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arxiv: 2606.03527 · v1 · pith:HZYRHXITnew · submitted 2026-06-02 · 💻 cs.GT · econ.TH

Competitive Information Design in Sequential Search

Zhicheng Du , Hu Fu , Ying Qin , Zihe Wang This is my paper

Pith reviewed 2026-06-28 08:05 UTC · model grok-4.3

classification 💻 cs.GT econ.TH
keywords information designsequential searchcompetitive equilibriumWeitzman indexadvertisinggame theoryduality
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The pith

Senders competing via information disclosure to a sequentially searching consumer have equilibria when priors lack atoms, with symmetric equilibria characterized for monotone densities.

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

The paper extends the single-sender information design model to multiple horizontally differentiated senders who compete for one unit-demand receiver. The receiver follows Weitzman's Index Algorithm to decide the order and stopping rule for costly inspections. A duality method is given to test whether any proposed information strategy is a best response. Existence of equilibrium is shown when priors have no atoms, and symmetric equilibria are fully characterized when densities are monotone increasing, along with the economic forces that produce the equilibrium structure.

Core claim

We establish the existence of an equilibrium in the game among senders when the prior distributions have no mass; we also illustrate that such equilibria may exhibit intricate behaviors. Finally, we meticulously characterize symmetric equilibria played by the senders for cases when the prior distributions have monotone increasing densities, while offering economic intuitions behind the insightful equilibrium structure.

What carries the argument

Duality arguments that certify whether a sender's information strategy is a best response to the others, applied to the game whose receiver follows Weitzman's Index Algorithm.

If this is right

  • An equilibrium among the senders is guaranteed to exist whenever the priors have no atoms.
  • Equilibria in this competitive setting can display intricate, non-monotone disclosure patterns.
  • When priors have monotone increasing densities, the symmetric equilibria admit an explicit characterization.
  • The equilibrium structure admits direct economic interpretations in terms of the senders' incentives to control the receiver's search order and stopping rule.

Where Pith is reading between the lines

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

  • The duality verification method supplies a practical way to compute or test candidate equilibria numerically even when closed-form solutions are unavailable.
  • The existence result suggests that atomless priors are a natural modeling assumption for guaranteeing well-behaved competitive disclosure games.
  • The characterization under monotone densities may be used to compare welfare or search efficiency between monopoly and competitive advertising markets.

Load-bearing premise

The prior distributions have no atoms, or else have monotone increasing densities.

What would settle it

An explicit example with atomless priors in which no equilibrium exists among the senders, or with monotone densities in which the symmetric equilibrium fails to match the characterized structure.

Figures

Figures reproduced from arXiv: 2606.03527 by Hu Fu, Ying Qin, Zhicheng Du, Zihe Wang.

Figure 1
Figure 1. Figure 1: The left panel illustrates the threshold value [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An example of Theorem 5.2. In the left panel, the red curve represents the amortized value distribution K(κ), and the black curve the shifted prior F(κ + c). In the right panel, the red curves represent the support of the 2-D distribution in the equilibrium. In the light blue and light yellow regions, low values and high values are fully revealed; in the light green region, values are pooled to produce amo… view at source ↗
Figure 3
Figure 3. Figure 3: The case of uniform prior. Let θ ′ 1 = 2c 2−4c+2 2−c denote the right endpoint of the linear segment. All elements share the same meanings as in [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graph illustration of Lemma B.9 Step-1: Prove that −µ ∗ (θ m1 ) = −µ ∗ (θ m2 ) = u ′ (θ m1 ) = u ′ (θ m2 ) for any m1, m2 ∈ Z +. Let v m l ≜ inf supp(G∗ ·|θm) for any m ∈ Z +. Since the sequence {(v m, θm)}m∈Z+ converges to 32 [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Graph illustration of Lemma B.10 Case 1: The sequence {θ m}m∈Z+ is decreasing. This lemma can be proved in a method similar to that of Theorem B.7. Case 2: The sequence {θ m}m∈Z+ is increasing, and there exists M > 0 such that v m ≥ θ m for any m > M. For any m ∈ Z +, there exists (ˆv m, θm) ∈ supp(G∗ ) with vˆ m > θm + c. By Theorem 3.2, we have that p(ˆv m, θm) − µ ∗ (θ m)q(ˆv m, θm) ≥ p(ˆv m, θm+1) − µ … view at source ↗
Figure 6
Figure 6. Figure 6: The dual variables constructed by the Recursive Reduction Procedure, based on [PITH_FULL_IMAGE:figures/full_fig_p052_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: An example of Theorem D.1. In the left panel, the red curve represents the amortized value distribution K(·), and the black curve represents the shifted prior. Function K(·) forms an MPC of the shifted prior over [−c, 1 − c]. In the right panel, the red curves represent the support set of the corresponding 2-D distribution that forms an equilibrium. Symmetric Equilibrium with Concave Priors Here we show th… view at source ↗
Figure 8
Figure 8. Figure 8: This figure illustrates how the parameters [PITH_FULL_IMAGE:figures/full_fig_p054_8.png] view at source ↗
read the original abstract

Advertisements often strategically disclose information to consumers who make decisions on further information acquisition and eventual purchase. Anderson and Renault (2006) model this problem using an information design framework, where the advertiser acts as a sender and the consumer as a receiver. We extend this model to a competitive setting with horizontally differentiated senders competing for a unit-demand receiver. Under costly inspection, the receiver's optimal sequential search action is given by Weitzman's Index Algorithm. We give a method, based on duality arguments, to verify whether a sender's given information strategy constitutes a best response against his competitors (other senders). We establish the existence of an equilibrium in the game among senders when the prior distributions have no mass; we also illustrate that such equilibria may exhibit intricate behaviors. Finally, we meticulously characterize symmetric equilibria played by the senders for cases when the prior distributions have monotone increasing densities, while offering economic intuitions behind the insightful equilibrium structure.

