Classification of Matrices by Means of Envelopes for Bicriteria Matrix Game pp. 371-378
Authors: Masakazu Higuchi and Tamaki Tanakarsch
Abstract: The study on games with real-valued payoff functions has been developed widely, and several kind of solution concepts and their useful properties have been analyzed and used; see  and references cited therein. On the other hand, multicriteria games with vector-valued payoff function have also been studied as strategic forms in ,  and references cited therein. This kind of games has multiple criteria situations and the plural number of incomparable optimal solutions, denoted by a set of efficient (or Pareto, or nondominated) points. Existence Results of several solutions for such games are studied, but we could not know the detail geometric information on each set of solutions except for , , . Hence, the aim of this paper is to observe properties of payoff functions for multicriteria games in detail. One of the most effective ways to study them is to analyze the set of minimax values and the set of maximin values for payoff functions, but it is not so easy as compare with single-criterion usual two-person zero-sum matrix games, because the representation for graphs or images of vector-valued payoff functions is too difficult for more than two dimensional case and also the drawing of minimax and maximin values for the images is impossible. Therefore, we restrict the dimension of the image space to two, that is, we consider a bicriteria two-person zero-sum matrix game with two 2 × 2 matrices. By using computational program on computers, we obtain the structure information for the image of the payoff functions, and characterize the payoff functions by classifying the images of the payoff functions.