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arxiv: 2604.04405 · v1 · submitted 2026-04-06 · 💰 econ.TH

Recognition: 2 theorem links

· Lean Theorem

Coarse Screening

Rui Sun, Yi Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:02 UTC · model grok-4.3

classification 💰 econ.TH
keywords screeninginformation designmechanism designlimited liabilityrational inattentionoptimal signalscoarse informationeffective policy dimension
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The pith

The optimal signal for investigating a buyer before pricing is always coarse, using at most three outcomes no matter how many types exist.

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

When a seller can acquire costly information about a buyer before choosing prices and quantities, the best signal uses at most three outcomes regardless of how many types the buyer might have. This follows because the seller must make two separate decisions once the signal arrives: whether to sell and what price to charge. Limited liability forces the seller to use the signal for both decisions instead of relying solely on payments to control information rents. As a result, every type of buyer trades with positive probability and the exclusion of low types from the classic model disappears. The same bound applies to any strictly convex cost of acquiring information.

Core claim

The seller designs both an information structure and a mechanism to maximize expected profit minus information cost. The central discovery is that an optimal information structure exists with at most three signal realizations for each buyer. This ternary bound equals the effective policy dimension, which counts the independent post-signal choices plus one. Screening involves the allocation decision and the pricing decision, and limited liability ensures these cannot be collapsed into one.

What carries the argument

The effective policy dimension, equal to the number of independent decisions the seller makes after receiving the signal plus one.

Load-bearing premise

The seller cannot impose arbitrarily large penalties on the buyer due to limited liability.

What would settle it

Solve the optimal signal for a continuous buyer type space under a quadratic information cost and check whether any optimal signal uses four or more realizations.

Figures

Figures reproduced from arXiv: 2604.04405 by Rui Sun, Yi Zhang.

Figure 1
Figure 1. Figure 1: The geometric mechanism behind the support bound. Panel a: the dual [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The function MΛ(z) and its concave envelope Mc Λ (z) for vL = 0.5, πH = 0.6, γ = 1/2. The optimal experiment places mass on at most three points; here two, corresponding to a binary signal. Strong duality holds because the screening problem, for fixed experiments, is a finite-dimensional linear program. The Lagrangian is convex in Λ as a pointwise supremum of affine functions. The proof, which extends the … view at source ↗
Figure 3
Figure 3. Figure 3: Support of the optimal experiment |supp(F ∗ )| as a function of πHvH and vL, with vH = 10/11 and γ = 1/2. Dark blue: degenerate, no investigation. Turquoise: binary signal. Yellow: ternary signal. The dashed line is vL = πHvH. with R z dF = 1, pays cost R ψ(z) dF(z) satisfying Assumption 2, and chooses action a ∈ A with A compact. Suppose the dual post-signal objective is PΛ(z) = sup a∈A  αΛ(a) + z βΛ(a) … view at source ↗
Figure 4
Figure 4. Figure 4: Seller benefit from information acquisition, [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: , where all three kinks are distinct and lie in a non-concave region containing z = 1 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: N-type convergence study. The support bound |supp(F ∗ )| ≤ 3 holds at every report for all N tested. Investigation concentrates in a narrow region. 20 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: N = 80-type mechanism with optimized multipliers, ˆγ = 0.5. Panel a: allocation schedule, nearly identical to Myerson except in the investigation region. Panel b: the transfer cutoff κp departs from 1 after outer optimization, shown as red squares, unlike the Myerson case where κp ≡ 1. Panel c: investigation cost shifts toward Φ(θ) = 0, marked by the gray dotted line. Panel d: support ≤ 2 everywhere. 21 [… view at source ↗
Figure 8
Figure 8. Figure 8: Allocation q ∗ (θ) = w(η(θ)) for uniform types with entropy cost and Myerson multiplier λ = 1 − θ, at three values of ˆγ. Dashed: Myerson benchmark. With this multiplier, η is uniformly small and q ∗ ≈ 1/2 for all types. The optimal multiplier λ ∗ concentrates investigation near θ0, as the numerical solution in Section 6.2 shows. For any given multiplier λ, the seller’s revenue under the investigation mech… view at source ↗
Figure 9
Figure 9. Figure 9: Investigation, allocation, and welfare for [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
read the original abstract

A seller investigates a buyer before setting prices, balancing the cost of acquiring information against the gain from tailoring the contract to the buyer's private type. The optimal signal is coarse: no matter how rich the type space, the seller never needs more than three outcomes per buyer. The bound equals the number of independent post-signal decisions plus one, a quantity we call the effective policy dimension. Screening involves two decisions, whether to allocate and what to charge, giving the ternary bound. Limited liability is the source: without it, the price is pinned by the envelope, only the allocation decision remains, and signals are binary as in monitoring. The Myerson exclusion rule is an artifact of not investigating. With investigation, every marginal buyer trades with positive probability, governed by a universal function that connects information design to rational inattention. The bound holds for any strictly convex information cost.

