Lower Bounds on the Complexity of Solving Two Classes of Non-cooperative Games

This paper studies the complexity of solving two classes of non-cooperative games in a distributed manner in which the players communicate with a set of system nodes over noisy communication channels. The complexity of solving each game class is defined as the minimum number of iterations required to find a Nash equilibrium (NE) of any game in that class with ǫ accuracy. First, we consider the class G of all N -player non-cooperative games with a continuous action space that admit at least one NE. Using information-theoretic inequalities, we derive a lower bound on the complexity of solving G that depends on the Kolmogorov 2ǫ-capacity of the constraint set and the total capacity of the communication channels. We also derive a lower bound on the complexity of solving games in G which depends on the volume and surface area of the constraint set. We next consider the class of all N -player non-cooperative games with at least one NE such that the players’ utility functions satisfy a certain (differential) constraint. We derive lower bounds on the complexity of solving this game class under both Gaussian and nonGaussian noise models. Our result in the non-Gaussian case is derived by establishing a connection between the Kullback-Leibler distance and Fisher information.