Common agency with informed principals: Menus and signals

I analyze common agency games in which the principals, and possibly the agent, have private information. I distinguish between games in which the principals delegate the final decisions to the agent, and games in which they retain some decision power after offering their mechanisms. I show that, in contrast with mechanism design models with one informed principal, Myerson's Inscrutability Principle fails when there are many informed principals. I also find that, in contrast with common agency models with uninformed principals, the Delegation Principle (Menu Theorem) fails when principals are informed. I then focus on Perfect Bayesian Equilibria in which principals offer their mechanisms without randomizing. I characterize the outcomes of arbitrary games with delegation as outcomes of a new game in which principals offer menus and send cheap-talk signals. Next, I characterize the outcomes of arbitrary games without delegation as outcomes of a new game in which principals offer menus of direct revelation mechanisms, to which they truthfully report their types.