Nash fuzzy equilibrium: Theory and application to a spatial duopoly

Consider a non-cooperative n-persons game. Each gambler has a set of mixed strategies at his disposal. The payoffs are some physical or immaterial objects. The game is a fuzzy game because (1) gamblers have more or less precise preferences for the payoffs and (2) the outcoming of payoffs is uncertain. The uncertainty can be expressed either by a distribution of possibility or by a distribution of probability. The product set of a gambler's mixed strategies is convex and compact and the payoff functions are continuous. Then a closed and convex fuzzy point-to-set mapping is defined on the product set of strategies and, by using a Butnariu theorem, the existence of a fixed point for this fuzzy point-to-set mapping is proved. The issue allows us to generalize a famous Nash result: a n-persons non-cooperative fuzzy game with mixed strategies has at least one equilibrium point. In the second part of the paper an economic application is devoted to the statement of the equilibrium existence conditions in a spatial duopoly. The model is not only more general than the classical ones, but also more relevant because new results are obtained.