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

Computing Approximate Pure Nash Equilibria in Shapley Value Weighted Congestion Games

We study the computation of approximate pure Nash equilibria in Shapley value (SV) weighted congestion games, introduced in [19]. This class of games considers weighted congestion games in which Shapley values are used as an alternative (to proportional shares) for distributing the total cost of each resource among its users. We focus on the interesting subclass of such games with polynomial resource cost functions and present an algorithm that computes approximate pure Nash equilibria with a polynomial number of strategy updates. Since computing a single strategy update is hard, we apply sampling techniques which allow us to achieve polynomial running time. The algorithm builds on the algorithmic ideas of [7], however, to the best of our knowledge, this is the first algorithmic result on computation of approximate equilibria using other than proportional shares as player costs in this setting. We present a novel relation that approximates the Shapley value of a player by her proportional share and vice versa. As side results, we upper bound the approximate price of anarchy of such games and significantly improve the best known factor for computing approximate pure Nash equilibria in weighted congestion games of [7].Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-71924-5_1

Deep Reinforcement Learning from Self-Play in Imperfect-Information Games

Many real-world applications can be described as large-scale games of imperfect information. To deal with these challenging domains, prior work has focused on computing Nash equilibria in a handcrafted abstraction of the domain. In this paper we introduce the first scalable end-to-end approach to learning approximate Nash equilibria without prior domain knowledge. Our method combines fictitious self-play with deep reinforcement learning. When applied to Leduc poker, Neural Fictitious Self-Play (NFSP) approached a Nash equilibrium, whereas common reinforcement learning methods diverged. In Limit Texas Holdem, a poker game of real-world scale, NFSP learnt a strategy that approached the performance of state-of-the-art, superhuman algorithms based on significant domain expertise.Comment: updated version, incorporating conference feedbac

Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games

We show that there is a polynomial-time approximation scheme for computing Nash equilibria in anonymous games with any fixed number of strategies (a very broad and important class of games), extending the two-strategy result of Daskalakis and Papadimitriou 2007. The approximation guarantee follows from a probabilistic result of more general interest: The distribution of the sum of n independent unit vectors with values ranging over {e1, e2, ...,ek}, where ei is the unit vector along dimension i of the k-dimensional Euclidean space, can be approximated by the distribution of the sum of another set of independent unit vectors whose probabilities of obtaining each value are multiples of 1/z for some integer z, and so that the variational distance of the two distributions is at most eps, where eps is bounded by an inverse polynomial in z and a function of k, but with no dependence on n. Our probabilistic result specifies the construction of a surprisingly sparse eps-cover -- under the total variation distance -- of the set of distributions of sums of independent unit vectors, which is of interest on its own right.Comment: In the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 200