Stopping game problem for dynamic fuzzy systems

A stopping game problem is formulated by cooperating with fuzzy stopping time in a decision environment. The dynamic fuzzy system is a fuzzification version of a deterministic dynamic system and the move of the game is a fuzzy relation connecting between two fuzzy states. We define a fuzzy stopping time using several degrees of levels and instances under a monotonicity property, then an "expectation" of the terminal fuzzy state via the stopping time. By inducing a scalarization function (a linear ranking function) as a payoff for the game problem we will evaluate the expectation of the terminal fuzzy state. In particular, a two-person zero-sum game is considered in case its state space is a fuzzy set and a payoff is ordered in a sense of the fuzzy max order. For both players, our aim is to find the equilibrium point of a payoff function. The approach depends on the interval analysis, that is, manipulating a class of sets arising from alpha-cut of fuzzy sets. We construct an equilibrium fuzzy stopping time under some conditions.