A unified distributed algorithm for non-cooperative games

Abstract

This chapter presents a unified framework for the design and analysis of distributed algorithms for computing first-order stationary solutions of non-cooperative games with non-differentiable player objective functions. These games are closely associated with multi-agent optimization wherein a large number of selfish players compete noncooperatively to optimize their individual objectives under various constraints. Unlike centralized algorithms that require a certain system mechanism to coordinate the players’ actions, distributed algorithms have the advantage that the players, either individually or in subgroups, can each make their best responses without full information of their rivals’ actions. These distributed algorithms by nature are particularly suited for solving hugesize games where the large number of players in the game makes the coordination of the players almost impossible. The distributed algorithms are distinguished by several features: parallel versus sequential implementations, scheduled versus randomized player selections, synchronized versus asynchronous transfer of information, and individual versus multiple player updates. Covering many variations of distributed algorithms, the unified algorithm employs convex surrogate functions to handle nonsmooth nonconvex functions and a (possibly multi-valued) choice function to dictate the players’ turns to update their strategies. There are two general approaches to establish the convergence of such algorithms: contraction versus potential based, each requiring different properties of the players’ objective functions. We present the details of the convergence analysis based on these two approaches and discuss randomized extensions of the algorithms that require less coordination and hence are more suitable for big data problems. Introduction Introduced by John von Neumann [1], modern-day game theory has developed into a very fruitful research discipline with applications in many fields. There are two major classifications of a game, cooperative versus non-cooperative. This chapter pertains to one aspect of non-cooperative games for potential applications to big data, namely, the computation of a “solution” to such a game by a distributed algorithm. In a (basic) non-cooperative game, there are finitely many selfish players/agents each optimizing a rival-dependent objective by choosing feasible strategies satisfying certain private constraints. Providing a solution concept to such a game, a Nash equilibrium (NE) [2, 3] is by definition a tuple of strategies, one for each player, such that no player will be better off by unilaterally deviating from his/her equilibrium strategy while the rivals keep executing their equilibrium strategies.