Best Response Against Stationary Opponents

A satisfactory multiagent learning algorithm should, {\em at a minimum}, learn to play optimally against stationary opponents and converge to a Nash equilibrium in self-play. The algorithm that has come closest, WoLF-IGA, has been proven to have these two properties in 2-player 2-action repeated games--assuming that the opponent's (mixed) strategy is observable. In this paper we present AWESOME, the first algorithm that is guaranteed to have these two properties in {\em all} repeated (finite) games. It requires only that the other players' actual actions (not their strategies) can be observed at each step. It also learns to play optimally against opponents that {\em eventually become} stationary. The basic idea behind AWESOME ({\em Adapt When Everybody is Stationary, Otherwise Move to Equilibrium}) is to try to adapt to the others' strategies when they appear stationary, but otherwise to retreat to a precomputed equilibrium strategy. The techniques used to prove the properties of AWESOME are fundamentally different from those used for previous algorithms, and may help in analyzing other multiagent learning algorithms also

BL-WoLF: A Framework For Loss-Bounded Learnability In Zero-Sum Games

We present BL-WoLF, a framework for learnability in repeated zero-sum games where the cost of learning is measured by the losses the learning agent accrues (rather than the number of rounds). The game is adversarially chosen from some family that the learner knows. The opponent knows the game and the learner's learning strategy. The learner tries to either not accrue losses, or to quickly learn about the game so as to avoid future losses (this is consistent with the Win or Learn Fast (WoLF) principle; BL stands for ``bounded loss''). Our framework allows for both probabilistic and approximate learning. The resultant notion of {\em BL-WoLF}-learnability can be applied to any class of games, and allows us to measure the inherent disadvantage to a player that does not know which game in the class it is in. We present {\em guaranteed BL-WoLF-learnability} results for families of games with deterministic payoffs and families of games with stochastic payoffs. We demonstrate that these families are {\em guaranteed approximately BL-WoLF-learnable} with lower cost. We then demonstrate families of games (both stochastic and deterministic) that are not guaranteed BL-WoLF-learnable. We show that those families, nevertheless, are {\em BL-WoLF-learnable}. To prove these results, we use a key lemma which we derive

Settling Some Open Problems on 2-Player Symmetric Nash Equilibria

Over the years, researchers have studied the complexity of several decision versions of Nash equilibrium in (symmetric) two-player games (bimatrix games). To the best of our knowledge, the last remaining open problem of this sort is the following; it was stated by Papadimitriou in 2007: find a non-symmetric Nash equilibrium (NE) in a symmetric game. We show that this problem is NP-complete and the problem of counting the number of non-symmetric NE in a symmetric game is #P-complete. In 2005, Kannan and Theobald defined the "rank of a bimatrix game" represented by matrices (A, B) to be rank(A+B) and asked whether a NE can be computed in rank 1 games in polynomial time. Observe that the rank 0 case is precisely the zero sum case, for which a polynomial time algorithm follows from von Neumann's reduction of such games to linear programming. In 2011, Adsul et. al. obtained an algorithm for rank 1 games; however, it does not solve the case of symmetric rank 1 games. We resolve this problem