A non-differentiable approach to revenue equivalence

We give a sufficient condition on the type space for revenue equivalence when the set of social alternatives consists of probability distributions over a finite set. Types are identified with real-valued functions that assign valuations to elements of this finite set, and the type space is equipped with the Euclidean topology. Our sufficient condition is stronger than connectedness but weaker than smooth arc-wise connectedness. Our result generalizes all existing revenue equivalence theorems when the set of social alternatives consists of probability distributions over a finite set. When the set of social alternatives is finite, we provide a necessary and sufficient condition. This condition is similar to, but slightly weaker than, connectedness.