  In this paper, we consider a distributed learning problem in a subnetwork
zero-sum game, where agents are competing in different subnetworks. These
agents are connected through time-varying graphs where each agent has its own
cost function and can receive information from its neighbors. We propose a
distributed mirror descent algorithm for computing a Nash equilibrium and
establish a sublinear regret bound on the sequence of iterates when the graphs
are uniformly strongly connected and the cost functions are convex-concave.
Moreover, we prove its convergence with suitably selected diminishing stepsizes
for a strictly convex-concave cost function. We also consider a constant
step-size variant of the algorithm and establish an asymptotic error bound
between the cost function values of running average actions and a Nash
equilibrium. In addition, we apply the algorithm to compute a mixed-strategy
Nash equilibrium in subnetwork zero-sum finite-strategy games, which have
merely convex-concave (to be specific, multilinear) cost functions, and obtain
a final-iteration convergence result and an ergodic convergence result,
respectively, under different assumptions.