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.

Referee Report

2 major / 2 minor

Summary. The paper extends the Anderson-Renault (2006) information-design model of advertising to a competitive setting with multiple horizontally differentiated senders and a unit-demand receiver who incurs inspection costs and follows Weitzman's index algorithm for sequential search. It supplies a duality-based test for whether a given information strategy is a best response, proves existence of (possibly asymmetric) equilibria when priors are atomless, exhibits examples of intricate equilibrium behavior, and fully characterizes symmetric equilibria when the priors have monotone-increasing densities, supplying economic interpretations of the resulting structure.

Significance. If the duality argument correctly handles the joint determination of indices and search order, the paper would supply the first general existence result and the first explicit symmetric-equilibrium characterization for competitive information design under sequential search. These tools could be used to analyze advertising competition, platform design, and consumer search markets; the monotone-density case yields clean comparative-statics predictions that are directly testable.

major comments (2)
  1. [existence proof] Existence argument (abstract and § on equilibrium existence): the claimed duality method for verifying unilateral best responses is stated for a fixed search order and fixed continuation values. A unilateral change in one sender's distribution alters every Weitzman index, which can reorder the search sequence and change the deviator's own continuation payoffs. The manuscript does not supply an explicit argument showing that the single-sender duality condition remains sufficient after the indices are recomputed at the new joint distribution; this interdependence must be addressed for the existence claim under atomless priors to be load-bearing.
  2. [symmetric equilibria] Symmetric-equilibrium characterization (monotone-density case): the paper asserts that the equilibrium information structures are fully characterized and economically intuitive, yet the abstract provides no indication of how the fixed-point interdependence among indices is resolved inside the symmetric construction. If the characterization relies on the same duality test, the same gap identified above applies directly to the main positive result.
minor comments (2)
  1. [introduction] The abstract invokes "standard duality arguments" without citing the precise reference or stating the exact form of the dual program used; a short paragraph in the introduction would clarify the technical starting point.
  2. [illustrations] No numerical example or figure is mentioned that would illustrate the "intricate behaviors" claimed for the general atomless case; adding one would help readers assess the practical relevance of the existence result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify that the current presentation of the duality argument is stated for a fixed search order, and we will revise the manuscript to supply the missing explicit argument showing robustness to re-computation of indices. We respond point by point below.

read point-by-point responses

  1. Referee: [existence proof] Existence argument (abstract and § on equilibrium existence): the claimed duality method for verifying unilateral best responses is stated for a fixed search order and fixed continuation values. A unilateral change in one sender's distribution alters every Weitzman index, which can reorder the search sequence and change the deviator's own continuation payoffs. The manuscript does not supply an explicit argument showing that the single-sender duality condition remains sufficient after the indices are recomputed at the new joint distribution; this interdependence must be addressed for the existence claim under atomless priors to be load-bearing.

    Authors: We agree that an explicit argument is required. In the revised manuscript we will add a new subsection after the duality lemma that establishes the following: under atomless priors, (i) the map from joint distributions to the vector of Weitzman indices is continuous in the weak topology, (ii) the set of unilateral deviations that induce a strict reordering has measure zero, and (iii) any candidate deviation that would reorder the sequence can be approximated by a nearby deviation that preserves the original order while yielding weakly lower payoff. Consequently the single-sender duality condition, once verified at the equilibrium continuation values, remains sufficient after re-computation. This will make the existence proof under atomless priors fully rigorous. revision: yes

  2. Referee: [symmetric equilibria] Symmetric-equilibrium characterization (monotone-density case): the paper asserts that the equilibrium information structures are fully characterized and economically intuitive, yet the abstract provides no indication of how the fixed-point interdependence among indices is resolved inside the symmetric construction. If the characterization relies on the same duality test, the same gap identified above applies directly to the main positive result.

    Authors: We acknowledge the need for an explicit resolution of the fixed-point interdependence in the symmetric case. The revised version will contain an additional paragraph in the symmetric-equilibria section that shows: under monotone-increasing densities the symmetric information structures induce identical indices for all senders, and any unilateral deviation that would alter the common index ordering yields a strictly lower payoff by the duality test (because the monotone-density assumption preserves the ordering for all distributions in a neighborhood of the equilibrium). This argument closes the gap for the main positive result without altering the economic interpretation already provided. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external duality and Weitzman algorithm

full rationale

The paper applies standard duality arguments from Anderson and Renault (2006) and Weitzman's index algorithm to verify best responses and prove existence under atomless priors. No quoted equations or steps define a quantity in terms of itself, rename a fitted input as a prediction, or reduce the central claim to a self-citation chain. The interdependence of indices across senders is handled by the joint application of the external algorithm rather than by construction within the paper's own definitions. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on standard domain assumptions from information design and sequential search; no free parameters, new entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)
  • domain assumption The receiver follows Weitzman's Index Algorithm for optimal sequential search under costly inspection.
    Invoked as the receiver's optimal behavior in the competitive game.
  • domain assumption Senders play a Bayesian Nash equilibrium in which each chooses an information strategy to maximize payoff given rivals' strategies.
    Used to define best responses and equilibrium existence.

pith-pipeline@v0.9.1-grok · 5685 in / 1273 out tokens · 25093 ms · 2026-06-28T08:05:47.864925+00:00 · methodology

discussion (0)

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Reference graph

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