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We address the fundamental question of whether or not there exist stable operating points in a network in which selfish nodes share a common channel, and if they exist, how the nodes behave at these stable operating points. We begin with a wireless communication network in which n nodes (agents), which might have different utility functions, contend for access on a common, wireless communication channel. We characterize this distributed multiple-access problem in terms of a one-shot random-access game, and then analyze the behavior of the nodes using the tools of game theory. We give necessary and sufficient conditions on nodes for the complete characterization of the Nash equilibria of this game for all n ges 2. We show that all centrally controlled optimal solutions are a subset of this game theoretic solution, and almost all (w.r.t. Lebesgue measure) transmission probability assignments chosen by a central authority are supported by the game theoretic solution. We analyze the behavior of the network throughput at Nash equilibria as a function of the costs of the transmitters incurred by failed transmissions. Finally, we conclude the paper with the asymptotic analysis of the system as the number of transmitters goes to infinity. We show that the asymptotic distribution of the packet arrivals converges in distribution to a Poisson random variable, and the channel throughput converges to -(c/(1+c))ln(c/(1+c)) with c > 0 being the cost of failed transmissions. We also give the best possible bounds on the rates of convergence of the packet arrival distribution and the channel throughput.