Abstract-We present a view of cooperative control using the language of learning in games. We review the game-theoretic concepts of potential and weakly acyclic games, and demonstrate how several cooperative control problems, such as consensus and dynamic sensor coverage, can be formulated in these settings. Motivated by this connection, we build upon game-theoretic concepts to better accommodate a broader class of cooperative control problems. In particular, we extend existing learning algorithms to accommodate restricted action sets caused by the limitations of agent capabilities and group-based decision making. Furthermore, we also introduce a new class of games called sometimes weakly acyclic games for time-varying objective functions and action sets, and provide distributed algorithms for convergence to an equilibrium.

We consider a continuous-time form of repeated matrix games in which player strategies evolve in reaction to opponent actions. Players observe each other's actions, but do not have access to other player utilities. Strategy evolution may be of the best response sort, as in fictitious play, or a gradient update. Such mechanisms are known to not necessarily converge. We introduce a form of "dynamic" fictitious and gradient play strategy update mechanisms. These mechanisms use derivative action in processing opponent actions and, in some cases, can lead to behavior converging to Nash equilibria in previously nonconvergent situations. We analyze convergence in the case of exact and approximate derivative measurements of the dynamic update mechanisms. In the ideal case of exact derivative measurements, we show that convergence to Nash equilibrium can always be achieved. In the case of approximate derivative measurements, we derive a characterization of local convergence that shows how the dynamic update mechanisms can converge if the traditional static counterparts do not. We primarily discuss two player games, but also outline extensions to multiplayer games. We illustrate these methods with convergent simulations of the well known Shapley and Jordan counterexamples.

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Abstract-We consider multi-player repeated games involving a large number of players with large strategy spaces and enmeshed utility structures. In these "large-scale" 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 "Fictitious Play" (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. Index Terms-Fictitious play (FP), joint strategy fictitious play (JSFP).

Abstract-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 algorithm is decentralized in that each decision maker has access to only its local information, the state information, and the local cost realizations; furthermore, it is completely oblivious to the presence of other decision makers. We show that these algorithms converge to equilibrium policies almost surely in large classes of stochastic games.