Screening Under Competition

Yu-Ting Ho

arxiv: 2606.22720 · v1 · pith:ZEKB2RAUnew · submitted 2026-06-21 · 💰 econ.TH

Screening Under Competition

Yu-Ting Ho This is my paper

Pith reviewed 2026-06-26 09:11 UTC · model grok-4.3

classification 💰 econ.TH

keywords screeningcompetitionposted pricesdensity-regularityoptimal controlmechanism designdifferentiated productsunit demand

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The pith

Under density-regularity on valuations, each firm's best response to any rival menus is a single posted-price contract.

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

The paper examines competition among firms selling differentiated goods to a unit-demand buyer whose valuations are private and drawn from a known distribution. It establishes that a condition called density-regularity on this distribution makes it optimal for every firm to respond to arbitrary opponent menus by offering only a posted price. The proof proceeds via an optimal-control formulation because the standard virtual-surplus argument from monopoly screening does not carry over once multiple firms and cross-firm preference heterogeneity are present. The condition is necessary for the result; the paper notes that posted prices can cease to be optimal when density-regularity fails.

Core claim

In a simultaneous-move game where firms post menus and a buyer selects one contract, density-regularity ensures each firm's revenue-maximizing reply to any fixed profile of opponent menus consists solely of a posted-price contract. This equivalence is obtained by recasting the firm's problem as an optimal-control program whose solution is the posted price.

What carries the argument

The density-regularity condition on the joint distribution of the buyer's valuations, which enables the optimal-control argument to rule out profitable deviations to richer menus.

If this is right

Any Nash equilibrium of the menu-posting game must involve only posted-price strategies when density-regularity holds.
Complex instruments such as quantity discounts or lotteries cannot improve a firm's revenue against arbitrary competition.
The competitive outcome can be analyzed by restricting attention to posted-price strategies without loss of generality.
The standard single-seller logic must be replaced by an optimal-control approach once competition and differentiated varieties are introduced.

Where Pith is reading between the lines

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

Market data showing only posted prices across competitors could be consistent with density-regularity even if richer menus are feasible.
The result suggests that menu-design regulations may be redundant in markets where buyer valuations are known to be density-regular.
Similar best-response simplifications might hold in dynamic or multi-buyer extensions provided an analogous regularity condition is imposed.

Load-bearing premise

The distribution of the buyer's valuations must satisfy density-regularity.

What would settle it

A valuation distribution that violates density-regularity together with an explicit rival menu profile against which some firm earns strictly higher revenue by posting a menu containing more than one contract.

read the original abstract

We study competition among multiple firms that offer differentiated varieties of the same good to a unit-demand agent. The agent has heterogeneous valuations for goods from different firms. Firms do not observe the agent's exact valuations, but they know their distribution. Firms simultaneously post menus of contracts, after which the agent chooses a firm and one of its contracts to maximize her utility. This defines a game in which firms aim to maximize expected revenue. We introduce a sufficient condition, density-regularity, under which each firm's best response to any arbitrary menu profile posted by its opponents is equivalent to posting a menu that contains only a posted-price contract. Our result is not a direct extension of the canonical Myersonian model with a single seller. The standard argument in the literature breaks down once heterogeneous preferences and competition are introduced. We therefore adopt an optimal-control approach, in which the density-regularity condition is essential for establishing the optimality of posted prices. When this condition fails, posted prices may fail to be a best response.

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

The paper supplies a density-regularity condition that makes posted prices a best response in competitive differentiated screening, using optimal control rather than virtual valuations.

read the letter

The central claim is that density-regularity on the valuation distribution turns each firm's best response into a single posted price, even when rivals post arbitrary menus. This is not presented as a routine extension of Myerson; the abstract notes the standard argument fails under heterogeneous preferences and competition, so the paper switches to an optimal-control setup where the condition becomes essential.

What works is the clear motivation for the new method and the explicit statement that the result is sufficient rather than general. The setup matches standard models of unit-demand buyers facing multiple differentiated sellers, and the claim that posted prices can emerge as equilibrium strategies under the condition is a concrete simplification for applied work.

