Damned if you do and damned if you don't: Two masters

We study common agency problems in which two principals (groups) make costly commitments to incentives that are conditioned on imperfect signals of the agent's action. Our framework allows for incentives to be either rewards or punishments. For our basic model we obtain a unique equilibrium, which typically involves randomization by both principals. Greater similarity between principals leads to more aggressive competition. The principals weakly prefer punishment to rewards, sometimes strictly. With rewards an agent voluntarily joins both groups; with punishment it depends on whether severe punishments are feasible and cheap for the principals. We study whether introducing an attractive compromise reduces competition between principals. Our framework of imperfect monitoring offers a natural perturbation of the standard common agency model of menu auctions, which results in sharper equilibrium predictions. The limit equilibrium prediction provides support to both truthful equilibria and the competing notion of natural equilibria, which unlike the former may be inefficient.