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To achieve full cooperative diversity gains while still maintaining spectral and energy efficiency, relay assignment schemes for cooperative communications have been extensively studied in recent research. These schemes select only the best relay from multiple relaying candidates to cooperate with a communication link. In this paper, we formulate the problem of relay assignment as a non-cooperative, mixed strategy, repeated game, where relaying candidates are modeled as rational players. We then propose a Game Theory based Relay Assignment scheme GTRA, in which each player plays against all the other players, and determines whether to cooperate with a communication link on a packet-by-packet basis in a distributed manner. To adapt to dynamic environments, an adaptive learning algorithm is utilized by players to learn optimal strategies of relay assignment, as well as orienting the game to converge to a set of correlated equilibriums. We compare GTRA with BR, a fictitious game based approach. The simulation results show that GTRA outperforms BR in terms of network throughput, especially in environments where the channel fading becomes severe. It is also shown that GTRA can converge to a correlated equilibrium in a short period that enables it to work well in dynamic environments.