The soft spot is the condition itself. It must hold for the optimal-control argument to deliver posted prices, and the paper acknowledges that without it the equivalence can break. How restrictive density-regularity is on common distributions, and whether the control problem is set up without hidden regularity assumptions, would need checking in the full derivation. The abstract alone does not let one verify the steps.

This is for theorists working on multi-principal mechanism design or IO models with asymmetric information and competition. A reader who needs to know when menus collapse to prices would find the condition and the control approach worth seeing. It is coherent on its own terms and deserves a serious referee to check the proof details and the scope of the condition.

Referee Report

0 major / 2 minor

Summary. The paper studies a game of competitive screening in which multiple firms simultaneously post menus of contracts to a unit-demand agent with heterogeneous valuations drawn from a known distribution. It introduces a sufficient condition called density-regularity under which each firm's best response to any arbitrary menu profile of its opponents is to post a menu containing only a single posted-price contract. The result is obtained via an optimal-control approach because the standard single-seller virtual-surplus argument fails once heterogeneous preferences across firms and competition are introduced; the paper explicitly states that the result is not a direct Myersonian extension.

Significance. If the density-regularity condition and the optimal-control argument establish the claimed equivalence, the result would supply a clean sufficient condition for posted-price optimality in multi-firm screening environments. This addresses a genuine modeling gap, since the canonical single-seller logic does not carry over, and could serve as a building block for further work on competitive mechanism design. The paper's careful framing that the condition is sufficient rather than necessary, and that a new method is required, is appropriate.

minor comments (2)

[Abstract] Abstract: the claim that the standard argument 'breaks down' is stated without even a one-sentence indication of the specific point of failure (e.g., violation of single-crossing or non-monotonicity of virtual values); adding this would help readers assess the necessity of the new approach.
[Abstract] The abstract refers to 'density-regularity' but supplies neither its formal definition nor the precise statement of the optimal-control problem; both should appear early in the introduction or in a dedicated preliminary section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the paper and for acknowledging the significance of providing a sufficient condition for posted-price optimality under competition, where standard single-seller arguments do not apply. The recommendation is listed as uncertain, but the report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces density-regularity as a new sufficient condition and uses an optimal-control formulation to prove that posted-price menus are best responses. The abstract explicitly states that the standard single-seller Myerson argument fails under competition and heterogeneous preferences, so the result is not obtained by re-labeling or extending prior results by definition. No equations reduce the claimed equivalence to a fitted parameter or self-referential definition, no load-bearing self-citations appear, and the derivation is self-contained against the stated modeling assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard domain assumptions about information and timing plus the newly introduced density-regularity condition as a sufficient mathematical restriction on valuation distributions.

axioms (3)

domain assumption Firms know only the distribution of the agent's valuations, not the exact realizations.
Stated directly in the abstract as the information structure of the game.
domain assumption Firms post menus simultaneously and the agent then selects a firm and contract to maximize utility.
Core timing and choice rule described in the abstract.
domain assumption The agent has unit demand.
Model primitive stated in the abstract.

pith-pipeline@v0.9.1-grok · 5683 in / 1485 out tokens · 30508 ms · 2026-06-26T09:11:45.567866+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references

[1]

The existence of such an equilib- rium follows from the proof of Corollary 3.1

be an equilibrium of the duopoly price-posting game. The existence of such an equilib- rium follows from the proof of Corollary 3.1. LetM ⋆ be the posted-price menu profile corresponding to (p ⋆ 1, p⋆ 2). By construction, each firm is best responding to the other in the price-posting game. Moreover, by the posted-price best-response result in Theorem 3.1,...
[2]

We can conclude that (q ′ 1, t′

The rest of inequalities come directly from the definition. We can conclude that (q ′ 1, t′
[3]

