Decomposing Common Agency

Zhiming Feng

arxiv: 2604.23971 · v1 · submitted 2026-04-27 · 💰 econ.TH

Decomposing Common Agency

Zhiming Feng This is my paper

Pith reviewed 2026-05-07 17:32 UTC · model grok-4.3

classification 💰 econ.TH

keywords common agencyscreeningmechanism designdelegationbundlingindirect utilityperfect Bayesian equilibriummenu offers

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The pith

Common agency equilibria can be constructed by solving each principal's problem as an independent screening task against the agent's indirect utility from rival offers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to decompose common agency games, where several principals each offer mechanisms to a single agent whose type is private. The central reduction turns each principal's best-response calculation into a standard monopoly screening problem, but with the agent's payoff from the best rival offer serving as her type-dependent outside option. When these individual solutions satisfy a compatibility condition that preserves the agent's utilities across principals, they form an equilibrium. The approach is illustrated in delegation and bundling models, where it produces concrete predictions about menu designs and market outcomes that would otherwise be hard to characterize directly.

Core claim

The 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.

What carries the argument

The agent's indirect utility, defined as the upper envelope of her payoff over rivals' offerings, which serves as the effective type-dependent outside option for each principal's screening problem.

If this is right

In the 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 the competitive bundling duopoly, the decomposition yields market-splitting equilibria 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.
The method 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduction suggests that numerical algorithms already used for single-principal screening can be reused to compute common-agency equilibria by iterating over candidate indirect-utility functions.
The exception for unused menu items indicates that some equilibria rely on off-path discipline that must still be checked separately after the main decomposition.
The method may extend to games with more than two principals provided the upper-envelope indirect utility can be computed iteratively without circularity.

Load-bearing premise

Each principal's payoff depends only on her own outcome and the agent's type, not on rivals' outcomes, and a utility-preserving recombination condition exists to align tie-breaking across principals.

What would settle it

A common agency game in which individually optimal mechanisms against the indirect utility envelope fail to form a perfect Bayesian equilibrium even after applying the utility-preserving recombination condition.

read the 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Feng gives a decomposition that reduces each principal's best response in common agency to a standard screening problem over the agent's indirect utility, then recombines them via a compatibility condition to form equilibria.

read the letter

The main takeaway is that this paper offers a way to break common agency problems into separate screening exercises for each principal. The reduction uses the agent's indirect utility—the upper envelope of what she gets from all offers—as the object of the screening problem. Best responses then assemble into equilibrium if they meet the utility-preserving recombination condition for consistent tie-breaking. Under non-indifference, it captures all but the equilibria that rely on unused off-path items to discipline rivals. When payoffs depend on the full allocation profile, it needs extra regularity such as Luce's axiom. The two applications show the method in action: one produces finite discrete regimes with piecewise delegation in a quadratic-loss model, and the other yields market splitting or asymmetric nested menus in competitive bundling. The paper states its scope and limits clearly from the start and sticks to standard mechanism-design building blocks without circular steps or free parameters. The soft spots are the ones the abstract already flags. The recombination condition must be verified in each application, and the decomposition does not cover every possible equilibrium when off-path items matter. For cases with full-profile payoffs, the required regularity may rule out some natural settings. These are not hidden flaws but explicit boundaries. The work is aimed at contract theorists and IO researchers who model multi-principal delegation or competition. Readers who need a structured way to locate equilibria without solving the joint game from scratch will get direct use from the examples and the stated conditions. It deserves a serious referee because the approach is grounded in existing tools, the qualifications are honest, and the applications are concrete enough to test the method. I would send it for peer review.

Referee Report

0 major / 3 minor

Summary. The 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 assemble into a pure-menu perfect Bayesian equilibrium when a compatibility condition (utility-preserving recombination) ensures aligned tie-breaking. Under a non-indifference condition, the decomposition recovers all equilibria except those sustained by off-path disciplining items. When principals' payoffs depend on the full allocation profile, the method requires additional regularity conditions such as Luce's choice axiom. Applications are provided to a quadratic-loss delegation model (yielding equilibria with one principal offering finite discrete regimes and the other piecewise full delegation) and a competitive bundling duopoly (yielding market splitting and asymmetric equilibria with a base contract plus nested/tree menus of upgrades).

