Money Burning Improves Mediated Communication

Yang Yu; Yi Liu

REVIEW 1 major objections 7 minor 35 references

Money Burning Improves Mediated Communication

T0 review · 1 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Money burning can strictly improve mediated communication.

desk verdict The main idea and value formula are genuinely new and likely right, but the proof of Theorem 1 has a real gap in the boundary-continuity lemma that needs repair before the result is fully convincing. read the letter →

arxiv 2411.19431 v2 pith:E5Y2RJBJ submitted 2024-11-29 econ.TH

Yi Liu , Yang Yu This is my paper · ORCID

classification econ.TH MSC 91A2891B44

keywords moneyburningmediatedcommunicationBayesianpersuasioncheaptalkcommitmentmechanismdesigntransparentmotivesincentivecompatibility

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether deliberately wasteful money burning can make an informed sender more persuasive when he can commit to a mediator but not to a full persuasion strategy. The answer is yes, under transparent motives: when the sender's payoff depends only on the receiver's action, mediated communication with report-contingent burning generically beats ordinary mediation for almost every prior belief at which commitment has any value. The paper characterizes the sender's optimal payoff as the best worst-case interim payoff over signaling schemes, which is also the payoff of a cautious sender or of a sender facing the worst subjective prior in Bayesian persuasion. If the characterization is right, burning money is a practical substitute for commitment power: it lets a sender without full commitment capture part of the Bayesian-persuasion surplus.

What carries the argument

The central object is a canonical mediated-communication-with-money-burning mechanism: the sender reports his type to a mediator, who publicly outputs a posterior belief from a pre-committed signaling scheme and burns a deterministic amount of money that depends only on that posterior. Because burning carries no information beyond the posterior, the receiver's obedience constraint and the sender's incentive constraints separate. The load-bearing identity is Proposition 3, $V^*(\mu_0)=\max_{\pi}\min_{\theta}V_\pi(\theta)=\min_{\lambda}\operatorname{cav}(\hat V_\lambda)(\mu_0)$, obtained by viewing the mechanism-design problem as a zero-sum game between the sender and a fictitious chooser of the subjective prior $\lambda$. The incentive-compatibility equation (5), $\int_{\mu}\left(\frac{\mu(\theta)}{\mu_0(\theta)}-\frac{\mu(\theta')}{\mu_0(\theta')}\right)(V(\mu)-x(\mu))\,dp(\mu)=0$, forces all types' net interim payoffs to be equal, and money burning $x(\mu)$ is the transfer that makes this equalization possible. Proposition 11's construction, which burns money only on messages that reveal the type with small unconditional probability, attains this max-min value in the limit.

What would settle it

A computational search over finite receiver payoff matrices that satisfy the generic condition could look for a positive-measure set of priors where commitment is valuable, $V^*_{CT}(\mu_0)<\max V$, yet $V^*_{MD}(\mu_0)=V^*(\mu_0)$; Theorem 1 asserts no such set exists. The paper's Example 3 shows the equality can occur outside the generic class, so the search must enforce Definition 3. A sharp experiment would compare sender payoffs under mediation with and without burning in a lab: under transparent motives the burning treatment should strictly dominate for almost all priors, and the gap should disappear when sender payoffs depend on his type.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes Theorem 1: under a generic condition on the receiver's payoff functions (any action that is optimal at a belief is uniquely optimal at another belief with the same support), for almost all priors either cheap talk already reaches the sender's maximal feasible payoff, so commitment has no value, or money burning strictly raises the value of mediated communication. The companion characterization is $V^*(\mu_0)=\max_{\pi}\min_{\theta}V_\pi(\theta)=\min_{\lambda\in\Delta(\Theta)}\operatorname{cav}(\hat V_\lambda)(\mu_0)$, where $V_\pi(\theta)$ is type $\theta$'s interim payoff under the signaling scheme $\pi$ and $\hat V_\lambda(\mu)=\sum_{\theta}\lambda(\theta)\frac{\mu(\theta)}{\mu_0(\theta)}V(\mu)$. This ties the money-burning value to cautious Bayesian persuasion and to Bayesian persuasion under the sender's worst subjective prior. The proof builds a revelation principle that separates a mechanism into a signaling scheme and a report-contingent burning schedule, then shows that whenever burning fails to beat ordinary mediation, cheap talk already attains the same value.

Load-bearing premise

The result assumes the sender's payoff depends only on the receiver's action, not on his private type; if payoffs are state-dependent, money burning can fail to improve mediated communication, as the paper's buyer-seller example shows.

