On revenue equivalence in truthful mechanisms

Introduction. In distributed systems, where problem solutions have to be jointly derived by several selfish agents and where problem data is spread over the agents as private information, mechanism design is used to motivate agents to reveal their private information truthfully and to obtain a good overall solution for the system. As a simple example, consider single item auctions, where several bidders are asked to reveal their valuation for a certain good. Dependent on the bids, the mechanism allocates the good to one of the bidders and the price of the good is designed such that agents have an incentive to bid their true valuation. We consider direct revelation mechanisms, which consist of an allocation rule that selects an allocation depending on the agents’ reports about their private information, and a payment scheme that assigns a payment to every agent. Allocation rules that give rise to a mechanism in which truth-telling is a dominant strategy for every agent are called truthfully implementable. Our concern is with the payment scheme that extends a truthfully implementable allocation rule to a truthful mechanism. The property of an allocation rule to have a unique payment scheme completing the allocation rule to a truthful mechanism is called revenue equivalence. We give a characterization for an allocation rule to satisfy revenue equivalence. In order to obtain this characterization, we prove a property on complete directed graphs and apply it to the so called allocation graph, which is defined by the allocation rule and the valuation function of an agent. The characterization holds for any (possibly infinite) outcome space. Furthermore, we give elementary and simple proofs for the uniqueness of the payment scheme in a truthful mechanism for the cases of finite and countably infinite outcome spaces under very weak assumptions. Many of the known results follow as immediate consequences of ours, e.g. results in Green and Laffont [2], Holmstrom [4], Krishna and Maenner [5], Milgrom and Segal [6], Suijs [10] and Chung and Olszewski [1]. For details and discussions, especially of the paper by Chung and Olszewski, we refer to the full version of