An investigation of strategy selection procedures in competitive situations

The problem of optimal strategy selection in the two person, zero sum matrix game is investigated. As part of the investigation, the usefulness of control charts in determining the selection procedure used by the opponent is demonstrated. Six basic decision criteria are examined and found to be applicable to the game problem when specific information about the opponent is available. The use of these criteria is shown to yield an increased payoff to a player when his opponent deviates from the optimal policy. This increased payoff is a direct result of effective use of knowledge about the opponent's selection procedure. Methods of calculating the particular strategy required by each of the decision criteria are described. Two methods of calculating the selections required by the Minimax Criterion are presented. One, an approximate method, is based on an iterative process. The other, an exact method, is based on the Simplex technique of Linear Programming. Computer programs, in the Fortran IV language, are presented for both methods. A variation of the iterative method is shown to apply to games with an unknown payoff matrix. A second variation of the iterative method is presented as a useful algorithm for the general problem of strategy selection, regardless of the selection procedure used by the opponent. Optimal betting strategies are presented for specific games of chance. In the case when the game is biased in favor of the player, the optimal strategy maximizes the exponential rate of increase of the player's assets. When the game is biased against the player, the optimal strategy yields an increase in the frequency of wins, in spite of the long term expected loss. Barriers to the widespread usefulness of the game model are also discussed. Among these are the successful solution of the n-person game and the accurate definition of the payoff matrix. Iterative methods are suggested as logical approaches to these problems.