In this paper, we investigate a distributed Nash equilibrium computationproblem for a time-varying multi-agent network consisting of two subnetworks,where the two subnetworks share the same objective function. We first propose asubgradient-based distributed algorithm with heterogeneous stepsizes to computea Nash equilibrium of a zero-sum game. We then prove that the proposedalgorithm can achieve a Nash equilibrium under uniformly jointly stronglyconnected (UJSC) weight-balanced digraphs with homogenous stepsizes. Moreover,we demonstrate that for weighted-unbalanced graphs a Nash equilibrium may notbe achieved with homogenous stepsizes unless certain conditions on theobjective function hold. We show that there always exist heterogeneousstepsizes for the proposed algorithm to guarantee that a Nash equilibrium canbe achieved for UJSC digraphs. Finally, in two standard weight-unbalancedcases, we verify the convergence to a Nash equilibrium by adaptively updatingthe stepsizes along with the arc weights in the proposed algorithm.
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