Editorial extensions

If this is right

At almost every prior where cheap talk cannot already deliver the sender's maximum feasible payoff, the optimal money-burning mechanism strictly outperforms ordinary mediation (Theorem 1).
Mediated communication with money burning has the same value as cautious Bayesian persuasion and as Bayesian persuasion under the sender's worst subjective prior, so robust-persuasion solution methods transfer directly (Corollaries 2 and 3).
With a finite burning budget, the value is $\min_{\lambda}\operatorname{cav}(\hat V_{\lambda,C})(\mu_0)$, and optimal mechanisms split messages into a costless persuasion group and a burning-for-credibility group (Proposition 10).
In smart-contract settings, commitment is valuable exactly when it is valuable in conventional communication, but the refined value of commitment is generically smaller (Corollary 4 and Theorem 1).
Money burning does not make cheap talk more credible; the credibility gain is specific to mediated communication (Section 6.2).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The equivalence with cautious Bayesian persuasion suggests money burning is a way to implement maxmin behavior without asking the sender to commit to ignoring his type: the burning schedule itself enforces equal interim payoffs.
Because the value is a minimum over subjective priors, algorithms that solve worst-prior persuasion could be repurposed to design optimal burning mechanisms, a computational shortcut the paper does not spell out.
A lab test could compare sender payoffs across cheap talk, mediation, and mediation with burning; under transparent motives the burning treatment should sit strictly between ordinary mediation and Bayesian persuasion for most priors, and the gap should disappear when sender payoffs are type-dependent.
The binary-type result (Proposition 6) suggests the worst subjective prior is always an extreme point in two-type environments; if that shortcut extends to small finite type spaces, optimal burning mechanisms become easy to compute.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Good paper to know about, but not one you can trust as written. The main idea is that a sender with state-independent preferences can use report-contingent money burning to gain commitment power, and the optimal value equals a robust Bayesian persuasion payoff: V*(µ0) = min_λ cav(^V_λ)(µ0). That is a clean, genuinely new characterization. The binary example is a nice illustration, and the connection to cautious Bayesian persuasion is a real contribution. The paper is also honest about its limits: transparent motives matter, and the quantitative magnitude of the improvement is left open.

The trouble is in the proof of Theorem 1. The stress-test note is on target. Lemma 5 claims that for a boundary belief µ and a direction µ1 supported inside supp(µ), the limit of V* at (1−ε)µ+εµ1 equals V*(µ) by Lemma 4. But Lemma 4 proves continuity only at full-support beliefs. When µ is on the boundary, (1−ε)µ+εµ1 has support supp(µ), a proper face, so Lemma 4 does not apply. There is no induction on support size, and the proof of Proposition 12's Case 2 explicitly needs continuity at a lower-support belief. As written, the almost-sure strict-improvement result is incomplete. This is a load-bearing gap, not a cosmetic one.

The bounded-budget section has a separate, smaller problem: Proposition 10 applies Sion's minimax with λ ranging over aff(Θ), which is not compact. Without a boundedness or coercivity argument, the interchange is not justified. The reader is right that this can likely be repaired, but it needs to be done.

What is good: the upper bound and the approximating construction are solid; the value formula in Proposition 3 is derived from primitives, not assumed; the generic condition is clearly stated, and the counterexamples in the appendix show the authors know where their assumptions bite. The self-citation to Liu-Wu is fine—it is a related implementation paper, not a hidden foundation.

Who should read it: anyone working on limited commitment, robust persuasion, or the Web3/blockchain communication story. It deserves a serious referee. My recommendation is to send it to peer review, but the referee should insist on a repaired proof of Lemma 5 / Proposition 7 before acceptance. If the gap cannot be closed, Theorem 1 as stated does not hold up.

Referee Report

1 major / 7 minor

Summary. The paper introduces "mediated communication with money-burning mechanisms" (MDMB), in which a Sender with state-independent preferences commits to a mediator who sends messages and burns money contingent on the Sender's report. It develops a revelation principle for this protocol, characterizes the Sender's value as V*(µ0) = max_π min_θ V_π(θ) = min_λ cav(V̂_λ)(µ0) (Proposition 3), and compares this value with the values of cheap talk, mediated communication without burning, and Bayesian persuasion. The main result, Theorem 1, claims that under a generic condition on Receiver payoffs, for almost all priors either cheap talk already attains the maximum possible Sender payoff or money burning strictly improves mediated communication. The paper also analyzes bounded burning budgets, relates MDMB to robust/cautious Bayesian persuasion, and discusses applications to Web 3.0 smart contracts.

