Eo∼F|a∗(b)[w(o)], this is equivalent to m(cid:96)(b) ≥ xb

(cid:96) · arg minw∈La∗(b)

Observation 3. The following linear program computes x from Claim 3 (2) in polynomial time.

(LP3)

(cid:96) ≤ 1, xb
xb

(cid:96) · ka∗(b) ≤ m(cid:96), 0 ≤ xb

(cid:96)

∀(cid:96) ∈ [n].

Lemma 3. If a setting satisﬁes ka∗(b) ≤(cid:80)

(cid:96)∈[n]

(cid:96)∈[n] m(cid:96)(b) ∀b ∈ V, Algorithm 1 computes the construc-

tion in Claim 3 in polynomial time.
Proof. The ﬁrst loop computes h(cid:96)(b−(cid:96)) ∀(cid:96) ∈ [n]. Line 4 computes wb, and line 6 computes x. The
last loop computes the exact payments according to Lemma 1. All linear programs are solvable in

polynomial time as argued above (Propositions 3, 4 and Observation 3).

Proof of Theorem 5. According to Lemma 2, Claim 3 and Lemma 3, ALG1 is an IIVCG contract

that satisﬁes LL and IR for every setting where such a contract exists. Truthfulness and social
Incomplete Information VCG Contracts for Common Agency