  This paper considers a class of noncooperative games in which the feasible
decision sets of all players are coupled together by a coupled inequality
constraint. Adopting the variational inequality formulation of the game, we
first introduce a new local edge-based equilibrium condition and develop a
distributed primal-dual proximal algorithm with full information. Considering
challenges when communication delays occur, we devise an asynchronous
distributed algorithm to seek a generalized Nash equilibrium. This asynchronous
scheme arbitrarily activates one player to start new computations independently
at different iteration instants, which means that the picked player can use the
involved out-dated information from itself and its neighbors to perform new
updates. A distinctive attribute is that the proposed algorithms enable the
derivation of new distributed forward-backward-like extensions. In theoretical
aspect, we provide explicit conditions on algorithm parameters, for instance,
the step-sizes to establish a sublinear convergence rate for the proposed
synchronous algorithm. Moreover, the asynchronous algorithm guarantees almost
sure convergence in expectation under the same step-size conditions and some
standard assumptions. An interesting observation is that our analysis approach
improves the convergence rate of prior synchronous distributed
forward-backward-based algorithms. Finally, the viability and performance of
the proposed algorithms are demonstrated by numerical studies on the networked
Cournot competition.
