on behalf of the agent:

of the other principals’ reports b−(cid:96), and (cid:80)

Lemma (Payment characterization, see Lemma 1). A contract t = (t1, . . . , t(cid:96)) is IIVCG if and only
if principal (cid:96)’s payment function t(cid:96) can be expressed using h(cid:96)(·) and w(cid:96), where h(cid:96)(·) is a function
(cid:96) w(cid:96) is a payment vector belonging to the polytope of

payments incentivizing the socially eﬃcient action.

This lemma is useful since it exactly deﬁnes our “design space”, and in particular translates to

linear constraints that are the cornerstone of our algorithmic constructions.

IIVCG contracts satisfy truthfulness and welfare maximization; what about the other desiderata
of IR and LL? Whether or not these properties hold depends on the choices of h(cid:96)(·) and w(cid:96) that
determine the payments for every principal (cid:96). A natural starting point is to choose these to make
Incomplete Information VCG Contracts for Common Agency