  We propose a fully asynchronous networked aggregative game (Asy-NAG) where
each player minimizes a cost function that depends on its local action and the
aggregate of all players' actions. In sharp contrast to the existing NAGs, each
player in our Asy-NAG can compute an estimate of the aggregate action at any
wall-clock time by only using (possibly stale) information from nearby players
of a directed network. Such an asynchronous update does not require any
coordination among players. Moreover, we design a novel distributed algorithm
with an aggressive mechanism for each player to adaptively adjust the
optimization stepsize per update. Particularly, the slow players in terms of
updating their estimates smartly increase their stepsizes to catch up with the
fast ones. Then, we develop an augmented system approach to address the
asynchronicity and the information delays between players, and rigorously show
the convergence to a Nash equilibrium of the Asy-NAG via a perturbed coordinate
algorithm which is also of independent interest. Finally, we evaluate the
performance of the distributed algorithm through numerical simulations.
