  We study contract design for welfare maximization in the well known "common
agency" model of [Bernheim and Whinston, 1986]. This model combines the
challenges of coordinating multiple principals with the fundamental challenge
of contract design: that principals have incomplete information of the agent's
choice of action. Motivated by the significant social inefficiency of standard
contracts for such settings (which we formally quantify using a price of
anarchy/stability analysis), we investigate whether and how a recent toolbox
developed for the first set of challenges under a complete-information
assumption, VCG contracts [Lavi and Shamash, 2019], can be extended to
incomplete information.
  We define and characterize the class of "incomplete information VCG contracts
(IIVCG)", and show it is the unique class guaranteeing truthfulness of the
principals and welfare maximization by the agent. Our results reveal an
inherent tradeoff between two important properties required to ensure
participation in the contract: individual rationality (for the principals) and
limited liability (for the agent). We design a polynomial-time algorithm for
determining whether a setting has an IIVCG contract with both properties. As
our main result we design a polynomial-time "algorithmic IIVCG" contract: given
valuation reports from the principals it returns, if possible for the setting,
a payment scheme for the agent that constitutes an IIVCG contract with all
desired properties. We also give a sufficient graph-theoretic condition on the
population of principals that ensures the existence of such an IIVCG contract.
