Informational size and two-stage mechanisms

We showed in McLean and Postlewaite (2014) that when agents are informationally small, there exist small modi…cations to VCG mechanisms in interdependent value problems that restore incentive compatibility. This paper presents a two-stage mechanism that similarly restores incentive compatibility. The …rst stage essentially elicits that part of the agents’private information that induces interdependence and reveals it to all agents, transforming the interdependent value problem into a private value problem. The second stage is a VCG mechanism for the now private value problem. Agents typically need to transmit substantially less information in the two stage mechanism than would be necessary for a single stage mechanism. Lastly, the …rst stage that elicits the part of the agents’ private information that induces interdependence can be used to transform certain other interdependent value problems into private value problems. Keywords: Auctions, Incentive Compatibility, Mechanism Design, Interdependent Values, Ex Post Incentive Compatibility, Informational Size JEL Classi…cations: C70, D44, D60, D82 1. Introduction The Vickrey-Clarke-Groves mechanism (hereafter VCG) for private values environments is a classic example of a mechanism for which truthful revelation is ex post incentive compatible. It is well-known, however, that truthful revelation is generally no longer incentive compatible when we move from a private values environment to an interdependent values environment. In McLean and Postlewaite (2014) (henceforth MP (2014)) we showed that, when agents are informationally small in the sense of McLean and Postlewaite (2002), there exists a modi…cation of a generalized VCG mechanism using small additional transfers that restores incentive compatibility. This paper presents an alternative, two-stage, mechanism that accomplishes the same goal –restoring incentive compatibility for interdependent value problems. The advantage of the two stage mechanism relative to a single stage mechanism is that, for typical problems, agents need to transmit substantially less information. We will explain intuitively the nature of the savings in transmitted information. Consider a problem in which there is uncertainty about the state of nature. An agent’s private information We thank the National Science Foundation for …nancial support. We thank Rakesh Vohra for helpful conversations and Zehao Hu for excellent proofreading. consists of a state dependent payo¤ function and a signal correlated with the state. A single stage mechanism that delivers an e¢ cient outcome for any realization of agents’ types must do two things. First, it must elicit the information agents have about the state of nature to determine the posterior probability distribution given that information. Second, it must elicit agents’privately known state dependent payo¤s. A two stage mechanism can separate the two tasks. First, elicit the information about the state of nature, but relay to agents the posterior distribution on the state of nature before collecting any additional information. When agents are induced to reveal their information about the state of nature truthfully, relaying the posterior distribution on the state of nature converts the interdependent value problem into a private value problem. When agents know the probability distribution on the set of states of nature, they need only report their expected utility for each possible social outcome rather than their utility for every social outcome in each of the states. Essentially, by moving from a one stage mechanism to a two stage mechanism, we can shift the job of computing expected utilities given the posterior from the mechanism to the agents. Doing this reduces the information that agents must report to the mechanism; we discuss this in the last section. In this paper, we construct a two stage game in which the second stage is modeled as a standard private values VCG mechanism. The basic mechanics of …rst eliciting the information correlated with the state of nature in order to convert an interdependent values problem into private values problem can be applied to certain other environments as well. In particular, our …rst stage can be combined with certain other mechanisms with desirable properties in private value problems to address implementation problems in the presence of interdependent values. We provide an example of how our mechanism works in the next section, and present the general mechanism after that. 2. Example The following single object auction example, a modi…cation of the example in McLean and Postlewaite (2004), illustrates our basic idea. An object is to be sold to one of three agents. There are two equally likely states of the world, 1 and 2, and an agent’s value for the object depends on the state of the world. Agent i’s state dependent utility function can written as vi = (v 1 i ; v 2 i ) = (vi( 1); vi( 2)) where v j i is his utility of the object in state j . An agent’s utility function is private information. In addition, each agent i receives a private signal si 2 fa1; a2g correlated with the state. These signals are independent conditional on the state and the conditional probabilities are as shown in the following table. signal a1 a2 state 1 1 2 1 where > 12 : Consequently, an agent’s private information, his type, is a pair (si; vi) and we make two assumptions. First, for any type pro…le (si; vi)i=1; the conditional distribution on the state space given (si; vi)i=1 depends only on the signals (s1; s2; s3): Therefore, the agents’utility functions provide no information relevant for predicting the state that is not already contained in the signal pro…le alone. Second, we assume that for any type (si; vi) of agent i, the conditional