  In this paper, we consider a continuous-type Bayesian Nash equilibrium (BNE)
seeking problem in subnetwork zero-sum games, which is a generalization of
deterministic subnetwork zero-sum games and discrete-type Bayesian zero-sum
games. In this continuous-type model, because the feasible strategy set is
composed of infinite-dimensional functions and is not compact, it is hard to
seek a BNE in a non-compact set and convey such complex strategies in network
communication. To this end, we design two steps to overcome the above
bottleneck. One is a discretization step, where we discretize continuous types
and prove that the BNE of the discretized model is an approximate BNE of the
continuous model with an explicit error bound. The other one is a communication
step, where we adopt a novel compression scheme with a designed sparsification
rule and prove that agents can obtain unbiased estimations through compressed
communication. Based on the above two steps, we propose a distributed
communication-efficient algorithm to practicably seek an approximate BNE, and
further provide an explicit error bound and an $O(\ln T/\sqrt{T})$ convergence
rate.
