Truthful germs are contagious: A local-to-global characterization of truthfulness.

We study the question of how to easily recognize whether a social choice function f from an abstract type space to a set of outcomes is truthful, i.e. implementable by a truthful mechanism. In particular, if the restriction of f to every "simple" subset of the type space is truthful, does it imply that f is truthful? Saks and Yu proved one such theorem: when the set of outcomes is finite and the type space is convex, a function f is truthful if its restriction to every 2-element subset of the type space is truthful, a condition called weak monotonicity. This characterization fails for infinite outcome sets. We provide a local-to-global characterization theorem for any set of outcomes (including infinite sets) and any convex space of types (including infinite-dimensional ones): a function f is truthful if its restriction to every sufficiently small 2-D neighborhood about each point is truthful. More precisely, f is truthful if and only if it satisfies local weak monotonicity and is vortex-free, meaning that the loop integral of f over every sufficiently small triangle vanishes. Our results apply equally well to multiple solution concepts, including dominant strategies, Nash and Bayes-Nash equilibrium, and to both deterministic and randomized mechanisms. When the type space is not convex, we show that f is truthful if and only if it extends to a truthful function on the convex hull of the original type space. We use our characterization theorem to give a simple alternate derivation of the Saks-Yu theorem. Generalizing this, we give a sufficient condition for constructing a truthful function by "stitching together" truthful subfunctions on different subsets of the domain.