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 / 3 minor

Summary. The paper analyzes a seller who acquires costly information about a buyer's private type before offering a contract, balancing information costs against gains from tailoring. It claims that the optimal signal is always coarse, requiring at most three outcomes per buyer for arbitrary type spaces. This ternary bound equals the effective policy dimension, defined as the number of independent post-signal decisions plus one; here, limited liability makes allocation and pricing independent, yielding dimension 2. The bound holds for any strictly convex information cost. Dropping limited liability reduces the problem to a single allocation decision and recovers binary signals. With investigation, the Myerson exclusion rule disappears and every marginal buyer trades with positive probability governed by a universal function linking information design to rational inattention.

Significance. If the dimensionality bound and its dependence on limited liability are established rigorously, the paper offers a sharp, portable result on the complexity of optimal signals in mechanism design with endogenous information. The effective-policy-dimension concept and the universal marginal-trading function provide a clean bridge between information design and rational inattention, while reframing the exclusion rule as an artifact of zero investigation. These contributions could shape subsequent work on coarse contracts and costly screening.

major comments (2)
  1. The central claim that the bound is exactly three outcomes rests on counting two independent decisions (allocation and pricing) under limited liability. The manuscript should explicitly derive why these decisions remain independent after the signal is realized and why no third decision (e.g., participation or type-dependent outside option) enters the count; a counter-example with a type-dependent participation constraint would falsify the dimension argument.
  2. The statement that the bound holds for arbitrary (possibly continuous) type spaces requires a precise statement of the information-cost functional and the topology on signals. If the proof relies on finite-type approximations, the passage to the continuum limit must be shown to preserve the three-outcome property; otherwise the result is limited to finite types.
minor comments (3)
  1. Define the effective policy dimension formally in the main text (rather than only in the abstract) and state its value for the baseline model before invoking it in the ternary-bound theorem.
  2. The universal function governing marginal-buyer trading probabilities should be stated explicitly (e.g., as an equation relating the posterior belief to the information cost) so that readers can verify its independence from the type distribution.
  3. Add a short discussion of how the result relates to the existing literature on rational inattention in mechanism design (e.g., references to Sims, Caplin-Dean, or Matějka-McKay) to clarify the incremental contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address the two major comments point by point below, indicating the revisions we will undertake to clarify the arguments.

read point-by-point responses

  1. Referee: The central claim that the bound is exactly three outcomes rests on counting two independent decisions (allocation and pricing) under limited liability. The manuscript should explicitly derive why these decisions remain independent after the signal is realized and why no third decision (e.g., participation or type-dependent outside option) enters the count; a counter-example with a type-dependent participation constraint would falsify the dimension argument.

    Authors: We agree that the independence of the two decisions requires a more explicit derivation in the text. In the revision we will add a dedicated paragraph immediately after the definition of effective policy dimension that solves the seller's post-signal problem explicitly: under limited liability the seller chooses an allocation probability q and a scalar price p for each realized signal, subject to the buyer's interim incentive and participation constraints. Because limited liability normalizes the outside option to zero independently of type, the participation constraint does not introduce an additional free decision variable; it is either slack or determines a boundary condition already embedded in the two-dimensional choice. We will also insert a remark stating that the result is specific to type-independent outside options and that a type-dependent participation constraint would indeed raise the dimension, but lies outside the maintained limited-liability environment of the model. revision: yes

  2. Referee: The statement that the bound holds for arbitrary (possibly continuous) type spaces requires a precise statement of the information-cost functional and the topology on signals. If the proof relies on finite-type approximations, the passage to the continuum limit must be shown to preserve the three-outcome property; otherwise the result is limited to finite types.

    Authors: The proof is formulated directly on general type spaces by treating the information cost as a strictly convex functional on the space of probability measures over the type space. We will revise the manuscript to state this functional and the weak topology explicitly in the model section and to note that the dimensionality argument bounding the number of signal outcomes relies only on the linear independence of the gradients of the two post-signal decisions, not on any finite-type approximation. Consequently the three-outcome bound carries over to continuous type spaces without additional limiting arguments. A short clarifying paragraph will be added to the main text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from model primitives

full rationale

The central claim derives the ternary bound on signal outcomes from the effective policy dimension, which equals the number of independent post-signal decisions (allocation plus pricing) plus one under limited liability. This is a direct counting argument from the contract structure and convex information costs, without any reduction of a prediction to a fitted input, self-referential definition, or load-bearing self-citation. When limited liability is removed the bound collapses to binary signals as stated, confirming the logic is driven by the primitives rather than by construction. The paper presents this as holding for arbitrary type spaces and any strictly convex cost, with no internal equations or steps that equate outputs to inputs by fiat.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard mechanism-design assumptions plus the new effective policy dimension; limited liability and strict convexity of information cost are invoked to obtain the bound.

axioms (2)
  • domain assumption Information cost is strictly convex
    Abstract states the bound holds for any strictly convex information cost.
  • domain assumption Limited liability constraint on transfers
    Abstract identifies limited liability as the source that leaves price and allocation as independent post-signal decisions.
invented entities (1)
  • effective policy dimension no independent evidence
    purpose: Quantity equal to number of independent post-signal decisions plus one that bounds optimal signal cardinality
    New concept introduced to generalize the ternary bound beyond screening

pith-pipeline@v0.9.0 · 5427 in / 1409 out tokens · 91863 ms · 2026-05-10T20:02:38.705356+00:00 · methodology

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Lean theorems connected to this paper

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