The Study Of Game With Payoffs Of Interval Numbers

When game theory is used to solve practical problems, more often than not, the player can not achieve determinate payoffs due to the uncertain game environment and the player’s fuzzy consideration for the problem. What is reached is only the range of variation for the payoffs. Therefore, the study to game whose payoffs are interval numbers should be valuable both in theory and in practice.The present paper will make a research on matrix game whose payoffs are interval numbers. And the research contents are included in the following aspects:I. Firstly, it is given that the definition of the possible degree. Based on the definition, an ordering is given, and this ordering is a total ordering.II. Worst elimination method and underline method are given to solve the interval-number matrix game. The applicability and limitations of the two methods are further analyzed and several examples are used to illustrate. If there is a solution of the interval-number matrix game by worst elimination method, this solution is the optimal strategy, but the solution may not exist. The applicability of underline method is wider than worst elimination method, but the solution obtained by underline method is not necessarily the optimal strategy.III. The definitions of the equilibrium point to interval-number matrix are given. Based on the definition, the existence theorem and the linear programming methods are reached. The linear programming methods are based on four order relations to solve interval-number zero-sum matrix game. Then, a numerical example is given to verify the validity and feasibility of various solutions.IV. Based on the definition of the equilibrium point of interval-number matrix, the existence theorems and the quadratic programming methods, which is similar to the method for solving interval-number zero-sum matrix game, are reached to solve interval-number matrix game on the basis of the four order relations. And a numerical example is given to verify the validity and feasibility of various solutions.