Decomposing Common Agency

TL;DR

This paper introduces a decomposition method for common agency games, simplifying equilibrium analysis by reducing principals' problems to standard screening problems and applying it to delegation and bundling models.

Contribution

It develops a novel decomposition approach for common agency games, enabling explicit equilibrium characterization in complex principal-agent interactions.

Findings

Equilibria include finite regimes and piecewise full delegation in delegation models.

Market splitting and asymmetric equilibria emerge in bundling duopoly.

Decomposition applies under regularity conditions, including Luce's choice axiom.

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.