As for allθ∈Θ ′ A \Θ A, they stick to the original contract (q 1, t1)

and (q 2(q1,θ), t 2(q1,θ)) are the optimal choice for allθ∈Θ A under M ′ 1, asM ′ 1 =M 1 ∪ {(q′ 1, t′ 1)}. As for allθ∈Θ ′ A \Θ A, they stick to the original contract (q 1, t1). Now, we consider any typeθ/∈Θ ′ A that originally chooses the contract (ˆq1, ˆt1)̸= (q 1, t1) under M, we must have, ˆq1θ1 − ˆt1 +q 2(ˆq1,θ)θ 2 −t 2(ˆq1,θ)> q 1θ1 −t 1 +q 2(q1,θ)θ...
[4]

Firm 1 can be strictly better off by deviating toM ′, which completes the proof thatMis not an equilibrium

and pay a higher transfer to firm 1, and for allθ̸∈Θ A will not change their original choice of contracts. Firm 1 can be strictly better off by deviating toM ′, which completes the proof thatMis not an equilibrium. B.3 Proof of Proposition 4.2 Let (M1, M2) be an equilibrium where each firm offers a finite menu of contracts. By Proposition 4.1, every equil...
[5]

Thus, p= ˆ∆(1−q k) +t k 1 +t k 2.(7) Since ∆k = ∆>0,q k <1, andt k 2 ≥0, we have ∆k(1−q k) +t k 2 >0

paired with (1−q k, tk 2): ˆ∆−p= ˆ∆qk −t k 1 −t k 2. Thus, p= ˆ∆(1−q k) +t k 1 +t k 2.(7) Since ∆k = ∆>0,q k <1, andt k 2 ≥0, we have ∆k(1−q k) +t k 2 >0. We can choose ˆ∆∈(∆ k−1,∆ k) sufficiently close to ∆ k such that ˆ∆(1−q k) +t k 2 >0 =⇒p−t k 1 = ˆ∆(1−q k) +t k 2 >0. 59 For any type ∆∈[∆ k−1,∆ k], the utility gain from (1, p) over (q k, tk
[6]

paired with (1−q k, tk
[7]

Hence, all types ∆> ˆ∆ strictly prefer (1, p), while all types ∆< ˆ∆ in the top interval do not prefer the new contract

is [∆−p]−[∆q k −t k 1 −t k 2] = (∆− ˆ∆)(1−q k). Hence, all types ∆> ˆ∆ strictly prefer (1, p), while all types ∆< ˆ∆ in the top interval do not prefer the new contract. We now show that no type below ∆k−1 wants to deviate to the new contract. For each contract (qj, tj
[8]

The utility gain from (1, p) over (q j, tj 1) paired with (1−q j, tj

withj≤k−1, consider types ∆∈[∆ j−1,∆ j]. The utility gain from (1, p) over (q j, tj 1) paired with (1−q j, tj
[9]

This expression is increasing in ∆, since 1−q j ≥0

is [∆−p]−[∆q j −t j 1 −t j 2] = ∆(1−q j)−p+t j 1 +t j 2. This expression is increasing in ∆, since 1−q j ≥0. Therefore, it is enough to evaluate it at the upper boundary ∆ = ∆ j ≤∆ k−1. By the chain of indifference conditions, at each threshold ∆l, forl=j, j+1, . . . , k−1, the agent is indifferent between consecutive contract pairs: ∆lql −t l 1 −t l 2 = ...
[10]

Summing these differences froml=jtok−1 yields ∆jqj −t j 1 −t j 2 = ∆jqk −t k 1 −t k 2 − k−1X l=j (∆j −∆ l)(ql+1 −q l)

= (∆j −∆ l)(ql+1 −q l). Summing these differences froml=jtok−1 yields ∆jqj −t j 1 −t j 2 = ∆jqk −t k 1 −t k 2 − k−1X l=j (∆j −∆ l)(ql+1 −q l). Therefore, the utility difference at ∆ = ∆ j is [∆j −p]−[∆ jqj −t j 1 −t j 2] 60 =∆j −p−  ∆jqk −t k 1 −t k 2 − k−1X l=j (∆j −∆ l)(ql+1 −q l)   =∆j −p−∆ jqk +t k 1 +t k 2 + k−1X l=j (∆j −∆ l)(ql+1 −q l) =(∆j − ˆ...
[11]

