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This paper investigates the distributed Nash equilibrium seeking problem fortwo-network zero-sum games with set constraints, where the two networks havethe opposite nonsmooth cost functions. The interaction of the agents in eachnetwork is characterized by an unbalanced directed graph. We are particularlyinterested in the case that the local cost function of each agent in the twonetworks is unknown in this paper. We first construct the stochasticsubgradient-free two-variable oracles based on the measurements of the localcost functions. Instead of using subgradients of the local cost functions, thesubgradient-free two-variable oracles are employed to design the distributedalgorithm for the agents to search the Nash equilibrium. Under the strongconnectivity assumption, it is shown that the proposed algorithm guaranteesthat the states of all the agents converge almost surely to the neighborhood ofthe Nash equilibrium, where the scale of the neighborhood can be arbitrarilysmall by selecting suitable smooth parameter in the oracles. Numericalsimulations are finally given to validate the theoretical results.

Distributed stochastic subgradient-free algorithm for Nash equilibrium seeking in two-network zero-sum games