A generalization of monotone comparative statics

In this paper, we generalize the lattice theoretical comparative statics by Li Calzi and Veinott, and Milgrom and Shannon. While their theorem is constructed on lattices, particularly on partially ordered sets, we do not require the antisymmetry on a binary relation defined on the set. On the basis of this result, we can deal with the comparative statics of constrained optimization problems, including the cases with nonlinear constraints, in a very intuitive, but considerably general fashion. Specifically, we can extend the "value order methods" proposed by Antoniadou and Mirman and Ruble in the context of consumer problems with linear constraints. It is also worth noting that our results on the value order can be applicable for any comparative criterion as long as it is a complete preorder on the domain of the objective function.