of the principals).

6In the general case, the convex domains are represented by polynomial-time membership oracles. Convex valua-

tion domains appear, e.g., as the main application in [17].

7

Contracts. A contract proﬁle t = (t1, ..., tn) (contract for short) includes a payment rule t(cid:96) for each
principal (cid:96). We diverge from the classic model in that the payment rules depend not only on the
outcome of the agent’s action, but also on the reported valuations of all principals: t(cid:96) : V ×O → R.
Such contracts are known as contractible. We denote principal (cid:96)’s report (a.k.a. bid ) by b(cid:96) ∈ V (cid:96)
(the report need not be truthful, i.e., b(cid:96) is not necessarily v(cid:96)).

Taken together, the n (publicly known) payment rules incentivize the agent when choosing

an action. This introduces a tension among the n noncooperative principals to coordinate the
Incomplete Information VCG Contracts for Common Agency