The change in firm 1’s expected revenue is ∆Π1 = (p−t k 1) H(∆ k)−H( ˆ∆)

to (1, p). The change in firm 1’s expected revenue is ∆Π1 = (p−t k 1) H(∆ k)−H( ˆ∆) . Sincep−t k 1 >0 and ˆ∆<∆ k, we have ∆Π1 >0. Thus firm 1 has a profitable deviation, contradicting the assumption that (M 1, M2) is an equilib- rium. Therefore, the largest active contract of firm 1 must satisfyq k = 1. B.4 Proof of Proposition 4.3 By Proposition 4.2, bot...
[12]

•Types with ∆>∆ K−1 buy exclusively from firm 1, choosing (1, p 1) from firm 1 and (0,0) from firm 2

from firm 2. •Types with ∆>∆ K−1 buy exclusively from firm 1, choosing (1, p 1) from firm 1 and (0,0) from firm 2. We denote the step sizes byα k =q k −q k−1 >0 fork= 1, . . . , K, and the total transfer for contractkbys k =t k 1 +t k
[13]

, K−1, the agent is indifferent between adjacent contractskandk+ 1: ∆k qk −s k = ∆k qk+1 −s k+1

We know that, at each boundary ∆ k fork= 0,1, . . . , K−1, the agent is indifferent between adjacent contractskandk+ 1: ∆k qk −s k = ∆k qk+1 −s k+1. This yields: sk+1 −s k = ∆k αk+1, k= 0,1, . . . , K−1.(8) Give the menu profile we can write firm 1’s expected revenue as Π1 = K−1X j=1 tj 1 H(∆ j)−H(∆ j−1) +p 1 1−H(∆ K−1) . Changingt k 1 shifts the boundari...
[14]

+ h(∆k) αk+1 (tk+1 1 −t k
[15]

By the same calculation, with ∂∆k−1 ∂tk 2 = 1 αk and ∂∆k ∂tk 2 = −1 αk+1 , setting ∂Π2 ∂tk 2 = 0 yields: H(∆ k)−H(∆ k−1) + h(∆k−1) αk (tk−1 2 −t k

= 0.(F1-k) Similarly, we can write firm 2’s expected revenue as Π2 =p 2 H(∆0) + K−1X j=1 tj 2 H(∆ j)−H(∆ j−1) . By the same calculation, with ∂∆k−1 ∂tk 2 = 1 αk and ∂∆k ∂tk 2 = −1 αk+1 , setting ∂Π2 ∂tk 2 = 0 yields: H(∆ k)−H(∆ k−1) + h(∆k−1) αk (tk−1 2 −t k
[16]

+ h(∆k) αk+1 (tk+1 2 −t k
[17]

= 0.(F2-k) Note that, since the given profile is an equilibrium, (F1-k) and (F2-k) must hold at the same time. Adding (F1-k) and (F2-k), and writings k =t k 1 +t k 2: 2 H(∆ k)−H(∆ k−1) + h(∆k−1) αk (sk−1 −s k) + h(∆k) αk+1 (sk+1 −s k) = 0.(9) Substituting the indifference conditions (8), which gives k−1 −s k =−∆ k−1αk ands k+1 −s k = ∆kαk+1: 2 H(∆ k)−H(∆ ...

[1] [1]

The existence of such an equilib- rium follows from the proof of Corollary 3.1

be an equilibrium of the duopoly price-posting game. The existence of such an equilib- rium follows from the proof of Corollary 3.1. LetM ⋆ be the posted-price menu profile corresponding to (p ⋆ 1, p⋆ 2). By construction, each firm is best responding to the other in the price-posting game. Moreover, by the posted-price best-response result in Theorem 3.1,...

