arxiv: 2605.09029 · v1 · submitted 2026-05-09 · 💰 econ.TH · cs.IT· math.IT

Recognition: 2 theorem links

· Lean Theorem

Secret Communication with Plausible Deniability

Michael P.H. Tam, Xiaoyu Cheng, Yonggyun Kim

Pith reviewed 2026-05-12 02:24 UTC · model grok-4.3

classification 💰 econ.TH cs.ITmath.IT

keywords secret communicationplausible deniabilitydirectional messagesinformation structuressingle-crossing preferencesfrontier communicationcutoff revelation

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The pith

Secret communication can provide plausible deniability only if directional messages reveal at most whether the state is above or below a cutoff.

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

This paper studies how a sender can transmit information secretly while ensuring the receiver's later actions do not betray that hidden information was used. Plausible deniability requires that every action taken after the secret message could be justified using only the receiver's public baseline knowledge, given single-crossing preferences. The authors characterize the information structures that achieve both secrecy, where the message itself is independent of the state, and this deniability condition. When the baseline message is directional, its likelihood changes monotonically with the state, and the extra secret layer is then limited to indicating whether the state exceeds some threshold. This bound matters because it identifies the exact informational ceiling on hidden influence in environments where the receiver must appear to act on public information alone.

Core claim

Under single-crossing preferences, plausible deniability holds when every action induced by the sender's message can be rationalized from the receiver's baseline information alone. When combined with secrecy, directional baseline messages restrict frontier communication to structures that reveal at most whether the state lies above or below a cutoff. The paper also identifies conditions under which a greatest feasible communication structure exists and can be constructed explicitly.

What carries the argument

Frontier communication structures that jointly satisfy secrecy (message independent of the state) and plausible deniability (actions rationalizable by baseline information alone) under directional baseline messages.

If this is right

Plausible deniability restricts additional communication exactly when the baseline message is directional.
For directional messages, the most informative feasible structures reveal only a binary cutoff signal about the state.
A greatest feasible communication structure exists and admits an explicit construction under the identified conditions.
Joint information structures satisfying both secrecy and plausible deniability are fully characterized.

Where Pith is reading between the lines

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

The cutoff restriction suggests that optimal secret signals in repeated interactions would take the form of simple threshold rules rather than richer partitions.
The same logic may apply to settings with multiple receivers who share different baseline information, limiting the feasible secret layers accordingly.
In mechanism design, contracts or policies could embed these structures to allow private adjustments while preserving the appearance of public rationality.

Load-bearing premise

Single-crossing preferences make every induced action rationalizable from baseline information without reference to the hidden message.

What would settle it

A concrete example, under single-crossing preferences, in which a directional secret message induces an action that cannot be justified using only the receiver's baseline information.

Figures

Figures reproduced from arXiv: 2605.09029 by Michael P.H. Tam, Xiaoyu Cheng, Yonggyun Kim.

**Figure 1.** Figure 1: Hierarchy of Information We refer to Section 5.2 for further discussion of our modeling assumptions, including the possibility of repeated message observations and the single-crossing restriction on the receiver’s utility function. 2.2 Secrecy and Plausible Deniability The sender’s communication strategy—his choice of the joint information structure h— is subject to two constraints: secrecy and plausible d… view at source ↗

**Figure 2.** Figure 2: A Signal Representation and Joint Information Structure [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: A direction-ordered signal representation with multiple messages in [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

**Figure 4.** Figure 4: A case of misalignment in ω2 Misalignment in ω3. Suppose that, in ω3, some i-region y lies to the left of some sregion y ′ , with y < y′ . Plausible deniability implies that y ′ must be assigned either d or s in both ω1 and ω2. Since ω1 and ω2 are already aligned, it follows that y must likewise be assigned either d or s in those states. We focus on the configuration shown in [PITH_FULL_IMAGE:figures/ful… view at source ↗

**Figure 5.** Figure 5: A case of misalignment in ω3 Fix any receiver’s utility function u ∈ U. Under the original structure, let ad, as, and ai denote the receiver’s optimal actions following messages (d, y), (s, y), and (i, y), respectively. The receiver’s optimal action following message (s, y′ ) is a0, since her posterior coincides with the prior. Suppose as ≥ a0; the opposite case is symmetric. Under the new structure, cons… view at source ↗

**Figure 6.** Figure 6: Counter example with multiple non-directional messages [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗

read the original abstract

Communication is secret if a message is independent of the state; however, the receiver's subsequent action may still reveal that she has acted on hidden information. This paper studies when secret communication can also provide plausible deniability: under single-crossing preferences, every action induced by the sender's message must be rationalizable using the receiver's baseline information alone. We characterize joint information structures that satisfy both secrecy and plausible deniability. We show that plausible deniability restricts communication exactly when the baseline message is directional -- meaning its likelihood is monotone in the state. Combining this restriction with secrecy, we show that, for directional messages, frontier communication reveals at most whether the state lies above or below a cutoff. Finally, we identify conditions under which a greatest feasible communication structure exists and can be constructed explicitly in a simple way.

