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B.3 A Trade-oﬀ between LL and IR: Proof of Theorem 3

Here, we formally prove Theorem 3.

Example 4. There are two actions a1, a2 and two possible outcomes o1, o2. The distributions
imposed by actions are: po∼F|a1
(o1) = 1. There is one principal with valuation
domain V 1 = R2≥0. The cost for each action is: ψ(a1) = 0, ψ(a2) = (cid:15).

2 , po∼F|a2

(o1) = 1

(cid:96)∈[n] t(cid:96)(b, o1) − (cid:15) ≥ (cid:80)

of payments minus cost, it must hold that (cid:80)
Incomplete Information VCG Contracts for Common Agency