  In this paper we present an averaging technique applicable to the design of
zeroth-order Nash equilibrium seeking algorithms. First, we propose a
multi-timescale discrete-time averaging theorem that requires only that the
equilibrium is semi-globally practically stabilized by the averaged system,
while also allowing the averaged system to depend on ``fast" states.
Furthermore, sequential application of the theorem is possible, which enables
its use for multi-layer algorithm design. Second, we apply the aforementioned
averaging theorem to prove semi-global practical convergence of the
zeroth-order information variant of the discrete-time projected pseudogradient
descent algorithm, in the context of strongly monotone, constrained Nash
equilibrium problems. Third, we use the averaging theory to prove the
semi-global practical convergence of the asynchronous pseudogradient descent
algorithm to solve strongly monotone unconstrained Nash equilibrium problems.
Lastly, we apply the proposed asynchronous algorithm to the connectivity
control problem in multi-agent systems.
