Matrix games with dense fuzzy payoffs

Abstract

A novel notion of dense fuzzy lock set was introduced as an extension of fuzzy sets. Learning experiences have a vital role in this fuzzy lock set instigation. The novel fuzzy lock set reduces the fuzziness of the situation. Occasionally, the players are bound to make little changes in their game strategies to reach their goal for some matrix game problems. It may lead to some changes in payoffs. In this scenario, a matrix game's payoffs are chosen as dense fuzzy lock sets to make the problem more realistic. This paper's prime intent is to develop a mathematical model of a matrix game that represents payoffs by triangular dense fuzzy lock sets. Initially, a new defuzzification function MagD(.) is defined to find a ranking order relation of the dense fuzzy lock sets. Then a pair of auxiliary dense fuzzy programming problems is established for two players. These two problems are transformed into two equivalent crisp linear programming problems applying the proposed defuzzification function and its linearity property. The reduced problems are solved using LINGO 17.0 software to determine each player's optimal strategies and the game values. One surprising fact of this approach is that the value of the game increases with the increment of the player's learning experience. The validity, applicability, and superiority of this proposed methodology are illustrated by considering a real-life media share problem.