  Matrix games constitute a fundamental problem of game theory and describe a
situation of two players with completely conflicting interests. We show how
methods from statistical mechanics can be used to investigate the statistical
properties of optimal mixed strategies of large matrix games with random payoff
matrices and derive analytical expressions for the value of the game and the
distribution of strategy strengths. In particular the fraction of pure
strategies not contributing to the optimal mixed strategy of a player is
calculated. Both independently distributed as well as correlated elements of
the payoff matrix are considered and the results compared with numerical
simulations.
