{"id":"b2591e67-ddeb-4c78-b602-9a7986543500","arxiv_id":"2502.01587","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"VBP solves Bayesian persuasion in natural language by treating LLMs as sender and receiver in a mediator-augmented game and searching prompt strategies with Prompt-PSRO.","lead":"This paper builds a framework, VBP, that lets two large language models play Bayesian persuasion games in natural language: a sender chooses a prompt strategy to influence a receiver's decisions, and a solver searches for equilibria. If it works, it would let economists and AI designers test persuasion strategies in realistic dialogue settings instead of toy numeric games.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.1's proof invokes a binary-search saddle-point algorithm that Algorithm 1 does not implement; the ε-equilibrium guarantee is thus unverified for the executed solver.","rationale":"Read in good faith, the paper aims to solve natural-language BP by combining a mediator-augmented game formulation with Prompt-PSRO. The strongest claim is the ε-equilibrium guarantee. My inspection found a concrete internal inconsistency: the proof in Appendix A uses a bilinear saddle-point reduction and a binary-search method from Zhang et al. (2024a), while Algorithm 1 is a standard PSRO loop without the Lagrangian terms or binary search. No argument is given that the PSRO loop with LLM approximate best responses converges to the same fixed point. The reader's weakest assumption (LLM Bayesian updating) is real and empirically important, but it is a behavioral assumption; even if the LLM receiver were perfectly rational, the formal guarantee would not follow from the presented proof. I therefore regard the proof-algorithm mismatch as more load-bearing. I give credit for the reproduced classic equilibria and the honesty/lying dynamics, which suggest the framework has empirical promise, but those experiments do not fill the theoretical gap. The condition for acceptance should be a corrected proof or a clear statement that Proposition 5.1 applies only to a conceptual mapping, not to the implemented solver. Thus I keep the CONDITIONAL verdict, which matches the reader's recommendation.","tokens_in":57062,"tokens_out":6182,"duration_ms":54746,"concrete_test":"Run Algorithm 1 with an exact, exhaustive best-response oracle on a finite-prompt version of the REL problem, and compare the resulting sender payoff and exploitability to the exact BCE derived in Appendix B. If the output is not ε-close (e.g., ε ≤ 0.1) to the BCE, Proposition 5.1 is false for Algorithm 1 even with perfect best responses, confirming the proof-algorithm gap. Alternatively, formally derive Algorithm 1's fixed point from Eq. 3; the absence of such a derivation would itself demonstrate that the guarantee is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline formal claim is Proposition 5.1 (Section 5.3): 'VBP returns an ε-approximate Bayes correlated equilibrium in static BP and an ε-approximate Bayes-Nash equilibrium in multistage BP.' Appendix A attempts a proof by rewriting Eq. 1 as a bilinear saddle-point problem (Eq. 3) involving a threshold τ and invoking 'Algorithm 1 in Zhang et al. (2024a)' — a binary-search saddle-point method that uses a Lagrangian and modifies utilities to zero-sum. It then states that the authors use prompt-space response oracle (OPRO/FunSearch) as approximate best-response oracles in that binary search. But the algorithm actually presented and implemented in the main text (Algorithm 1) is a generic PSRO loop: initial finite prompt sets, payoff tensor, meta-strategy, LLM best responses, expand sets, recompute payoff tensor, update meta-strategy. There is no Lagrangian multiplier, no binary search over τ, and no zero-sum utility transformation. The proof therefore refers to a different algorithm than the one evaluated. For Proposition 5.1 to be valid, either Algorithm 1 must be shown to solve (or approximate) the saddle-point problem Eq. 3, or a separate convergence proof for the PSRO variant with LLM approximate best responses must be supplied. Neither appears in the paper. This is an internal inconsistency, not a disagreement with the literature. It matters because the convergence guarantee is the main theoretical contribution and the basis for the 'first general framework' claim. Even if the receiver LLM were a perfect Bayesian agent, the algorithm's output would not be known to be an ε-approximate equilibrium from the given proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces VBP (Verbalized Bayesian Persuasion), a framework that recasts Bayesian persuasion as a verbalized mediator-augmented extensive-form game in which both the sender and the receiver are implemented as LLMs. The sender's signaling scheme is represented by prompt categories (e.g., tone, detail level), and strategy optimization is performed via a prompt-space response-oracle (Prompt-PSRO) combined with OPRO and FunSearch. The authors claim in Proposition 5.1 that VBP returns an epsilon-approximate Bayes correlated equilibrium in static BP and an epsilon-approximate Bayes-Nash equilibrium in multistage BP. Experiments on recommendation-letter, courtroom, and law-enforcement scenarios show that VBP approaches the computed BCE baselines in terms of sender and receiver rewards, with exploitability decreasing to about 0.1, and that it produces interesting phenomena such as honesty oscillations and bargaining-like dynamics in the multistage setting.","tokens_in":57315,"tokens_out":3759,"duration_ms":35132,"significance":"If the convergence guarantee were substantiated, VBP would be a meaningful step toward applying information design to natural-language settings, a domain where existing numeric BP methods do not apply. The paper includes several strengths: it provides detailed prompt templates in Appendix C.4 that aid reproducibility, it evaluates on three classic BP scenarios plus a multistage variant, and it documents qualitatively interesting sender behaviors that connect to honesty and deception. However, the headline formal claim is currently not supported by the proof as written, and the experimental validation is partly built into the receiver's prompt rather than emergent from the model. The framework itself is promising, but the gap between the claimed guarantee and the implemented algorithm must be resolved before the central contribution can be accepted.","major_comments":[{"comment":"Proposition 5.1 asserts an epsilon-approximate equilibrium guarantee, but the proof in Appendix A invokes 'Algorithm 1 in Zhang et al. (2024a)', a binary-search saddle-point method with a Lagrangian, a zero-sum utility transformation, and a threshold tau. The algorithm actually presented and evaluated in the main text (Algorithm 1) is a generic PSRO loop: it maintains finite prompt sets, computes a payoff tensor, finds LLM best responses, expands the sets, and updates a meta-strategy. There is no binary search over tau, no Lagrangian multiplier, and no zero-sum conversion. The convergence theorem therefore applies to a different algorithm than the one whose results are reported. Either Algorithm 1 must be shown to solve or approximate the saddle-point problem in Eq. (3), or a separate convergence proof for the PSRO variant with LLM approximate best responses must be supplied. Without this, the central theoretical claim is unverified for the executed solver.","section":"§5.3 and Appendix A, Eq. (3)"},{"comment":"The receiver's prompt explicitly instructs the LLM to 'first guess the student's quality given the sent recommendation letter according to the Bayesian decision rule' and embeds the sender's writing style and its probability into the receiver's context as the commitment assumption. This means the empirical reproduction of the BCE baselines in Figures 4 and 5 is partly by construction: the experiments demonstrate that the LLM follows the prescribed Bayesian update and best-response rule, not that the VBP framework induces this behavior from a more neutral prompt. To support the claim that VBP solves real BP problems, the authors should evaluate the receiver's behavior without the explicit Bayesian-rule instruction, or at least report a control condition that measures how much of the observed proximity to BCE is due to the prompt content versus the learning dynamics.","section":"§5.1 (Verbalized Commitment Assumption) and Appendix C.4"},{"comment":"The convergence guarantee and the experimental results are restricted to a finite set of prompt categories (e.g., tone, detail level, specificity) that the Prompt-PSRO loop is allowed to optimize. The paper assumes, without proof, that this low-dimensional prompt space spans the signaling space well enough to approximate optimal persuasion. The exploitability values in Figure 6 are computed within this restricted meta-game, not against the full space of natural-language signals, so they do not directly quantify the gap to an equilibrium of the original verbalized game. The claims about solving 'real, non-abstract BP problems' should be scoped to the expressiveness of the chosen prompt categories, and the paper should either justify the choice of categories or discuss the sensitivity of the results to that choice.","section":"§5.1 (Action A) and §6.4"}],"minor_comments":[{"comment":"The word 'mutlistage' appears in the sentence 'In mutlistage BP, this significantly restricts...'; it should be 'multistage'.","section":"§5.3"},{"comment":"In the last sentence of the proof, 'strategties' is a typo for 'strategies', and 'utilty' is a typo for 'utility'.","section":"Appendix