Multidimensional Monotonicity and Economic Applications

We characterize the extreme points of multidimensional monotone functions from [0,1]^n to [0,1], as well as the extreme points of the set of one-dimensional marginals of these functions. These characterizations lead to new results in various mechanism design and information design problems, including public good provision with interdependent values; interim efficient bilateral trade mechanisms; asymmetric reduced form auctions; and optimal private private information structure. As another application, we also present a mechanism anti-equivalence theorem for two-agent, two-alternative social choice problems: A mechanism is payoff-equivalent to a deterministic DIC mechanism if and only if they are ex-post equivalent.