Competitive Sequential Screening

Deniz Kattwinkel; Ian Ball; Jan Knoepfle

arxiv: 2602.08144 · v4 · submitted 2026-02-08 · 💰 econ.TH

Competitive Sequential Screening

Ian Ball , Deniz Kattwinkel , Jan Knoepfle This is my paper

Pith reviewed 2026-05-16 06:12 UTC · model grok-4.3

classification 💰 econ.TH

keywords sequential screeningoption contractsHotelling duopolyconsumer surpluslock-incompetitive contractingexclusive contracts

0 comments

The pith

In a Hotelling duopoly, sufficiently early contracting raises consumer surplus relative to spot pricing and reverses the monopoly ranking.

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

This paper examines competition between two firms offering contracts to consumers before those consumers learn their product preferences. Firms screen via menus of option contracts in a linear city model. Consumers select from both firms but become locked into the firm with the lower strike price option, producing inefficient consumption. Less informed consumers appear more homogeneous to the firms, so earlier contracting stiffens price competition. Sufficiently early timing improves consumer surplus over spot pricing, unlike in monopoly, and exclusive contracting raises surplus further by heightening rivalry.

Core claim

We characterize the unique equilibrium in which consumers select contracts from both firms. Each consumer is endogenously locked into the firm from which he chooses an option with a lower strike price. Lock-in yields inefficient consumption. Yet earlier contracting stiffens competition because less informed consumers are more homogeneous. Sufficiently early contracting raises consumer surplus relative to spot pricing -- reversing the ranking under monopoly. Exclusive contracting further increases consumer surplus by intensifying competition.

What carries the argument

Menus of option contracts that produce endogenous lock-in to the firm with the lower strike price, operating inside a Hotelling linear-city duopoly with quasi-linear preferences.

If this is right

Contract lock-in produces inefficient product consumption.
Earlier contracting homogenizes consumers and intensifies price competition.
Sufficiently early timing raises consumer surplus above the level under spot pricing.
Exclusive contracting arrangements increase consumer surplus by strengthening competitive pressure.

Where Pith is reading between the lines

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

The reversal of the monopoly ranking may extend to other competitive markets that allow advance contracting, such as insurance or subscription services.
Antitrust or consumer-protection rules could usefully distinguish between monopoly and duopoly when evaluating early-commitment practices.
Lab experiments that vary the timing of contract offers while holding the Hotelling structure fixed could directly test the predicted surplus ordering.

Load-bearing premise

Consumers choose contracts from both firms and lock into the lower-strike-price firm, with the linear-city structure causing earlier contracting to homogenize consumers.

What would settle it

A laboratory experiment or field data set comparing consumer surplus under early option contracting versus spot pricing in a controlled duopoly setting with measurable preference uncertainty.

Figures

Figures reproduced from arXiv: 2602.08144 by Deniz Kattwinkel, Ian Ball, Jan Knoepfle.

**Figure 1.** Figure 1: Consumer’s valuations along the Hotelling line consumer learns his position θ = γ + ε, where ε is a taste shock that is drawn independently of γ from an integrable, fullsupport distribution F, which has continuous, strictly positive density f : R → (0, ∞).11 Denote the set of consumer positions by Θ = R. Without loss, we assume E[ε] = 0. Thus, E[θ|γ] = γ, so the consumer’s type γ equals his expected posi… view at source ↗

**Figure 2.** Figure 2: Equilibrium with the uniform type distribution more premium than another if it has a lower strike price and a higher subscription fee. More rightward types expect to have stronger preferences for firm B over firm A, 26 so they select more premium subscriptions from firm B and less premium subscriptions from firm A. Intuitively, more rightward types are willing to pay more for a reduction in the strike pric… view at source ↗

**Figure 3.** Figure 3: Exclusive equilibrium allocation with the uniform type distribution derived from a first-order condition for each firm, as we discuss below. The assumption that v0 ≥ 1/g(γ † ) + |γ † | ensures that the consumer always selects a contract in the first period. Nevertheless, for each consumer type, there is a positive probability of not purchasing a product in the second period. The assumption that ¯ γ < 0 < … view at source ↗

