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arxiv: 2206.09918 · v4 · submitted 2022-06-20 · 💰 econ.TH

Withholding Verifiable Information

Denis Shishkin , Maria Titova , Kun Zhang This is my paper

Pith reviewed 2026-05-24 12:01 UTC · model grok-4.3

classification 💰 econ.TH
keywords disclosure gamesinformation designsender-receiver gameslaminar partitionscommitment powerequilibrium payoffsverifiable informationpooling
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The pith

Any equilibrium payoff in these disclosure games can be achieved by a laminar partition that pools nonadjacent states.

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

This paper studies finite-action disclosure games where the sender's payoff depends only on the action chosen by the receiver, not on the true state, and the receiver's optimal action depends only on the expected state. Full revelation is receiver-preferred, but the authors focus on characterizing all other equilibria. They establish that every equilibrium payoff can be replicated exactly by a disclosure strategy whose support is a laminar partition, a nested family of sets that can pool non-consecutive states. The laminar form is then used to derive conditions under which the sender gains nothing from committing in advance to a disclosure rule.

Core claim

We show that any equilibrium payoff can be obtained with a disclosure strategy corresponding to a partition with a laminar structure that allows pooling of nonadjacent states. In a sender-preferred equilibrium, such a structure balances inducing more sender-favorable actions with deterring deviations. Leveraging this insight, we identify conditions under which the sender does not benefit from commitment power. We then apply these results to study selling with quality disclosure and influencing voters.

What carries the argument

Laminar partition of the state space: a collection of intervals in which any two are either disjoint or one contains the other, used to define the sender's disclosure strategy that replicates any equilibrium payoff.

If this is right

  • Sender-preferred equilibria can always be supported by laminar disclosure rules that pool some nonadjacent states.
  • There exist conditions on the action space and payoffs under which the sender's ability to commit to a disclosure policy yields no gain.
  • The same laminar characterization applies directly to quality-disclosure models in selling and to models of voter persuasion.

Where Pith is reading between the lines

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

  • Disclosure policies that appear complex may often be replaced by simpler nested pooling rules without changing the equilibrium outcome.
  • The restriction to laminar structures may help identify which verifiable-information environments allow tractable computation of equilibrium sets.
  • Regulators interested in limiting sender influence could focus on whether observed disclosure patterns are laminar or require non-laminar pooling.

Load-bearing premise

The sender's payoff depends only on the receiver's action and not on the true state, while the receiver's optimal action depends only on the conditional expectation of the state.

What would settle it

An equilibrium payoff vector in one of these games that cannot be produced by any disclosure strategy whose induced partition is laminar.

Figures

Figures reproduced from arXiv: 2206.09918 by Denis Shishkin, Kun Zhang, Maria Titova.

Figure 1
Figure 1. Figure 1: An illustration for Sender’s value function, which is a function of posterior mean [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A deterministic representation (upper panel) and a canonical representation (lower [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Violation of the conditions in Proposition 3.5: in the left panel condition (i) is violated because 2 is skipped, but 2 < 1 where 1 is a “lower” action that is not skipped; in the right panel condition (ii) is violated since {2, 3} is a nested intervals representation, but 2 > 3. action is skipped, it must be that no “lower” action is recommended in any state in which is optimal under complete information.… view at source ↗
Figure 4
Figure 4. Figure 4: The two cases when 3 is a pooling interval with 3 > 4. In the left panel there exists ∈ 3 with < 3, and in the right panel 2 ∉ 3. To get a closer look into how Condition (2) works, consider the following example. Let {} −1 =0 be a canonical representation of a bi-pooling solution, and suppose 3 is a pooling interval with 3 > 4. 20 By Proposition 3.5, this bi-pooling solution cannot be implemented; I show t… view at source ↗
Figure 5
Figure 5. Figure 5: The left panel illustrates the geometric interpretation of inequality [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The deterministic representation in the upper panel is not a laminar representation [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Voter ’s utilities. subsets of the state space that contains the true state. Assume that the expert’s preferences satisfy () > () > (∅) = 0; that is, the expert strictly prefers the bill to the amended bill, and the amended bill is strictly preferred to no bill. Because the voting rule satises the Condorcet winner criterion, and the preferences are single-peaked, the Condorcet winner is the median voter’s… view at source ↗
Figure 8
Figure 8. Figure 8: An illustration for Receiver’s optimal actions when [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
read the original abstract

