Local incentive compatibility in non-convex type-spaces

We explore the equivalence of local incentive compatibility (LIC) (Carroll (2012)) and incentive compatibility (IC) in non-convex type-spaces. We provide a sufficient condition on a type-space called minimal richness for the said equivalence. Using this result, we show that LIC and IC are equivalent on large class of non-convex type-spaces such as type-spaces perturbed by modularity and concave-modularity. The gross substitutes type-space and the generalized gross substitutes and complements type-space are important examples of type-spaces perturbed by modularity and concave-modularity, respectively. Finally, we provide a geometric property consisting of three conditions for the equivalence of LIC and IC, and show that all the conditions are indispensable.