Abstract

In this paper, the two player zero sum games with diagonal game matrixes, TPZSGD, are analyzed. Many important applications of this particular class of games are found in military decision problems, in customs and immigration strategies and police work. Explicit functions are derived that give the optimal frequences of different decisions and the expected results of relevance to the different decision makers. Arbitrary numbers of decision alternatives are covered. It is proved that the derived optimal decision frequency formulas correspond to the unique optimization results of the two players. It is proved that the optimal solutions, for both players, always lead to a unique completely mixed strategy Nash equilibrium. For each player, the optimal frequency of a particular decision is strictly greater than 0 and strictly less than 1. With comparative statics analyses, the directions of the changes of optimal decision frequences and expected game values as functions of changes in different parameter values, are determined. The signs of the optimal changes of the decision frequences, of the different players, are also determined as functions of risk in different parameter values. Furthermore, the directions of changes of the expected optimal value of the game, are determined as functions of risk in the different parameter values. Finally, some of the derived formulas are used to confirm earlier game theory results presented in the literature. It is demonstrated that the new functions can be applied to solve common military problems.

Keywords

optimal decisions, completely mixed strategy nash equilibrium, zero sum game theory, stochastic games, typical, applications, description, dynamic, programming, stochastic, horizon, treatment, convergence, economically, positive