Abstract: In the cooperative game with strategic substitute,the best-response of each player is decreasing in actions of the other players. The famous Takisi fixed point theorem of the increasing mapping can do nothing about it. It brings difficulties to the study of comparative statics for Nash equilibrium in the cooperative games with strategic substitutes.The paper studies the comparative statics of a decreasing mapping by cone theory and some fixed point theorems are obtained.The result reduces the convexity, compactness and lattice structure of space. To demonstrate the applicability of our result, we apply it to study a problem of comparative statics for Nash equilibrium in the cooperative games with strategic substitutes. This result has no the requirement of convexity, compactness and lattice structure.

Key words: decreasing mapping, games with strategic substitutes, Nash equilibrium, Cone