In this contribution, a power allocation game for multiple input multiple output multiple access channels is provided. Considering competing transmitting users, equipped with several antennas each and common multiple antennas at the receiver (base station), a game theoretic framework is conducted to analyze the optimum precoding matrices (power allocation and eigenvector transmit structure) such that each user maximizes selfishly his own rate under a power constraint (assuming single user decoding at the receiver). Interestingly, as the dimensions of the system grow i.e the numbers of transmitting and receiving antennas go to infinity but their ratio stays constant, a Nash equilibrium is shown to exist and be unique. The results are based on random matrix theory and provide, in the asymptotic case, a closed-form expression of the Nash equilibrium operating point. Each terminal can compute the power allocation independently based only on the knowledge of the statistics of the channel (spatial correlation structure at the transmitter and the receiver) and not its instantaneous realizations. This reduces dramatically the downlink overhead signaling
protocol, which becomes important as the number of users grow. The asymptotic claims are then validated through simulations using only a finite number of antennas.