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

On Similarities between Inference in Game Theory and Machine Learning

In this paper, we elucidate the equivalence between inference in game theory and machine learning. Our aim in so doing is to establish an equivalent vocabulary between the two domains so as to facilitate developments at the intersection of both fields, and as proof of the usefulness of this approach, we use recent developments in each field to make useful improvements to the other. More specifically, we consider the analogies between smooth best responses in fictitious play and Bayesian inference methods. Initially, we use these insights to develop and demonstrate an improved algorithm for learning in games based on probabilistic moderation. That is, by integrating over the distribution of opponent strategies (a Bayesian approach within machine learning) rather than taking a simple empirical average (the approach used in standard fictitious play) we derive a novel moderated fictitious play algorithm and show that it is more likely than standard fictitious play to converge to a payoff-dominant but risk-dominated Nash equilibrium in a simple coordination game. Furthermore we consider the converse case, and show how insights from game theory can be used to derive two improved mean field variational learning algorithms. We first show that the standard update rule of mean field variational learning is analogous to a Cournot adjustment within game theory. By analogy with fictitious play, we then suggest an improved update rule, and show that this results in fictitious variational play, an improved mean field variational learning algorithm that exhibits better convergence in highly or strongly connected graphical models. Second, we use a recent advance in fictitious play, namely dynamic fictitious play, to derive a derivative action variational learning algorithm, that exhibits superior convergence properties on a canonical machine learning problem (clustering a mixture distribution)

From Weak Learning to Strong Learning in Fictitious Play Type Algorithms

The paper studies the highly prototypical Fictitious Play (FP) algorithm, as well as a broad class of learning processes based on best-response dynamics, that we refer to as FP-type algorithms. A well-known shortcoming of FP is that, while players may learn an equilibrium strategy in some abstract sense, there are no guarantees that the period-by-period strategies generated by the algorithm actually converge to equilibrium themselves. This issue is fundamentally related to the discontinuous nature of the best response correspondence and is inherited by many FP-type algorithms. Not only does it cause problems in the interpretation of such algorithms as a mechanism for economic and social learning, but it also greatly diminishes the practical value of these algorithms for use in distributed control. We refer to forms of learning in which players learn equilibria in some abstract sense only (to be defined more precisely in the paper) as weak learning, and we refer to forms of learning where players' period-by-period strategies converge to equilibrium as strong learning. An approach is presented for modifying an FP-type algorithm that achieves weak learning in order to construct a variant that achieves strong learning. Theoretical convergence results are proved.Comment: 22 page

On Robustness Properties in Empirical Centroid Fictitious Play

Empirical Centroid Fictitious Play (ECFP) is a generalization of the well-known Fictitious Play (FP) algorithm designed for implementation in large-scale games. In ECFP, the set of players is subdivided into equivalence classes with players in the same class possessing similar properties. Players choose a next-stage action by tracking and responding to aggregate statistics related to each equivalence class. This setup alleviates the difficult task of tracking and responding to the statistical behavior of every individual player, as is the case in traditional FP. Aside from ECFP, many useful modifications have been proposed to classical FP, e.g., rules allowing for network-based implementation, increased computational efficiency, and stronger forms of learning. Such modifications tend to be of great practical value; however, their effectiveness relies heavily on two fundamental properties of FP: robustness to alterations in the empirical distribution step size process, and robustness to best-response perturbations. The main contribution of the paper is to show that similar robustness properties also hold for the ECFP algorithm. This result serves as a first step in enabling practical modifications to ECFP, similar to those already developed for FP.Comment: Submitted for publication. Initial Submission: Mar. 201

Convergent learning algorithms for potential games with unknown noisy rewards

In this paper, we address the problem of convergence to Nash equilibria in games with rewards that are initially unknown and which must be estimated over time from noisy observations. These games arise in many real-world applications, whenever rewards for actions cannot be prespecified and must be learned on-line. Standard results in game theory, however, do not consider such settings. Specifically, using results from stochastic approximation and differential inclusions, we prove the convergence of variants of fictitious play and adaptive play to Nash equilibria in potential games and weakly acyclic games, respectively. These variants all use a multi-agent version of Q-learning to estimate the reward functions and a novel form of the e-greedy decision rule to select an action. Furthermore, we derive e-greedy decision rules that exploit the sparse interaction structure encoded in two compact graphical representations of games, known as graphical and hypergraphical normal form, to improve the convergence rate of the learning algorithms. The structure captured in these representations naturally occurs in many distributed optimisation and control applications. Finally, we demonstrate the efficacy of the algorithms in a simulated ad hoc wireless sensor network management problem

Joint strategy fictitious play with inertia for potential games

We consider multi-player repeated games involving a large number of players with large strategy spaces and enmeshed utility structures. In these ldquolarge-scalerdquo games, players are inherently faced with limitations in both their observational and computational capabilities. Accordingly, players in large-scale games need to make their decisions using algorithms that accommodate limitations in information gathering and processing. This disqualifies some of the well known decision making models such as ldquoFictitious Playrdquo (FP), in which each player must monitor the individual actions of every other player and must optimize over a high dimensional probability space. We will show that Joint Strategy Fictitious Play (JSFP), a close variant of FP, alleviates both the informational and computational burden of FP. Furthermore, we introduce JSFP with inertia, i.e., a probabilistic reluctance to change strategies, and establish the convergence to a pure Nash equilibrium in all generalized ordinal potential games in both cases of averaged or exponentially discounted historical data. We illustrate JSFP with inertia on the specific class of congestion games, a subset of generalized ordinal potential games. In particular, we illustrate the main results on a distributed traffic routing problem and derive tolling procedures that can lead to optimized total traffic congestion