**Figure 4.** Figure 4: Limiting equilibrium under early contracting firm posts a price. For each θ, valuations are common knowledge, so there is an associated complete-information (asymmetric) Bertrand game. If vi(θ) > v−i(θ)+, then firm i charges vi(θ)−v−i(θ)+ and firm −i charges 0; the consumer purchases product i. The resulting allocation is efficient. The consumer’s ex-ante expected payment to firm i is Eθ∼F [(vi(θ)−v−i(θ)+… view at source ↗

**Figure 5.** Figure 5: Consumer utility functions in the three competitive settings, with a uniform type distribution and a standard normal taste shock consumer’s interim utility function [PITH_FULL_IMAGE:figures/full_fig_p038_5.png] view at source ↗

read the original abstract

We study competition between firms that contract with consumers before the consumers fully learn their product preferences. In a Hotelling duopoly, firms screen consumers by offering menus of option contracts. We characterize the unique equilibrium. Consumers select contracts from both firms. Each consumer is endogenously locked into the firm from which he chooses an option with a lower strike price. Lock-in yields inefficient consumption. Yet earlier contracting stiffens competition because less informed consumers are more homogeneous. Sufficiently early contracting raises consumer surplus relative to spot pricing -- reversing the ranking under monopoly. Exclusive contracting further increases consumer surplus by intensifying competition.

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

Early contracting in this Hotelling duopoly raises consumer surplus over spot pricing because less-informed buyers become more homogeneous and competition on option menus intensifies, reversing the monopoly ranking.

read the letter

This paper characterizes the unique equilibrium of a competitive sequential-screening game in which two Hotelling firms offer option menus before consumers learn their locations. Consumers pick from both firms and lock into the firm with the lower strike price, which creates some consumption inefficiency. The main finding is that sufficiently early contracting still raises ex-ante consumer surplus relative to spot pricing because the uninformed types are more similar, forcing tighter menu competition that outweighs the lock-in cost. Exclusive contracting strengthens the effect further by sharpening rivalry. The derivation uses the standard quasi-linear Hotelling primitives and delivers an explicit equilibrium in which both firms' options are chosen and the lock-in rule is incentive-compatible for all types. That equilibrium characterization is the cleanest part of the work and supplies a concrete mechanism linking contract timing, competition intensity, and welfare. The monopoly contrast is used only to highlight the difference, not as a formal benchmark. The result sits squarely inside industrial-organization contract theory. A reader who already works on screening under competition or on the welfare effects of early contracting will see a clear new interaction. The limitation is that the welfare reversal is tied to the linear-city homogenization effect; other preference distributions or transport-cost structures could easily make lock-in dominate. The paper does not test robustness on that margin, so the quantitative claim stays local to these assumptions. I would send it to a serious referee. The equilibrium construction is explicit enough to be checked, and the mechanism is worth having on record even if later work relaxes the Hotelling setup.

Referee Report

3 major / 2 minor

Summary. The paper studies competition between two firms in a Hotelling duopoly that offer menus of option contracts to consumers before preferences are fully learned. It claims to characterize the unique equilibrium in which consumers select contracts from both firms and are endogenously locked into the firm offering the lower strike price, producing inefficient consumption. The central result is that sufficiently early contracting raises consumer surplus relative to spot pricing (reversing the monopoly ranking) because less-informed consumers are more homogeneous and competition intensifies; exclusive contracting further increases consumer surplus.

