Methods for solving matrix games with cross-evaluated payoffs

In the traditional fuzzy matrix game, given a pair of strategies, the payoffs of one player are usually associated with themselves, but not linked to the payoffs of the other player. Such payoffs can be called self-evaluated payoffs. However, according to the regret theory, the decision makers may care more about what they might get than what they get. Therefore, one player in a matrix game may pay more attention to the payoffs of the other player than his/her payoffs. In this paper, motivated by the pairwise comparison matrix, we allow the players to compare their payoffs and the other ones to provide their relative payoffs, which can be called the cross-evaluated payoffs. Moreover, the players' preference about the cross-evaluated payoffs is usually distributed asymmetrically according to the law of diminishing utility. Then, the cross-evaluated payoffs of players can be expressed by using the asymmetrically distributed information, i.e., the interval-valued intuitionistic multiplicative number. Comparison laws are developed to compare the cross-evaluated payoffs of different players, and aggregation operators are introduced to obtain the expected cross-evaluated payoffs of players. Based on minimax and maximin principles, several mathematical programming models are established to obtain the solution of a matrix game with cross-evaluated payoffs. It is proved that the solution of a matrix game with cross-evaluated payoffs can be obtained by solving a pair of primal-dual linear-programming models and can avoid some unreasonable results. Two examples are finally given to illustrate that the proposed method is based on the cross-evaluated payoffs of players, and can directly provide the priority degree that one player is preferred to the other player in winning the game.