Determining Strategies in Game-Theoretic Shadowed Sets

A three-way approximation of shadowed sets maps the membership grades of all objects into a three-value set with a pair of thresholds. The game-theoretic shadowed sets (GTSS) determine and interpret a pair of thresholds of three-way approximations based on a principle of tradeoff with games. GTSS formulate competitive games between the elevation and reduction errors. The players start from the initial thresholds (1,0) and perform the certain strategies to change the thresholds in the game. The games are repeated with the updated thresholds to gradually reach the suitable thresholds. However, starting from a pair of randomly selected non-(1,0) thresholds is not examined in GTSS. We propose a game approach to make it possible for GTSS starting from a pair of randomly selected thresholds and then determine the strategies associated with them. In particular, given a pair of randomly chosen initial thresholds, we use a game mechanism to determine the change directions that players prefer to make on the initial thresholds. The proposed approach supplements the GTSS, and can be added in the game formulation and repetition learning phases. We explain the game formulation, equilibrium analysis, and the determination of strategies in this paper. An example demonstrates how the proposed approach can supplement GTSS to obtain the thresholds of three-way approximations of shadowed sets when starting from randomly selected thresholds.