Significance. If the equilibrium characterization and the net welfare comparison hold, the paper offers a precise mechanism by which pre-contracting timing can reverse standard monopoly results on consumer surplus under competition. The explicit use of the Hotelling linear-city structure to generate endogenous homogeneity and lock-in provides a clean, falsifiable prediction that distinguishes competitive sequential screening from both monopoly screening and spot-market competition.

major comments (3)

[§3] §3 (equilibrium characterization): the claim that consumers always select contracts from both firms and that the lower-strike lock-in rule is incentive-compatible for all types on the Hotelling line is asserted but not shown to survive unilateral deviation by a firm to a single-contract menu; without an explicit check that the participation and incentive constraints bind uniformly across the type interval, the uniqueness result is not load-bearing.
[§4] §4 (consumer-surplus comparison): the reversal of the CS ranking versus spot pricing is stated to occur for sufficiently early contracting, yet the net effect of homogenization-driven competition intensification versus lock-in inefficiency is not derived in closed form or via explicit integration over the type distribution; the threshold at which the reversal occurs therefore cannot be verified from the primitives.
[§5] §5 (exclusive contracting): the claim that exclusive contracting further raises CS by intensifying competition rests on the same lock-in rule, but no comparative-static exercise shows that the welfare gain survives when the exclusivity constraint is imposed on the equilibrium menus derived in §3.

minor comments (2)

[preceding Eq. (3)] The definition of the strike-price lock-in rule in the text preceding Eq. (3) should explicitly state the tie-breaking rule when strike prices are equal.
[Figure 2] Figure 2 (welfare plots) uses the same line style for monopoly and duopoly CS; different dashing or color would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the equilibrium characterization, welfare comparisons, and exclusive contracting. We address each point below and will revise the manuscript to incorporate the suggested clarifications and additional derivations.

read point-by-point responses

Referee: [§3] §3 (equilibrium characterization): the claim that consumers always select contracts from both firms and that the lower-strike lock-in rule is incentive-compatible for all types on the Hotelling line is asserted but not shown to survive unilateral deviation by a firm to a single-contract menu; without an explicit check that the participation and incentive constraints bind uniformly across the type interval, the uniqueness result is not load-bearing.

Authors: We agree that an explicit verification is required for rigor. The current derivation assumes bilateral menu selection and derives the lock-in rule from incentive compatibility, but we will add a dedicated subsection in §3 that (i) computes firm profits under unilateral deviation to a single-contract menu and shows strict dominance of the candidate equilibrium, and (ii) explicitly solves the participation and incentive constraints for every type t ∈ [0,1] to confirm uniform binding. These additions will establish that the equilibrium survives deviations and that uniqueness holds. revision: yes
Referee: [§4] §4 (consumer-surplus comparison): the reversal of the CS ranking versus spot pricing is stated to occur for sufficiently early contracting, yet the net effect of homogenization-driven competition intensification versus lock-in inefficiency is not derived in closed form or via explicit integration over the type distribution; the threshold at which the reversal occurs therefore cannot be verified from the primitives.

Authors: The comparison is obtained by integrating equilibrium consumer utilities over the type distribution, with the threshold expressed in terms of the information-precision parameter. To make the net effect fully transparent, we will insert the explicit integral expressions for consumer surplus under sequential screening and under spot pricing, derive the closed-form difference, and solve for the critical timing threshold directly in terms of the Hotelling transportation cost and type distribution primitives. revision: yes
Referee: [§5] §5 (exclusive contracting): the claim that exclusive contracting further raises CS by intensifying competition rests on the same lock-in rule, but no comparative-static exercise shows that the welfare gain survives when the exclusivity constraint is imposed on the equilibrium menus derived in §3.

Authors: We will add a new comparative-static subsection that imposes the exclusivity constraint on the §3 menus, re-solves for the resulting equilibrium contracts, and recomputes consumer surplus. The exercise will show that the competition-intensification effect continues to dominate lock-in inefficiency for sufficiently early contracting, preserving the welfare ranking. The revised analysis will report the modified equilibrium conditions and the associated surplus comparison. revision: yes

