  In this paper, we investigate a distributed Nash equilibrium computation
problem for a time-varying multi-agent network consisting of two subnetworks,
where the two subnetworks share the same objective function. We first propose a
subgradient-based distributed algorithm with heterogeneous stepsizes to compute
a Nash equilibrium of a zero-sum game. We then prove that the proposed
algorithm can achieve a Nash equilibrium under uniformly jointly strongly
connected (UJSC) weight-balanced digraphs with homogenous stepsizes. Moreover,
we demonstrate that for weighted-unbalanced graphs a Nash equilibrium may not
be achieved with homogenous stepsizes unless certain conditions on the
objective function hold. We show that there always exist heterogeneous
stepsizes for the proposed algorithm to guarantee that a Nash equilibrium can
be achieved for UJSC digraphs. Finally, in two standard weight-unbalanced
cases, we verify the convergence to a Nash equilibrium by adaptively updating
the stepsizes along with the arc weights in the proposed algorithm.
