  This paper investigates the distributed Nash equilibrium seeking problem for
two-network zero-sum games with set constraints, where the two networks have
the opposite nonsmooth cost functions. The interaction of the agents in each
network is characterized by an unbalanced directed graph. We are particularly
interested in the case that the local cost function of each agent in the two
networks is unknown in this paper. We first construct the stochastic
subgradient-free two-variable oracles based on the measurements of the local
cost functions. Instead of using subgradients of the local cost functions, the
subgradient-free two-variable oracles are employed to design the distributed
algorithm for the agents to search the Nash equilibrium. Under the strong
connectivity assumption, it is shown that the proposed algorithm guarantees
that the states of all the agents converge almost surely to the neighborhood of
the Nash equilibrium, where the scale of the neighborhood can be arbitrarily
small by selecting suitable smooth parameter in the oracles. Numerical
simulations are finally given to validate the theoretical results.
