A.3 Existence of IIVCG for Examples 1 and 2

Proposition 6. There exists an IIVCG contract for the setting in Example 1.

Proof. We must show that ka∗(b) ≤(cid:80)
1. a∗(b) = a1. Note that ka1 = 0, and that m(cid:96)(b) ≥ 0 ∀(cid:96) ∈ [n], b ∈ V. Thus, ka∗(b) ≤(cid:80)

(cid:96)∈[n] m(cid:96)(b) ∀b ∈ V. We use the following case analysis:

(cid:96)∈[n] m(cid:96)(b).
2. a∗(b) = a2. Note that ka2 = (cid:15) by using w = (0, (cid:15), 0). Further, since a∗(b) = a2, and by the
deﬁnition of the domain, Wela∗(b)(b) = b2(o2) − (cid:15). Thus, Wela∗(b)(b−1, 0) = Wela∗(b)(b−(cid:96), 0) =
b2(o2)−(cid:15) ∀(cid:96) > 2, and min˜v1∈V 1 Wela∗(b−1,˜v1)(b−1, ˜v1), min˜v(cid:96)∈V (cid:96) Wela∗(b−(cid:96),˜v(cid:96))(b−(cid:96), ˜v(cid:96)) ≥ b2(o2)−
(cid:15) ∀(cid:96) ∈ [n]. This implies that m1(b) = m(cid:96)(b) = 0 ∀(cid:96) > 2. Further, Wela∗(b)(b−2, 0) = −(cid:15).

Thus, m2(b) ≥ (cid:15). It follows that ka∗(b) ≤(cid:80)

(cid:96)∈[n] m(cid:96)(b).
Incomplete Information VCG Contracts for Common Agency