A brief resume of the role of linear utility functions in Game Theory is given. The point is made that, in practical applications, a knowledge of these functions is usually not available, and hence much of the rationale of the game theoretic approach to competitive problems is lost. The random character of the real payoff of a matrix game is then discussed, and the probability distribution function of the payoff is derived. The dependence of this distribution function upon the mixed strategies of the players is shown. Criteria are developed to provide definition of optimal mixed strategy in terms of the effect on the distribution function. The mathematical formulation of the solution is given for each criterion discussed. The reader will require knowledge of the elements of Probability Theory, Game Theory, Utility Theory, and Linear Programming.