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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 consider how the system behaves asymptotically at these stable operating points as n approaches infinity. We begin with a wireless communication network in which n identical nodes (agents) contend for access on a common, wireless communication channel. We characterize this distributed multiple access problem in terms of a homogeneous one-shot random access game, and then analyze the behavior of the nodes using the tools of game theory. We completely characterize the Nash equilibria of this game for all n ≤ 2, and show that there exists a unique fully-mixed Nash equilibrium (FMNE) that is also a focal equilibrium . We show that all centrally controlled optimal solutions are a subset of this game theoretic solution. We then conclude with results about the asymptotic behavior of the nodes as n → ∞, including a bound on the rate of convergence.