[2] [2]

We can conclude that (q ′ 1, t′

The rest of inequalities come directly from the definition. We can conclude that (q ′ 1, t′

[3] [3]

As for allθ∈Θ ′ A \Θ A, they stick to the original contract (q 1, t1)

and (q 2(q1,θ), t 2(q1,θ)) are the optimal choice for allθ∈Θ A under M ′ 1, asM ′ 1 =M 1 ∪ {(q′ 1, t′ 1)}. As for allθ∈Θ ′ A \Θ A, they stick to the original contract (q 1, t1). Now, we consider any typeθ/∈Θ ′ A that originally chooses the contract (ˆq1, ˆt1)̸= (q 1, t1) under M, we must have, ˆq1θ1 − ˆt1 +q 2(ˆq1,θ)θ 2 −t 2(ˆq1,θ)> q 1θ1 −t 1 +q 2(q1,θ)θ...

[4] [4]

Firm 1 can be strictly better off by deviating toM ′, which completes the proof thatMis not an equilibrium

and pay a higher transfer to firm 1, and for allθ̸∈Θ A will not change their original choice of contracts. Firm 1 can be strictly better off by deviating toM ′, which completes the proof thatMis not an equilibrium. B.3 Proof of Proposition 4.2 Let (M1, M2) be an equilibrium where each firm offers a finite menu of contracts. By Proposition 4.1, every equil...

[5] [5]

Thus, p= ˆ∆(1−q k) +t k 1 +t k 2.(7) Since ∆k = ∆>0,q k <1, andt k 2 ≥0, we have ∆k(1−q k) +t k 2 >0

paired with (1−q k, tk 2): ˆ∆−p= ˆ∆qk −t k 1 −t k 2. Thus, p= ˆ∆(1−q k) +t k 1 +t k 2.(7) Since ∆k = ∆>0,q k <1, andt k 2 ≥0, we have ∆k(1−q k) +t k 2 >0. We can choose ˆ∆∈(∆ k−1,∆ k) sufficiently close to ∆ k such that ˆ∆(1−q k) +t k 2 >0 =⇒p−t k 1 = ˆ∆(1−q k) +t k 2 >0. 59 For any type ∆∈[∆ k−1,∆ k], the utility gain from (1, p) over (q k, tk

[6] [6]

paired with (1−q k, tk

[7] [7]

Hence, all types ∆> ˆ∆ strictly prefer (1, p), while all types ∆< ˆ∆ in the top interval do not prefer the new contract

is [∆−p]−[∆q k −t k 1 −t k 2] = (∆− ˆ∆)(1−q k). Hence, all types ∆> ˆ∆ strictly prefer (1, p), while all types ∆< ˆ∆ in the top interval do not prefer the new contract. We now show that no type below ∆k−1 wants to deviate to the new contract. For each contract (qj, tj

[8] [8]

The utility gain from (1, p) over (q j, tj 1) paired with (1−q j, tj

withj≤k−1, consider types ∆∈[∆ j−1,∆ j]. The utility gain from (1, p) over (q j, tj 1) paired with (1−q j, tj

[9] [9]

This expression is increasing in ∆, since 1−q j ≥0

is [∆−p]−[∆q j −t j 1 −t j 2] = ∆(1−q j)−p+t j 1 +t j 2. This expression is increasing in ∆, since 1−q j ≥0. Therefore, it is enough to evaluate it at the upper boundary ∆ = ∆ j ≤∆ k−1. By the chain of indifference conditions, at each threshold ∆l, forl=j, j+1, . . . , k−1, the agent is indifferent between consecutive contract pairs: ∆lql −t l 1 −t l 2 = ...

