Rough interval (RI) is an appropriate generalization of the crisp interval and is very much useful to express uncertain parameters or partially unknown variables when they contain dual-layer information. In real-life problems, there are many situations where players of a matrix game cannot assess their pay-offs by using fuzzy sets/intuitionistic fuzzy sets or ordinary intervals. RIs are used as an excellent tool to handle such situations. This paper explores matrix games with RI pay-offs and investigates two different solution methodologies to solve such a game. In the first approach, a pair of auxiliary linear programming problems with RI coefficients for the players has been constructed. Then each of the two auxiliary programming problems is converted into two linear programming problems with interval coefficients (LPPICs) using lower and upper approximation intervals of the RI. Finally from each LPPIC, two classical linear programming problems (LPPs) are constructed. In the second approach, the expected value technique for RI is used for transforming auxiliary mathematical programming models under the RI environment to crisp LPP. A case study on the telecom market share problem is considered to show the applicability of the proposed approaches and results are compared and analyzed with an existing method.

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.