We consider finite state, finite action, stochastic games over an infinite time horizon. We survey algorithms for the computation of minimax optimal stationary strategies in the zerosum case, and of Nash equilibria in stationary strategies in the nonzerosum case. We also survey those theoretical results that pave the way towards future development of algorithms.

Algorithms for stochastic games — A survey

: Learning behaviors in a multiagent environment are crucial for developing and adapting multiagent systems. Reinforcement learning techniques have addressed this problem for a single agent acting in a stationary environment, which is modeled as a Markov decision process (MDP). But, multiagent environments are inherently non-stationary since the other agents are free to change their behavior as they also learn and adapt. Stochastic games, first studied in the game theory community, are a natural extension of MDPs to include multiple agents. In this paper we contribute a comprehensive presentation of the relevant techniques for solving stochastic games from both the game theory community and reinforcement learning communities. We examine the assumptions and limitations of these algorithms, and identify similarities between these algorithms, single agent reinforcement learners, and basic game theory techniques.

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An Analysis of Stochastic Game Theory for Multiagent Reinforcement Learning

Conditional independence and chain event graphs

There are only a few learning algorithms applicable to stochastic dynamic teams and games which generalize Markov decision processes to decentralized stochastic control problems involving possibly self-interested decision makers. Learning in games is generally difficult because of the non-stationary environment in which each decision maker aims to learn its optimal decisions with minimal information in the presence of the other decision makers who are also learning. In stochastic dynamic games, learning is more challenging because, while learning, the decision makers alter the state of the system and hence the future cost. In this paper, we present decentralized Q-learning algorithms for stochastic games, and study their convergence for the weakly acyclic case which includes team problems as an important special case. The algorithms are decentralized in that each decision maker has access only to its own decisions and cost realizations as well as the state transitions; in particular, each decision maker is completely oblivious to the presence of the other decision makers. We show that these algorithms converge to equilibrium policies almost surely in large classes of stochastic games.

Decentralized Q-Learning for Stochastic Teams and Games

We prove the existence of e-(Nash) equilibria in two-person non-zerosum limiting average repeated games with absorbing states. These are stochastic games in which all states but one are absorbing. A state is absorbing if the probability of ever leaving that state is zero for all available pairs of actions.

P. W. Jones

Papers

Stochastic Games with Finite State and Action Spaces.

Readings in Decision Analysis