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This paper presents a novel game-theoretic design to optimize the performance of medium access control (MAC) in wireless networks. The nodes of the network are modeled as selfish and rational players of a non-cooperative game. We define novel utility functions to capture their gain from channel access. We characterize the Nash equilibrium of the game and show that it is unique and non-trivial. This ensures a stable operating point from which no player has an incentive to deviate unilaterally and where every player has an equal non-trivial share of the transmission channel. Thus the selfish behavior of the nodes is used to ensure desirable properties of the network as a whole. The nodes follow a distributed update mechanism to reach the equilibrium. They need no message passing or network-wide information. We implement its asynchronous version in NS-2 and study the dynamics of the game. We compare, via simulations, our game model with the distributed coordination function (DCF) in IEEE 802.11 and a comparable game model in the literature. We observe that our design outperforms both these designs and provides much higher throughput and lower collision overhead over a very wide range of network sizes.