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 shows secrecy plus plausible deniability restricts directional messages to at most a cutoff revelation and gives an explicit construction for the largest feasible structure.

read the letter

The core result is that secrecy and plausible deniability together force directional messages to reveal only whether the state is above or below a cutoff. This follows directly from the definitions once single-crossing preferences are imposed and the baseline information is fixed. The paper also identifies when a greatest feasible communication structure exists and shows how to build it in a simple way. That explicit construction is the part most likely to be picked up by others working on constrained information design. The argument stays deductive and avoids circularity or fitted parameters, which keeps it clean. The restriction to directional messages is stated upfront, so the claim does not overreach. The plausible-deniability definition, requiring every induced action to be rationalizable from baseline information alone, is strong but transparent and load-bearing for the cutoff result. No obvious gaps appear in the logic given in the abstract and stress-test note. The work sits squarely in information economics and signaling games. Readers who already use information-design tools and want to add deniability or secrecy constraints will find the characterization usable. It is not a broad overhaul of the field, but it adds a precise limit that prior results did not isolate. I would send it to peer review because the central characterization is internally consistent and the construction is concrete enough to be checked and extended.

Referee Report

0 major / 2 minor

Summary. The paper studies secret communication (messages independent of the state) that also satisfies plausible deniability under single-crossing preferences, where every action induced by the sender's message is rationalizable using only the receiver's baseline information. It characterizes the admissible joint information structures satisfying both conditions. For directional baseline messages (monotone likelihood ratios in the state), the combination restricts frontier communication to revealing at most whether the state lies above or below a cutoff. The paper further identifies conditions for existence of a greatest feasible communication structure and provides an explicit construction.

Significance. If the characterizations hold, this provides a precise delineation of the limits imposed by secrecy and deniability on strategic communication, which is a useful contribution to information design and mechanism design in economics. The explicit construction of the maximal feasible structure is a particular strength, as it offers a concrete and potentially implementable object. The result on cutoff revelations for directional messages yields a clean, testable restriction.

minor comments (2)

The definition of plausible deniability (every induced action rationalizable from baseline information alone) is load-bearing; a short remark comparing it to related notions in the cheap-talk or information-design literature would aid readers.
The abstract states the characterizations and the cutoff result without proof sketches; ensure the main text or appendix makes the deductive steps from single-crossing and monotonicity fully explicit for verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. The referee's summary accurately reflects the paper's focus on characterizing joint information structures that satisfy both secrecy and plausible deniability under single-crossing preferences, as well as the key results on directional baseline messages and the explicit construction of maximal feasible structures. We appreciate the positive assessment of the contribution to information design.

Circularity Check

0 steps flagged

No significant circularity; derivation is deductive from definitions

full rationale

The paper derives characterizations of admissible joint information structures directly from its explicit definitions of secrecy and plausible deniability, together with the single-crossing preference assumption. The key result—that directional messages under these constraints reveal at most a cutoff—follows by logical implication from the stated conditions without any parameter fitting, self-referential equations, or load-bearing self-citations. The explicit construction of the greatest feasible structure is obtained by applying the identified restrictions, keeping the argument self-contained and independent of the target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on domain assumptions standard in information economics and the specific definition of plausible deniability; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Single-crossing preferences
Invoked to ensure actions are rationalizable from baseline information alone under the plausible-deniability definition.

pith-pipeline@v0.9.0 · 5436 in / 1055 out tokens · 25996 ms · 2026-05-12T02:24:26.843069+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
We show that plausible deniability restricts communication exactly when the baseline message is directional—meaning its likelihood is monotone in the state. ... for directional messages, frontier communication reveals at most whether the state lies above or below a cutoff.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear
Lemma 2. ... If ϕ(z|ω) is decreasing and non-constant in ω, R_ϕ(z) = {a : a ≤ a0}.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Subversive Con- versations,

Antic, Nemanja, Archishman Chakraborty, and Rick Harbaugh (2025) “Subversive Con- versations,”Journal of Political Economy, 133 (5), 1621–1660, 10.1086/734130. Athey, Susan and Jonathan Levin (2018) “The value of information in monotone decision problems,”Research in Economics, 72 (1), 101–116. B¨orgers, Tilman, Angel Hernando-Veciana, and Daniel Kr¨ahmer...

work page doi:10.1086/734130 2025
[2]

Monotone Comparative Statics,

Milgrom, Paul and Chris Shannon (1994) “Monotone Comparative Statics,”Econometrica, 62 (1), 157–180,http://www.jstor.org/stable/2951479. Shannon, C. E. (1949) “Communication Theory of Secrecy Systems,”The Bell System Technical Journal, 28 (4), 656–715, 10.1002/j.1538-7305.1949.tb00928.x. Strack, Philipp and Kai Hao Yang (2024) “Privacy-Preserving Signals,...

work page doi:10.1002/j.1538-7305.1949.tb00928.x 1994
[3]

Inferentially-Private Private Information,

46 Wang, Shuaiqi, Shuran Zheng, Zinan Lin, Giulia Fanti, and Zhiwei Steven Wu (2024) “Inferentially-Private Private Information,”arXiv preprint arXiv:2410.17095. Xu, Zhang and Wei Zhao (2025a) “Privacy-Constrained Signals,”arXiv preprint arXiv:2511.21196. (2025b) “Privacy Structure and Blackwell Frontier,”arXiv preprint arXiv:2511.10226. Zhu, Shuguang (20...

work page arXiv 2024