[10] [10]

Summing these differences froml=jtok−1 yields ∆jqj −t j 1 −t j 2 = ∆jqk −t k 1 −t k 2 − k−1X l=j (∆j −∆ l)(ql+1 −q l)

= (∆j −∆ l)(ql+1 −q l). Summing these differences froml=jtok−1 yields ∆jqj −t j 1 −t j 2 = ∆jqk −t k 1 −t k 2 − k−1X l=j (∆j −∆ l)(ql+1 −q l). Therefore, the utility difference at ∆ = ∆ j is [∆j −p]−[∆ jqj −t j 1 −t j 2] 60 =∆j −p−  ∆jqk −t k 1 −t k 2 − k−1X l=j (∆j −∆ l)(ql+1 −q l)   =∆j −p−∆ jqk +t k 1 +t k 2 + k−1X l=j (∆j −∆ l)(ql+1 −q l) =(∆j − ˆ...

[11] [11]

The change in firm 1’s expected revenue is ∆Π1 = (p−t k 1) H(∆ k)−H( ˆ∆)

to (1, p). The change in firm 1’s expected revenue is ∆Π1 = (p−t k 1) H(∆ k)−H( ˆ∆) . Sincep−t k 1 >0 and ˆ∆<∆ k, we have ∆Π1 >0. Thus firm 1 has a profitable deviation, contradicting the assumption that (M 1, M2) is an equilib- rium. Therefore, the largest active contract of firm 1 must satisfyq k = 1. B.4 Proof of Proposition 4.3 By Proposition 4.2, bot...

[12] [12]

•Types with ∆>∆ K−1 buy exclusively from firm 1, choosing (1, p 1) from firm 1 and (0,0) from firm 2

from firm 2. •Types with ∆>∆ K−1 buy exclusively from firm 1, choosing (1, p 1) from firm 1 and (0,0) from firm 2. We denote the step sizes byα k =q k −q k−1 >0 fork= 1, . . . , K, and the total transfer for contractkbys k =t k 1 +t k

[13] [13]

, K−1, the agent is indifferent between adjacent contractskandk+ 1: ∆k qk −s k = ∆k qk+1 −s k+1

We know that, at each boundary ∆ k fork= 0,1, . . . , K−1, the agent is indifferent between adjacent contractskandk+ 1: ∆k qk −s k = ∆k qk+1 −s k+1. This yields: sk+1 −s k = ∆k αk+1, k= 0,1, . . . , K−1.(8) Give the menu profile we can write firm 1’s expected revenue as Π1 = K−1X j=1 tj 1 H(∆ j)−H(∆ j−1) +p 1 1−H(∆ K−1) . Changingt k 1 shifts the boundari...

[14] [14]

+ h(∆k) αk+1 (tk+1 1 −t k

[15] [15]

By the same calculation, with ∂∆k−1 ∂tk 2 = 1 αk and ∂∆k ∂tk 2 = −1 αk+1 , setting ∂Π2 ∂tk 2 = 0 yields: H(∆ k)−H(∆ k−1) + h(∆k−1) αk (tk−1 2 −t k

= 0.(F1-k) Similarly, we can write firm 2’s expected revenue as Π2 =p 2 H(∆0) + K−1X j=1 tj 2 H(∆ j)−H(∆ j−1) . By the same calculation, with ∂∆k−1 ∂tk 2 = 1 αk and ∂∆k ∂tk 2 = −1 αk+1 , setting ∂Π2 ∂tk 2 = 0 yields: H(∆ k)−H(∆ k−1) + h(∆k−1) αk (tk−1 2 −t k

[16] [16]

+ h(∆k) αk+1 (tk+1 2 −t k

[17] [17]

= 0.(F2-k) Note that, since the given profile is an equilibrium, (F1-k) and (F2-k) must hold at the same time. Adding (F1-k) and (F2-k), and writings k =t k 1 +t k 2: 2 H(∆ k)−H(∆ k−1) + h(∆k−1) αk (sk−1 −s k) + h(∆k) αk+1 (sk+1 −s k) = 0.(9) Substituting the indifference conditions (8), which gives k−1 −s k =−∆ k−1αk ands k+1 −s k = ∆kαk+1: 2 H(∆ k)−H(∆ ...