Significance. If the main characterization and Theorem 1 are correct, the paper makes a substantive theoretical contribution: it identifies a limited-commitment communication protocol whose value has a clean geometric formula, and it shows a generic strict-improvement result relative to ordinary mediation. The proof strategy is largely constructive, and the connection to cautious Bayesian persuasion and heterogeneous-prior persuasion is insightful. The paper is also commendably explicit about the scope of its conclusions: Section 7 states the transparent-motives assumption, and Appendix A.4 gives a counterexample without it. The formal apparatus, including the new revelation principle and the minimax argument, is a serious derivation rather than a list of assertions. The paper does not provide machine-checked proofs or computational verification, but the written proofs are extensive.

major comments (1)

[Appendix B.3, Lemma 5] The first equality V*(µ0) = max_π min_{θ∈Θ} V_π(θ), together with the definition of V̂_λ in Eq. (10), is not well-posed for priors with zero-probability types. The ratio µ(θ)/µ0(θ) is undefined when µ0(θ) = 0, and for such types π(·|θ) is unconstrained by the model yet still enters the minimum over Θ. The statement should either restrict attention to supp(µ0), restrict λ to be supported on supp(µ0), or explicitly assume a full-support prior. Lemma 2, which derives the upper bound, has the same issue: incentive compatibility is only required for types with positive prior probability.

minor comments (7)

[Section 4.1] In the paragraph after Proposition 3, "min-mas" should read "min-max".
[Section 6.3] The word "Lagragian" appears twice and should be "Lagrangian".
[Appendix A.4] The sentence beginning "Because we can merge the posteriors that induce the same action..." is repeated verbatim in consecutive paragraphs; please delete the duplicate.
[Appendix B.4] The proof of Proposition 9 refers to "Cauchy inequality" without qualification; this should be "Cauchy-Schwarz inequality".
[Appendix B] Several proofs cite "Proposition 9 of the working paper version of Kamenica and Gentzkow [2011]" with no formal reference in the bibliography; please provide a citable published version or state the result explicitly.
[Section 5] The sentence "Technically, we provide an example in Appendix A. Example 3 shows..." is awkwardly punctuated and could be read as referring to two different examples; please rephrase for clarity.
[Figures 2-5] The figure captions are minimal; please label all curves and indicate which lines correspond to which protocol or value function, as this would substantially improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the value formula and strict-improvement theorem are derived from model primitives, with external benchmarks used as comparisons and only a non-load-bearing self-citation.

full rationale

Proposition 3 is derived from the primitives rather than assumed: Propositions 1 and 2 provide a canonical revelation principle that converts the MDMB equilibrium problem into incentive-compatibility, obedience, and Bayes-plausibility constraints; Lemma 2 gives the upper bound V*(μ0) ≤ maxπ minθ Vπ(θ) from nonnegative burning; Proposition 11 constructs a burning scheme attaining minθ Vπ(θ) in the limit; and Sion's minimax theorem yields V*(μ0) = minλ cav(V̂λ)(μ0). None of these steps presupposes the strict-improvement conclusion. Theorem 1 is a comparative statement built on external benchmarks from Salamanca (2021), Corrao and Dai (2023), and Lipnowski and Ravid (2020), and on the generic condition of Definition 3 taken from Lipnowski et al. (2024); no parameter is fitted to data and no benchmark is the paper's own conclusion. The only self-citation, Liu and Wu (2024), is described as related work on implementation with outcome-contingent transfers and is not an input to Proposition 3, Proposition 7, or Theorem 1, so it does not make the derivation circular. One non-circular completeness concern should be noted: in Appendix B.3, Lemma 5's first case invokes Lemma 4 for boundary beliefs with supp(μ1) ⊆ supp(μ), while Lemma 4 is proved only for full-support beliefs; this is a proof gap requiring an additional support-size argument rather than a circular reduction of the theorem to its own assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper is a pure theory paper: no parameters are fitted to data. The value formula and theorems are derived from the model. The main additional assumptions beyond standard game theory are transparent motives and the generic condition on receiver payoffs, both explicitly stated. No new physical or empirical entities are posited.

