Every setting is a tuple (A, ψ,O,V, F ) as follows. A is a set of q actions available to the agent.
ψ maps an agent’s action to its cost (we assume for simplicity no two actions have the same cost).
O is a set of m outcomes. Each principal (cid:96) ∈ [n] has a multiparameter valuation function v(cid:96) : O →
R≥0, which maps an outcome to a non-negative value, and can also be thought of as a vector in Rm≥0
(valuations in our model are thus succinct objects). Valuation v(cid:96) belongs to a domain V (cid:96), where
for simplicity we assume V (cid:96) is a convex, polynomially-described polytope; all our results extend to
general convex domains.6 Valuation proﬁle v is the tuple (v1, . . . , vn), and V is the tuple of domains
(V 1, ...,V n). F is a set of distributions; an action a induces a probability distribution F|a over the
outcomes (with probability po∼F|a(o) for outcome o). Throughout, principals are indexed by (cid:96),
Incomplete Information VCG Contracts for Common Agency