Circularity Check

0 steps flagged

No significant circularity in equilibrium derivation

full rationale

The paper derives its central results from an explicit, self-contained characterization of the unique Nash equilibrium in a Hotelling duopoly with quasi-linear preferences and linear city structure. Firms offer menus of option contracts; consumers choose from both firms and endogenously lock into the lower-strike option; earlier contracting homogenizes types and intensifies menu competition. Consumer-surplus comparisons to spot pricing and monopoly follow directly from solving these equilibrium conditions rather than from any fitted parameter, self-referential definition, or load-bearing self-citation. The reversal of the CS ranking is a computed outcome of the model primitives, not a renaming or smuggling of prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model relies on standard Hotelling linear transportation costs, quasi-linear consumer utility, and the assumption that consumers can costlessly choose one option from each firm; no free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption Consumers have quasi-linear preferences and can select one contract from each firm before learning their types.
Stated in the abstract as the basis for endogenous lock-in.
standard math Firms compete in a Hotelling duopoly with fixed locations.
Classic setup invoked to generate competition intensity that varies with information.

pith-pipeline@v0.9.0 · 5381 in / 1414 out tokens · 63155 ms · 2026-05-16T06:12:38.205830+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

firms screen consumers by offering menus of option contracts... Consumers select contracts from both firms. Each consumer is endogenously locked into the firm from which he chooses an option with a lower strike price.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

earlier contracting stiffens competition because less informed consumers are more homogeneous... Sufficiently early contracting raises consumer surplus relative to spot pricing—reversing the ranking under monopoly.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Competitive Price Discrimination,

[5] Armstrong, M. and J. Vickers(2001): “Competitive Price Discrimination,” RAND Journal of Economics, 32, 579–605. [6, 26] ——— (2010): “Competitive Non-linear Pricing and Bundling,”Review of Economic Studies, 77, 30–60. [6, 18] Armstrong, M. and J. Zhou(2022): “Consumer Information and the Limits to Competition,”American Economic Review, 112, 534–577. [5...

work page doi:10.2139/ssrn.5361609 2001
[2]

Nonlinear Pricing in Oligopoly: How Brand Preferences Shape Market Outcomes,

[6] Gomes, R., J.-M. Lozachmeur, and L. Maestri(2022): “Nonlinear Pricing in Oligopoly: How Brand Preferences Shape Market Outcomes,” TSE Working Paper

work page 2022
[3]

Cellular Service Demand: Biased Beliefs, Learning, and Bill Shock,

[6] Grubb, M. D. and M. Osborne(2015): “Cellular Service Demand: Biased Beliefs, Learning, and Bill Shock,”American Economic Review, 105, 234–271. [6] Guerrieri, V., R. Shimer, and R. Wright(2010): “Adverse Selection in Com- petitive Search Equilibrium,”Econometrica, 78, 1823–1862. [6] Kr¨ahmer, D. and R. Strausz(2015): “Optimal Sales Contracts with Withd...

work page 2015
[4]

For each typeγ∈Γ, the setP(γ;s A, sB)is a nonempty, complete sublattice

work page
[5]

Part 1 ensures that the infimum and supremum in part 2 exist

For all typesγ, γ ′ ∈Γ, ifγ > γ ′, theninfP(γ;s A, sB)⪰supP(γ ′;s A, sB). Part 1 ensures that the infimum and supremum in part 2 exist. Part 2 implies that every selection from the correspondenceP(·;s A, sB) is weakly increasing (with 58 respect to⪰). Intuitively, more rightward types choose higher strike prices at firmA and lower strike prices at firmB. ...

work page
[6]

For each typeγ∈Γand pricep B ∈[0,∞], the setP A(pB, γ;s A)is nonempty and closed

work page
[7]

For all typesγ∈Γ, the mapp B 7→P A(pB, γ;s A)is weakly decreasing (in the strong set order). 59

work page
[8]

For all typesγ, γ ′ ∈Γand pricesp B, p′ B ∈[0,∞], ifγ > γ ′ andp B ≤p ′ B, then infP A(pB, γ;s A)≥supP A(p′ B, γ′;s A). Given an arbitrary subscription schedules A offered by firmA, we next show that firmBcan, through its choice of subscription schedule, induce the consumer to follow any monotone price-selection rulep B : Γ→[0,∞] at firmB. An analogous re...

