equilibrium welfare).

3.2 Deﬁnition, Properties and Uniqueness of the IIVCG Family

As our main conceptual contribution, we propose to overcome the ineﬃciency of ﬁrst price contracts

by introducing the following family of contracts, which we show is (uniquely) truthful and welfare-

maximizing.

Deﬁnition 2. A contract t = (t1, ..., tn) is called Incomplete Information VCG (IIVCG) if the
following two properties hold:

1. Declared welfare maximization. Given contract t, for every bid proﬁle b ∈ V, the agent

chooses to maximize the declared social welfare, that is, x∗(b) = a∗(b).

2. Expected payment characterization. There exist functions h1, . . . , hn where h(cid:96) : V−(cid:96) →
R, such that for every bid proﬁle b ∈ V, principal (cid:96)’s expected payment Eo∼F|x∗(b)[t(cid:96)(b, o)] for
action x∗(b) = a∗(b) is equal to h(cid:96)(b−(cid:96)) − Wela∗(b)(b−(cid:96), 0) .

The ﬁrst property of IIVCG contracts requires that the agent maximizes the declared social
Incomplete Information VCG Contracts for Common Agency