A Unified View of Large-scale Zero-sum Equilibrium Computation

The task of computing approximate Nash equilibria in large zero-sum extensive-form games has received a tremendous amount of attention due mainly to the Annual Computer Poker Competition. Immediately after its inception, two competing and seemingly different approaches emerged---one an application of no-regret online learning, the other a sophisticated gradient method applied to a convex-concave saddle-point formulation. Since then, both approaches have grown in relative isolation with advancements on one side not effecting the other. In this paper, we rectify this by dissecting and, in a sense, unify the two views.Comment: AAAI Workshop on Computer Poker and Imperfect Informatio

Applying machine learning techniques to an imperfect information game

The game of poker presents a challenging game to Artificial Intelligence researchers because it is a complex asymmetric information game. In such games, a player can improve his performance by inferring the private information held by the other players from their prior actions. A novel connectionist structure was designed to play a version of poker (multi-player limit Hold‟em). This allows simple reinforcement learning techniques to be used which previously not been considered for the game of multi-player hold‟em. A related hidden Markov model was designed to be fitted to records of poker play without using any private information. Belief vectors generated by this model provide a more convenient and flexible representation of an opponent‟s action history than alternative approaches. The structure was tested in two settings. Firstly self-play simulation was used to generate an approximation to a Nash equilibrium strategy. A related, but slower, rollout strategy that uses Monte-Carlo samples was used to evaluate the performance. Secondly the structure was used to model and hence exploit a population of opponents within a relatively small number of games. When and how to adapt quickly to new opponents are open questions in poker AI research. A opponent model with a small number of discrete types is used to identify the largest differences in strategy between members of the population. A commercial software package (Poker Academy) was used to provide a population of sophisticated opponents to test against. A series of experiments was conducted to compare adaptive and static systems. All systems showed positive results but surprisingly the adaptive systems did not show a significant improvement over similar static systems. The possible reasons for this result are discussed. This work formed the basis of a series of entries to the computer poker competition hosted at the annual conferences of the Association for the Advancement of Artificial Intelligence (AAAI). Its best rankings were 3rd in the 2006 6-player limit hold‟em competition and 2nd in the 2008 3-player limit hold‟em competition

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

Arena: A General Evaluation Platform and Building Toolkit for Multi-Agent Intelligence

Learning agents that are not only capable of taking tests, but also innovating is becoming a hot topic in AI. One of the most promising paths towards this vision is multi-agent learning, where agents act as the environment for each other, and improving each agent means proposing new problems for others. However, existing evaluation platforms are either not compatible with multi-agent settings, or limited to a specific game. That is, there is not yet a general evaluation platform for research on multi-agent intelligence. To this end, we introduce Arena, a general evaluation platform for multi-agent intelligence with 35 games of diverse logics and representations. Furthermore, multi-agent intelligence is still at the stage where many problems remain unexplored. Therefore, we provide a building toolkit for researchers to easily invent and build novel multi-agent problems from the provided game set based on a GUI-configurable social tree and five basic multi-agent reward schemes. Finally, we provide Python implementations of five state-of-the-art deep multi-agent reinforcement learning baselines. Along with the baseline implementations, we release a set of 100 best agents/teams that we can train with different training schemes for each game, as the base for evaluating agents with population performance. As such, the research community can perform comparisons under a stable and uniform standard. All the implementations and accompanied tutorials have been open-sourced for the community at https://sites.google.com/view/arena-unity/

Imperfect-Recall Abstractions with Bounds in Games

Imperfect-recall abstraction has emerged as the leading paradigm for practical large-scale equilibrium computation in incomplete-information games. However, imperfect-recall abstractions are poorly understood, and only weak algorithm-specific guarantees on solution quality are known. In this paper, we show the first general, algorithm-agnostic, solution quality guarantees for Nash equilibria and approximate self-trembling equilibria computed in imperfect-recall abstractions, when implemented in the original (perfect-recall) game. Our results are for a class of games that generalizes the only previously known class of imperfect-recall abstractions where any results had been obtained. Further, our analysis is tighter in two ways, each of which can lead to an exponential reduction in the solution quality error bound. We then show that for extensive-form games that satisfy certain properties, the problem of computing a bound-minimizing abstraction for a single level of the game reduces to a clustering problem, where the increase in our bound is the distance function. This reduction leads to the first imperfect-recall abstraction algorithm with solution quality bounds. We proceed to show a divide in the class of abstraction problems. If payoffs are at the same scale at all information sets considered for abstraction, the input forms a metric space. Conversely, if this condition is not satisfied, we show that the input does not form a metric space. Finally, we use these results to experimentally investigate the quality of our bound for single-level abstraction

Solving Games with Functional Regret Estimation

We propose a novel online learning method for minimizing regret in large extensive-form games. The approach learns a function approximator online to estimate the regret for choosing a particular action. A no-regret algorithm uses these estimates in place of the true regrets to define a sequence of policies. We prove the approach sound by providing a bound relating the quality of the function approximation and regret of the algorithm. A corollary being that the method is guaranteed to converge to a Nash equilibrium in self-play so long as the regrets are ultimately realizable by the function approximator. Our technique can be understood as a principled generalization of existing work on abstraction in large games; in our work, both the abstraction as well as the equilibrium are learned during self-play. We demonstrate empirically the method achieves higher quality strategies than state-of-the-art abstraction techniques given the same resources.Comment: AAAI Conference on Artificial Intelligence 201