work page
[9]

Sinceu(·, γ) is supermodular, the setP(γ) is a sublattice of [0,∞] 2

Fixγ∈Γ. Sinceu(·, γ) is supermodular, the setP(γ) is a sublattice of [0,∞] 2. Sinceu(·, γ) is upper semicontinuous and [0,∞] 2 is complete, we conclude that P(γ) is complete. 63

work page
[10]

163), but we include the proof for completeness

This part essentially follows from the monotone selection theorem (Milgrom and Shannon, 1994, Theorem 4’, p. 163), but we include the proof for completeness. Fix typesγ, γ ′ withγ > γ ′. Write ¯ p(γ) = infP(γ) and ¯p(γ ′) = supP(γ ′). Since ¯p(γ′) is inP(γ ′), we haveu(¯p(γ′), γ′)≥u( ¯ p(γ)∧¯p(γ ′), γ′). Sinceu(·, γ ′) is supermodular, it follows thatu( ¯...

work page 1994
[11]

For each typeγ∈[ ¯ γ,¯γA), we havep A(γ)<∞andp B(γ) =∞

work page
[12]

For each typeγ∈(¯γ A, ¯ γB), we havep A(γ) =∞andp B(γ) =∞

work page
[13]

With these preliminaries established, we turn to the main proof

For each typeγ∈( ¯ γB,¯γ], we havep B(γ)<∞andp A(γ) =∞. With these preliminaries established, we turn to the main proof. Denote the strategy profile in Proposition 4 by (s E A, sE B, pE, qE).62 The proof has three parts. To show that (s E A, sE B, pE, qE) is an equilibrium, we check that the consumer is playing a best response, and then we check that each...

work page 2002
[14]

Therefore, we may assume thatε A < γ0 −γ

In this case, (64) is satisfied. Therefore, we may assume thatε A < γ0 −γ. Hence, εA < γ 0 −γ < ε B. If ∆(γ)≥κ, thenC(κ)∆(γ)≥2f(0)≥2f(γ 0 −γ), so (64) is immediate. Suppose that ∆(γ)< κ. We have 2f(γ 0 −γ)−(f(ε B) +f(ε A)) ≤ Z γ0−γ εA |f ′(z)|dz+ Z εB γ0−γ |f ′(z)|dz = Z γ0−γ εA |f ′(z)| f(z) f(z) dz+ Z εB γ0−γ |f ′(z)| f(z) f(z) dz ≤sup{|f ′(ε)|/f(ε) :ε ...

work page

[1] [1]

Competitive Price Discrimination,

[5] Armstrong, M. and J. Vickers(2001): “Competitive Price Discrimination,” RAND Journal of Economics, 32, 579–605. [6, 26] ——— (2010): “Competitive Non-linear Pricing and Bundling,”Review of Economic Studies, 77, 30–60. [6, 18] Armstrong, M. and J. Zhou(2022): “Consumer Information and the Limits to Competition,”American Economic Review, 112, 534–577. [5...

work page doi:10.2139/ssrn.5361609 2001

[2] [2]

Nonlinear Pricing in Oligopoly: How Brand Preferences Shape Market Outcomes,

[6] Gomes, R., J.-M. Lozachmeur, and L. Maestri(2022): “Nonlinear Pricing in Oligopoly: How Brand Preferences Shape Market Outcomes,” TSE Working Paper

work page 2022

[3] [3]

Cellular Service Demand: Biased Beliefs, Learning, and Bill Shock,

[6] Grubb, M. D. and M. Osborne(2015): “Cellular Service Demand: Biased Beliefs, Learning, and Bill Shock,”American Economic Review, 105, 234–271. [6] Guerrieri, V., R. Shimer, and R. Wright(2010): “Adverse Selection in Com- petitive Search Equilibrium,”Econometrica, 78, 1823–1862. [6] Kr¨ahmer, D. and R. Strausz(2015): “Optimal Sales Contracts with Withd...