assumptions (5)

standard math Sion's minimax theorem and its extension by Arandjelović apply to the relevant payoff functions and domains.
Used in Proposition 3 to exchange max over signaling schemes and min over subjective priors, and in Proposition 10. The Proposition 10 application is flagged because aff(Theta) is not compact.
domain assumption Sender has state-independent preferences, so v(a) does not depend on theta.
This is central to the incentive-compatibility equality and to Theorem 1. Section 2.1 and Section 7; Example 4 shows the strict-improvement result can fail without it.
domain assumption Generic condition on the Receiver's payoff function in Definition 3: for any belief and any optimal action, there is a same-support belief at which that action is uniquely optimal.
Needed for Proposition 7 and Theorem 1. Example 3 shows the comparison between MDMB and MD can differ when the condition fails.
standard math Known properties of quasi-concave envelopes and concavifications from Lipnowski and Ravid [2020] and Kamenica and Gentzkow [2011], including piecewise constant qcav for finite actions and collinearity of concavifying supports.
Used in the proof of Theorem 1 and in Propositions 6 and 9 without re-derivation.
domain assumption Perfect Bayesian equilibrium and Bayes-plausible updating are the solution concept, with finite message, action, and burning sets.
Modeling framework in Section 2.1; the revelation principle Proposition 1 relies on this equilibrium concept.

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Cite this review

Pith. "Pith review of Money Burning Improves Mediated Communication." pith.science (2026). https://pith.science/paper/E5Y2RJBJ

@misc{pith2026241119431,
  author       = {Pith},
  title        = {Pith review of: Money Burning Improves Mediated Communication},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5Y2RJBJ}},
  note         = {Machine review of arXiv:2411.19431}
}

read the original abstract

Can wasteful money burning improve strategic communication? We show that it can, but only with intermediate commitment. In mediated communication with report-contingent burning, the mediator can use costly messages to discipline deviations and make persuasive messages credible. Under transparent motives, increasing the burning budget strictly raises the Sender's payoff once the budget is large enough, unless mediated communication with money burning collapses to cheap talk. With an unbounded budget, the value equals a robust Bayesian persuasion payoff, or equivalently the payoff of a cautious Sender. The framework clarifies commitment through smart contracts and Web 3.0 mediation.

Figures

Figures reproduced from arXiv: 2411.19431 by the authors.

**Figure 1.** Revelation principle In the rest of this section, we explain how to apply Proposition 1 to simplify the optimal MDMB problem. We first define a sequence of necessary concepts related to the belief-based approach. Then, we explain how to apply Proposition 1 and the belief approach to 1) convert Sender’s optimality constraints to incentive-compatible constraints and 2) convert the Bayes updating and Receiver’s optimal… view at source ↗

**Figure 2.** Vˆ θH (µ) and Vˆ θL (µ). µ0 V ∗ 0 0.5 1 1 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 4.** The geometric interpretation. According to Proposition 3, we can geometrically characterize the value of MDMB and explain how the money burning mechanism reshapes the value function. As depicted in [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: This figure compares the optimal payoffs of different prot [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]

**Figure 6.** Figure 6: Results of Example 2. A.3 The Necessity of Generic Condition Example 3. We present an abstract setting in this example, where we only specify the belief-value function and ensure the existence of the basic settings of A, u, v, Θ by imposing the upper-semi continuity of…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bayesian persuasion with heterogeneous priors

Ricardo Alonso and Odilon C \^a mara. Bayesian persuasion with heterogeneous priors. Journal of Economic Theory, 165: 0 672--706, 2016

work page 2016

[2] [2]

An extension of the sion's minimax theorem

Ivan Arandjelovi \'c . An extension of the sion's minimax theorem. Zbornik radova Filozofskog fakulteta u Ni s u. Serija Matematik , pages 1--3, 1992

work page 1992

[3] [3]

Cheap talk and burned money

David Austen-Smith and Jeffrey S Banks. Cheap talk and burned money. Journal of Economic Theory, 91 0 (1): 0 1--16, 2000

work page 2000

[4] [4]

Information design: A unified perspective

Dirk Bergemann and Stephen Morris. Information design: A unified perspective. Journal of Economic Literature, 57 0 (1): 0 44--95, 2019

2019

[5] [5]

The design and price of information

Dirk Bergemann, Alessandro Bonatti, and Alex Smolin. The design and price of information. American Economic Review, 108 0 (1): 0 1--48, 2018

work page 2018

[6] [6]

Contracting with imperfect commitment and the revelation principle: the single agent case

Helmut Bester and Roland Strausz. Contracting with imperfect commitment and the revelation principle: the single agent case. Econometrica, 69 0 (4): 0 1077--1098, 2001

work page 2001

[7] [7]

Persuasion by cheap talk

Archishman Chakraborty and Rick Harbaugh. Persuasion by cheap talk. American Economic Review, 100 0 (5): 0 2361--2382, 2010

work page 2010

[8] [8]

Mediation markets: The case of soft information

Roberto Corrao. Mediation markets: The case of soft information. Working Paper, 2023

work page 2023