Weak Monotone Comparative Statics

We develop a theory of monotone comparative statics based on weak set order -- in short, weak monotone comparative statics -- and identify the enabling conditions in the context of individual choices, Pareto optimal choices% for a coalition of agents, Nash equilibria of games, and matching theory. Compared with the existing theory based on strong set order, the conditions for weak monotone comparative statics are weaker, sometimes considerably, in terms of the structure of the choice environments and underlying preferences of agents. We apply the theory to establish existence and monotone comparative statics of Nash equilibria in games with strategic complementarities and of stable many-to-one matchings in two-sided matching problems, allowing for general preferences that accommodate indifferences and incompleteness.