work page 2015

[4] [4]

For each typeγ∈Γ, the setP(γ;s A, sB)is a nonempty, complete sublattice

work page

[5] [5]

Part 1 ensures that the infimum and supremum in part 2 exist

For all typesγ, γ ′ ∈Γ, ifγ > γ ′, theninfP(γ;s A, sB)⪰supP(γ ′;s A, sB). Part 1 ensures that the infimum and supremum in part 2 exist. Part 2 implies that every selection from the correspondenceP(·;s A, sB) is weakly increasing (with 58 respect to⪰). Intuitively, more rightward types choose higher strike prices at firmA and lower strike prices at firmB. ...

work page

[6] [6]

For each typeγ∈Γand pricep B ∈[0,∞], the setP A(pB, γ;s A)is nonempty and closed

work page

[7] [7]

For all typesγ∈Γ, the mapp B 7→P A(pB, γ;s A)is weakly decreasing (in the strong set order). 59

work page

[8] [8]

For all typesγ, γ ′ ∈Γand pricesp B, p′ B ∈[0,∞], ifγ > γ ′ andp B ≤p ′ B, then infP A(pB, γ;s A)≥supP A(p′ B, γ′;s A). Given an arbitrary subscription schedules A offered by firmA, we next show that firmBcan, through its choice of subscription schedule, induce the consumer to follow any monotone price-selection rulep B : Γ→[0,∞] at firmB. An analogous re...

work page

[9] [9]

Sinceu(·, γ) is supermodular, the setP(γ) is a sublattice of [0,∞] 2

Fixγ∈Γ. Sinceu(·, γ) is supermodular, the setP(γ) is a sublattice of [0,∞] 2. Sinceu(·, γ) is upper semicontinuous and [0,∞] 2 is complete, we conclude that P(γ) is complete. 63

work page

[10] [10]

163), but we include the proof for completeness

This part essentially follows from the monotone selection theorem (Milgrom and Shannon, 1994, Theorem 4’, p. 163), but we include the proof for completeness. Fix typesγ, γ ′ withγ > γ ′. Write ¯ p(γ) = infP(γ) and ¯p(γ ′) = supP(γ ′). Since ¯p(γ′) is inP(γ ′), we haveu(¯p(γ′), γ′)≥u( ¯ p(γ)∧¯p(γ ′), γ′). Sinceu(·, γ ′) is supermodular, it follows thatu( ¯...

work page 1994

[11] [11]

For each typeγ∈[ ¯ γ,¯γA), we havep A(γ)<∞andp B(γ) =∞

work page

[12] [12]

For each typeγ∈(¯γ A, ¯ γB), we havep A(γ) =∞andp B(γ) =∞

work page

[13] [13]

With these preliminaries established, we turn to the main proof

For each typeγ∈( ¯ γB,¯γ], we havep B(γ)<∞andp A(γ) =∞. With these preliminaries established, we turn to the main proof. Denote the strategy profile in Proposition 4 by (s E A, sE B, pE, qE).62 The proof has three parts. To show that (s E A, sE B, pE, qE) is an equilibrium, we check that the consumer is playing a best response, and then we check that each...

work page 2002

[14] [14]

Therefore, we may assume thatε A < γ0 −γ

In this case, (64) is satisfied. Therefore, we may assume thatε A < γ0 −γ. Hence, εA < γ 0 −γ < ε B. If ∆(γ)≥κ, thenC(κ)∆(γ)≥2f(0)≥2f(γ 0 −γ), so (64) is immediate. Suppose that ∆(γ)< κ. We have 2f(γ 0 −γ)−(f(ε B) +f(ε A)) ≤ Z γ0−γ εA |f ′(z)|dz+ Z εB γ0−γ |f ′(z)|dz = Z γ0−γ εA |f ′(z)| f(z) f(z) dz+ Z εB γ0−γ |f ′(z)| f(z) f(z) dz ≤sup{|f ′(ε)|/f(ε) :ε ...

work page