Decomposing Common Agency

Original Abstract

This paper develops a decomposition methodology for common agency games in which each principal's payoff depends on her own outcome and the agent's type, but not on rivals' outcomes. The key step reduces each principal's best-response problem to a standard screening problem defined over the agent's indirect utility -- the upper envelope of her payoff over rivals' offerings. Individually best-responding mechanisms then assemble into a pure-menu perfect Bayesian equilibrium when a compatibility condition (utility-preserving recombination) ensures aligned tie-breaking across principals. Under a non-indifference condition, the decomposition recovers all equilibria except those sustained by menu items that no type of the agent actually selects but which nevertheless discipline the rival's screening problem. When principals' payoffs depend on the full allocation profile, the decomposition adapts only under substantive regularity conditions on the agent's off-path choice behavior, one of which coincides with Luce's choice axiom. I apply the methodology to two settings. In a quadratic-loss delegation model, equilibria feature one principal offering a finite menu of discrete ``regimes'' while the other receives piecewise full delegation within each regime. In a competitive bundling duopoly under intrinsic common agency, the decomposition yields equilibria exhibiting market splitting, in which firms specialize in complementary bundles, and asymmetric equilibria with a take-it-or-leave-it base contract paired with a nested or tree menu of upgrades.