Nash equilibria of games with increasing best replies

The intuitive idea of two activities being complements, for example tea and lemon, is that increasing the level of one makes somehow desirable to increase the level of the other (Samuelson, 1974). Hence complementarity, in its very nature, is a sensitivity property of the set of solutions to an optimization problem. In the context of games, complementarity should then be captured by properties of the joint best reply. We introduce notions of increasingness for the joint best reply which capture properly this intuitive idea of complementarity among players' strategies. We show, by generalizing the fixpoint theorems of Veinott (1992) and Zhou (1994), that the Nash sets of our games are nonempty complete lattices. Hence we extend the class of games with strategic complementarities