Distributed projection subgradient algorithm for two-network zero-sum game with random sleep scheme

In this paper, a zero-sum game Nash equilibrium computation problem with a common constraint set is investigated under two time-varying multi-agent subnetworks, where the two subnetworks have opposite payoff function. A novel distributed projection subgradient algorithm with random sleep scheme is developed to reduce the calculation amount of agents in the process of computing Nash equilibrium. In our algorithm, each agent is determined by an independent identically distributed Bernoulli decision to compute the subgradient and perform the projection operation or to keep the previous consensus estimate, it effectively reduces the amount of computation and calculation time. Moreover, the traditional assumption of stepsize adopted in the existing methods is removed, and the stepsizes in our algorithm are randomized diminishing. Besides, we prove that all agents converge to Nash equilibrium with probability 1 by our algorithm. Finally, a simulation example verifies the validity of our algorithm.