We study a class of finite-action disclosure games in which the sender's preferences are state-independent and the receiver's optimal action depends only on the expected state. While receiver-preferred equilibria in these games involve full revelation, other equilibria are less well understood. We show that any equilibrium payoff can be obtained with a disclosure strategy corresponding to a partition with a laminar structure that allows pooling of nonadjacent states. In a sender-preferred equilibrium, such a structure balances inducing more sender-favorable actions with deterring deviations. Leveraging this insight, we identify conditions under which the sender does not benefit from commitment power. We then apply these results to study selling with quality disclosure and influencing voters.

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

1 major / 1 minor

Summary. The paper examines a restricted class of finite-action disclosure games in which the sender has state-independent preferences and the receiver's optimal action depends only on the expected state. It claims to characterize all equilibrium payoffs as achievable via disclosure strategies corresponding to laminar partitions that permit pooling of nonadjacent states. The authors then apply the structure to sender-preferred equilibria (balancing favorable actions against deviation deterrence), conditions under which the sender gains no value from commitment, and applications to quality disclosure in sales and voter influence.

Significance. If the laminar-structure characterization holds, the result supplies a concrete structural tool for analyzing equilibria in this class of games, which may simplify identification of sender-preferred outcomes and commitment value. The explicit scoping to state-independent sender preferences and expectation-based receiver actions is a modeling choice that renders the claim tractable; the paper correctly grounds the result in these assumptions rather than claiming generality.

major comments (1)
  1. [Abstract / main characterization theorem] Abstract and the statement of the main characterization result: the claim that any equilibrium payoff is attainable via a laminar partition is asserted, but the manuscript must supply the full derivation of the partition construction together with verification that the resulting payoffs are exactly equilibrium payoffs (including any bounds or error terms). Without this, the central claim cannot be assessed.
minor comments (1)
  1. [Introduction / Section 2] The definition of laminar structure and the precise sense in which nonadjacent states may be pooled should be stated with an explicit example early in the text to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and for identifying the need for greater explicitness in the central characterization. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract / main characterization theorem] Abstract and the statement of the main characterization result: the claim that any equilibrium payoff is attainable via a laminar partition is asserted, but the manuscript must supply the full derivation of the partition construction together with verification that the resulting payoffs are exactly equilibrium payoffs (including any bounds or error terms). Without this, the central claim cannot be assessed.

    Authors: We agree that the derivation of the laminar-partition construction and the verification that it yields exactly the equilibrium payoffs should be presented more explicitly. The current manuscript contains the inductive construction and incentive-compatibility argument in Section 3, but we will move the full step-by-step derivation into the main text (or a dedicated subsection) and add an appendix containing a formal verification that the constructed strategy is incentive-compatible for both players and that the resulting payoffs coincide with any given equilibrium payoff vector. Because the action and state spaces are finite, no approximation bounds or error terms arise; the equality is exact. This revision will make the central claim directly verifiable without altering any results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; characterization result is self-contained within stated model

full rationale

The paper presents a characterization theorem for equilibrium payoffs in a explicitly restricted class of finite-action disclosure games (state-independent sender preferences; receiver action depends only on expected state). The central claim—that any equilibrium payoff is achievable via a laminar partition disclosure strategy—is derived directly from the game primitives and equilibrium conditions without reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. No equations or steps in the provided abstract or reader summary equate a derived object to its own inputs by construction. The result is scoped at the outset to the modeled setting and does not invoke uniqueness theorems or ansatzes from prior author work as external justification. This is a standard non-circular theoretical characterization.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions from Bayesian game theory rather than new free parameters or invented entities.

axioms (2)
  • domain assumption Sender's preferences are state-independent
    Explicitly stated as part of the class of finite-action disclosure games studied.
  • domain assumption Receiver's optimal action depends only on the expected state
    Explicitly stated as part of the class of finite-action disclosure games studied.

pith-pipeline@v0.9.0 · 5629 in / 1269 out tokens · 29344 ms · 2026-05-24T12:01:24.464980+00:00 · methodology

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

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