An official website of the United States government
Consider a set of n players. We suppose that each game involves two players, that there is some unknown player who wins each game it plays with a probability greater than 1/2, and that our objective is to determine this best player. Under the requirement that the policy employed guarantees a correct choice with a probability of at least some specified value, we look for a policy that has a relatively small expected number of games played before decision. We consider this problem both under the assumption that the best player wins each game with a probability of at least some specified value >1/2, and under a Bayesian assumption that the probability that player i wins a game against player j is its value divided by the sum of the values, where the values are the unknown values of n independent and identically distributed exponential random variables. In the former case, we propose a policy where chosen pairs play a match that ends when one of them has had a specified number of wins more than the other; in the latter case, we propose a Thompson sampling type rule.
more »
« less
Model-Free Reinforcement Learning for Stochastic Parity Games
This paper investigates the use of model-free reinforcement learning to compute the optimal value in two-player stochastic games with parity objectives. In this setting, two decision makers, player Min and player Max, compete on a finite game arena - a stochastic game graph with unknown but fixed probability distributions - to minimize and maximize, respectively, the probability of satisfying a parity objective. We give a reduction from stochastic parity games to a family of stochastic reachability games with a parameter ε, such that the value of a stochastic parity game equals the limit of the values of the corresponding simple stochastic games as the parameter ε tends to 0. Since this reduction does not require the knowledge of the probabilistic transition structure of the underlying game arena, model-free reinforcement learning algorithms, such as minimax Q-learning, can be used to approximate the value and mutual best-response strategies for both players in the underlying stochastic parity game. We also present a streamlined reduction from 1 1/2-player parity games to reachability games that avoids recourse to nondeterminism. Finally, we report on the experimental evaluations of both reductions
more »
« less
- Award ID(s):
- 2009022
- PAR ID:
- 10237400
- Date Published:
- Journal Name:
- 31st International Conference on Concurrency Theory (CONCUR 2020)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Team-PSRO for Learning Approximate TMECor in Large Team Games via Cooperative Reinforcement LearningRecent algorithms have achieved superhuman performance at a number of twoplayer zero-sum games such as poker and go. However, many real-world situations are multi-player games. Zero-sum two-team games, such as bridge and football, involve two teams where each member of the team shares the same reward with every other member of that team, and each team has the negative of the reward of the other team. A popular solution concept in this setting, called TMECor, assumes that teams can jointly correlate their strategies before play, but are not able to communicate during play. This setting is harder than two-player zerosum games because each player on a team has different information and must use their public actions to signal to other members of the team. Prior works either have game-theoretic guarantees but only work in very small games, or are able to scale to large games but do not have game-theoretic guarantees. In this paper we introduce two algorithms: Team-PSRO, an extension of PSRO from twoplayer games to team games, and Team-PSRO Mix-and-Match which improves upon Team PSRO by better using population policies. In Team-PSRO, in every iteration both teams learn a joint best response to the opponent’s meta-strategy via reinforcement learning. As the reinforcement learning joint best response approaches the optimal best response, Team-PSRO is guaranteed to converge to a TMECor. In experiments on Kuhn poker and Liar’s Dice, we show that a tabular version of Team-PSRO converges to TMECor, and a version of Team PSRO using deep cooperative reinforcement learning beats self-play reinforcement learning in the large game of Google Research Football.more » « less
-
Policy gradient methods enjoy strong practical performance in numerous tasks in reinforcement learning. Their theoretical understanding in multiagent settings, however, remains limited, especially beyond two-player competitive and potential Markov games. In this paper, we develop a new framework to characterize optimistic policy gradient methods in multi-player Markov games with a single controller. Specifically, under the further assumption that the game exhibits an equilibrium collapse, in that the marginals of coarse correlated equilibria (CCE) induce Nash equilibria (NE), we show convergence to stationary ϵ-NE in O(1/ϵ2) iterations, where O(⋅) suppresses polynomial factors in the natural parameters of the game. Such an equilibrium collapse is well-known to manifest itself in two-player zero-sum Markov games, but also occurs even in a class of multi-player Markov games with separable interactions, as established by recent work. As a result, we bypass known complexity barriers for computing stationary NE when either of our assumptions fails. Our approach relies on a natural generalization of the classical Minty property that we introduce, which we anticipate to have further applications beyond Markov games.more » « less
-
The problem of two-player zero-sum Markov games has recently attracted increasing interests in theoretical studies of multi-agent reinforcement learning (RL). In particular, for finite-horizon episodic Markov decision processes (MDPs), it has been shown that model-based algorithms can find an ϵ-optimal Nash Equilibrium (NE) with the sample complexity of O(H3SAB/ϵ2), which is optimal in the dependence of the horizon H and the number of states S (where A and B denote the number of actions of the two players, respectively). However, none of the existing model-free algorithms can achieve such an optimality. In this work, we propose a model-free stage-based Q-learning algorithm and show that it achieves the same sample complexity as the best model-based algorithm, and hence for the first time demonstrate that model-free algorithms can enjoy the same optimality in the H dependence as model-based algorithms. The main improvement of the dependency on H arises by leveraging the popular variance reduction technique based on the reference-advantage decomposition previously used only for single-agent RL. However, such a technique relies on a critical monotonicity property of the value function, which does not hold in Markov games due to the update of the policy via the coarse correlated equilibrium (CCE) oracle. Thus, to extend such a technique to Markov games, our algorithm features a key novel design of updating the reference value functions as the pair of optimistic and pessimistic value functions whose value difference is the smallest in the history in order to achieve the desired improvement in the sample efficiency.more » « less
-
null (Ed.)A two-player finite game is represented by two payoff matrices (A, B), one for each player. Imitation games are a subclass of two-player games in which B is the identity matrix, implying that the second player gets a positive payoff only if she "imitates" the first. Given that the problem of computing a Nash equilibrium (NE) is known to be provably hard, even to approximate, we ask if it is any easier for imitation games. We show that much like the general case, for any c > 0, computing a 1 over n^c -approximate NE of imitation games remains PPAD-hard, where n is the number of moves available to the players. On the other hand, we design a polynomial-time algorithm to find ε-approximate NE for any given constant ε > 0 (PTAS). The former result also rules out the smooth complexity being in P, unless PPAD ⊂ RP.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
