Dynamic Mechanism Collapse: A Boundary Characterization

TL;DR

This paper characterizes when dynamic mechanisms in Bayesian environments simplify to static ones, identifying key conditions and boundary cases where dynamic value is genuinely present or absent.

Contribution

It provides a necessary and sufficient condition for the collapse of dynamic mechanisms to static designs in Bayesian settings with public signals.

Findings

Collapse occurs when a global affine shadow value supports the revenue frontier.

Failure of the condition indicates the presence of genuine dynamic value.

A boundary statistic identifies when dynamic mechanisms do not collapse.

Abstract

When are dynamics valuable? In Bayesian environments with public signals and no intertemporal commitment, we study a seller who allocates an economically single-shot resource over time. We provide necessary and sufficient conditions under which the optimal dynamic mechanism collapses to a simple terminal design: a single public experiment at date 0 followed by a posterior-dependent static mechanism executed at a deterministic date, with no further disclosure. The key condition is the existence of a global affine shadow value that supports the posterior-based revenue frontier and uniformly bounds all history-dependent revenues. When this condition fails, a collapse statistic pinpoints the dates and public state variables that generate genuine dynamic value. The characterization combines martingale concavification on the belief space with an affine-support duality for concave envelopes.