Optimistic and Pessimistic Solutions of Single and Multi-Objective Matrix Games with Fuzzy Payoffs and Analysis of Some Military Cases

A new method for solving single-objective and multi-objective two-person zero-sum game problems with fuzzy payoffs is proposed in this paper. ýThe single-objective game problem with fuzzy payoffs is converted to a single-objective game problem with interval payoffs by considering the concept of nearest interval approximation of fuzzy numbersý, and a pair of linear programming problems is obtained to compute the optimistic and pessimistic solutions for each of the players. ýBy the strong duality theorem of linear programmingý, ýit is proved that the optimistic value of Player I is equal to the pessimistic value of Player II and also, the pessimistic value of Player I is equal to the optimistic value of Player II in interval-valued matrix gameý. ýThen, two multiobjective linear programming problems are introduced to compute the optimistic and pessimistic values of interval-valued multiobjective game and their corresponding Pareto optimal strategies for each of the players. As an application, the battle between U.S. and Germany forces in Avranches Gap in World War II is discussed by game theory and is concluded that the obtained optimal strategies of model by the mentioned method for commanders is identical with the analysis of the U.S. military doctrine. Finally, an example of a military battle is considered in which each of the commanders has two objectives.