Significance. If the central reduction and compatibility condition are rigorously verified, the decomposition provides a useful methodological advance for common agency analysis by converting multi-principal best-response problems into standard single-agent screening problems. This could facilitate equilibrium characterization in mechanism design settings that were previously difficult to handle. The applications generate concrete, falsifiable equilibrium predictions (e.g., market splitting in bundling and regime-based delegation), and the paper's explicit qualification of scope (payoff dependence only on own outcome, non-indifference, and regularity conditions) is a strength that avoids overgeneralization. No ad-hoc parameters or invented entities are apparent from the abstract.

minor comments (3)

Abstract: the term 'pure-menu perfect Bayesian equilibrium' is used without a brief definition or forward reference; adding one sentence would improve accessibility for readers outside the immediate subfield.
Abstract: the non-indifference condition and the precise statement of the utility-preserving recombination condition are referenced but not stated even at a high level; including short parenthetical descriptions would make the summary self-contained.
The applications section would benefit from a short table comparing equilibrium features across the two models (e.g., menu structure, delegation extent, and symmetry properties) to aid reader comprehension.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our manuscript, including the recognition of the decomposition's potential as a methodological tool and the explicit scope qualifications. The minor revision recommendation is appreciated.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central reduction maps each principal's best-response to a standard screening problem over the agent's indirect utility (upper envelope of payoffs). This construction is a direct application of existing mechanism-design techniques rather than a self-referential definition or fitted input renamed as prediction. The compatibility condition for equilibrium assembly is stated explicitly as an additional requirement, not derived from the decomposition itself. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation appear in the derivation chain. The two applications follow from the general framework without renaming known empirical patterns. The overall argument therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mechanism design and game-theoretic assumptions rather than introducing new free parameters or invented entities; the central advance is the decomposition procedure itself.

axioms (2)

standard math Standard assumptions of common knowledge of payoffs, type distributions, and rational play in perfect Bayesian equilibrium.
Invoked to define best responses and equilibrium assembly in the common agency setting.
domain assumption The agent's indirect utility is well-defined as the upper envelope of payoffs from rivals' mechanisms.
Central to reducing each principal's problem to a screening task.

pith-pipeline@v0.9.0 · 5516 in / 1365 out tokens · 63904 ms · 2026-05-07T17:32:59.039487+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

there exists an integrableh: [a, b]→R + such that|g ′(·)| ≤h(·)a.e.,

work page
[2]

≤” is by the condition 2 in Proposition 9, and the “=

if for anyj,f j is increasing on[a, b], thengis also increasing on[a, b]. The result still holds when J is an infinite set, {fj}j∈J is a set of equi-Lipschitz functions, and g(·) := sup j∈J fj(·)<∞. Proof. For any j,f j ∈C 1([a, b]) implies that fj is Lipschitz on [a, b]. Let the Lipschitz constant for fj beL j ∈R +. For anyx, y∈[a, b], |g(x)−g(y)|=|max j...

work page 2002
[3]

In addition, φ∗ i(t, ot −i) =o t i ∈argmax o∈M1×...×Mn V(o, t) Oi , which directly gives for anyt ′, o′ −i, vi(φ∗ i(t, ot −i), t|M−i)≥v i(φ∗ i(t′, o′ −i), t|M−i)

For anyt, sinceσ A is a pure-strategy, suppσA(M1, ...,Mn, t) O−i ={o t −i}. In addition, φ∗ i(t, ot −i) =o t i ∈argmax o∈M1×...×Mn V(o, t) Oi , which directly gives for anyt ′, o′ −i, vi(φ∗ i(t, ot −i), t|M−i)≥v i(φ∗ i(t′, o′ −i), t|M−i). Suppose the constructed φ∗ i does not solve the problem in Proposition 13, i.e., there exists φ′ i that is feasible an...

work page
[4]

Let O1 ={a 1, a2, a3}, O 2 ={b 1, b2, b3}. The agent’s utility is given by the following tables: b1 b2 b3 a1 1 2 0 a2 1 1 0 a3 2 1 0 t1 b1 b2 b3 a1 1 1 2 a2 2 1 0 a3 0 1 0 t2 b1 b2 b3 a1 1 2 0 a2 2 2 2 a3 2 2 0 t3 Principal 1’s utility, which is only(ai, ti)-dependent, is a1 a2 a3 t1 2 1 0 t2 0 2 0 t3 1 0 0 15 and principal 2’s utility is b1 b2 b3 t1 0 1 ...

work page
[5]

outside options

Let O1 ={a, b, c}, O 2 ={A, B, C}. The agent’s utility is given by the following tables: A B C a 5 2 0 b 1 7 6 c 4 8 3 t1 A B C a 5 8 7 b 2 3 4 c 6 1 0 t2 For each type, all nine utility levels are distinct. Hence(non-indifference)holds. Principal 1’s utility is u1(a, t1) = 4, u 1(b, t1) = 0, u 1(c, t1) = 3, u1(a, t2) = 1, u 1(b, t2) = 0, u 1(c, t2) = 4, ...

work page

[1] [1]

there exists an integrableh: [a, b]→R + such that|g ′(·)| ≤h(·)a.e.,

work page

[2] [2]

≤” is by the condition 2 in Proposition 9, and the “=

if for anyj,f j is increasing on[a, b], thengis also increasing on[a, b]. The result still holds when J is an infinite set, {fj}j∈J is a set of equi-Lipschitz functions, and g(·) := sup j∈J fj(·)<∞. Proof. For any j,f j ∈C 1([a, b]) implies that fj is Lipschitz on [a, b]. Let the Lipschitz constant for fj beL j ∈R +. For anyx, y∈[a, b], |g(x)−g(y)|=|max j...

work page 2002

[3] [3]

In addition, φ∗ i(t, ot −i) =o t i ∈argmax o∈M1×...×Mn V(o, t) Oi , which directly gives for anyt ′, o′ −i, vi(φ∗ i(t, ot −i), t|M−i)≥v i(φ∗ i(t′, o′ −i), t|M−i)

For anyt, sinceσ A is a pure-strategy, suppσA(M1, ...,Mn, t) O−i ={o t −i}. In addition, φ∗ i(t, ot −i) =o t i ∈argmax o∈M1×...×Mn V(o, t) Oi , which directly gives for anyt ′, o′ −i, vi(φ∗ i(t, ot −i), t|M−i)≥v i(φ∗ i(t′, o′ −i), t|M−i). Suppose the constructed φ∗ i does not solve the problem in Proposition 13, i.e., there exists φ′ i that is feasible an...

work page

[4] [4]

Let O1 ={a 1, a2, a3}, O 2 ={b 1, b2, b3}. The agent’s utility is given by the following tables: b1 b2 b3 a1 1 2 0 a2 1 1 0 a3 2 1 0 t1 b1 b2 b3 a1 1 1 2 a2 2 1 0 a3 0 1 0 t2 b1 b2 b3 a1 1 2 0 a2 2 2 2 a3 2 2 0 t3 Principal 1’s utility, which is only(ai, ti)-dependent, is a1 a2 a3 t1 2 1 0 t2 0 2 0 t3 1 0 0 15 and principal 2’s utility is b1 b2 b3 t1 0 1 ...

work page

[5] [5]

outside options

Let O1 ={a, b, c}, O 2 ={A, B, C}. The agent’s utility is given by the following tables: A B C a 5 2 0 b 1 7 6 c 4 8 3 t1 A B C a 5 8 7 b 2 3 4 c 6 1 0 t2 For each type, all nine utility levels are distinct. Hence(non-indifference)holds. Principal 1’s utility is u1(a, t1) = 4, u 1(b, t1) = 0, u 1(c, t1) = 3, u1(a, t2) = 1, u 1(b, t2) = 0, u 1(c, t2) = 4, ...

work page