A"},{"comment":"The caption contains 'comparision' instead of 'comparison'.","section":"Figure 7 caption"},{"comment":"The text says 'the table columns in Figure 9 reflect this structure', but Figure 9 appears to be a screenshot or diagram rather than a formatted table; please clarify the layout or refer to it consistently as a table.","section":"§6.4"},{"comment":"The statement that 'the probabilities in Figure 9 are computed as the average probability of selecting each prompt from the strategy pool across iterations' is ambiguous: it is unclear whether the reported probabilities are averaged over all iterations or taken from the final converged meta-strategy, and the figure itself should state which iteration or time window is shown.","section":"§6.4"},{"comment":"For the BCE and MARL baselines in Figure 4, the paper plots 'Probability of lie' and 'Probability of honest' without specifying how these probabilities are derived from the computed equilibrium or the MARL policy; a brief definition or reference would improve interpretability.","section":"§6.2"}],"recommendation":"major_revision","confidential_remarks":"The paper's main formal claim is not backed by the proof as written, because the proof describes a binary-search saddle-point algorithm that the implemented Algorithm 1 does not execute. This is a load-bearing mismatch, not a stylistic issue. I would ask the authors to either supply a correct convergence proof for the PSRO variant actually used, or weaken Proposition 5.1 to a statement about the restricted meta-game. Additionally, the empirical validation is weakened by the receiver prompt explicitly encoding the Bayesian decision rule; a control experiment without that instruction would strengthen the claim that the framework is solving BP rather than following a script. The paper is well-written and the experimental setup is thoughtful, but the theoretical gap and the prompt-construction circularity need to be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it takes the classic Bayesian persuasion setup and makes it verbal, letting an LLM sender optimize natural-language prompts against an LLM receiver. The three classic experiments (recommendation letter, courtroom, law enforcement) reproduce the right qualitative behavior, including the strategic mix of honesty and lying, and the unaligned LLaMA ablation is a thoughtful control. The step-wise conditional Prompt-PSRO for multistage settings is a sensible extension, and the generated prompt functions in the appendix show the framework is actually doing something non-trivial. This is a solid systems/empirical contribution worth building on.\n\nThe soft spots are real, though. The stress-test note is correct: Proposition 5.1's proof in Appendix A invokes a binary-search saddle-point algorithm from Zhang et al. (2024a), but Algorithm 1 in the main text is a generic PSRO loop—no Lagrangian multipliers, no binary search over tau, no zero-sum utility transform. That means the ε-equilibrium guarantee is simply unverified for the algorithm the authors actually evaluated. This is an internal inconsistency, not a disagreement with the literature. The authors need either to show that Algorithm 1 solves (or approximates) the saddle-point problem, or give a separate PSRO convergence proof with LLM approximate best responses.\n\nThe circularity concern also lands, though with less force. The receiver prompt explicitly instructs Bayesian updating and includes the sender's writing style and its probability, so the obedience constraints are partly built in. The sender's side is genuinely optimized, but the claim that LLM agents \"solve\" BP in a fully emergent sense is overstated. Relatedly, the experiments would be much more convincing with an LLM-agent baseline and error bars, and the code is not released—that should be required for a framework paper.\n\nThe citation pattern looks fine, and the limitations section is honest about LLM fidelity and computational cost. The paper deserves a serious referee, but it needs major revision before acceptance: fix the proof gap, add baselines and variance, and give the reader a clear picture of which components are learned and which are prompted. This is a conditional accept in spirit, not a reject.","headline":"A useful empirical integration of Prompt-PSRO and Bayesian persuasion, but the headline convergence guarantee is unverified for the algorithm actually run.","tokens_in":57913,"tokens_out":2412,"would_cite":true,"duration_ms":24951,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A26","91A10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that Bayesian persuasion, traditionally solved only for small discrete games, can be solved in natural language by representing both players as LLMs and searching over prompt strategies until an approximate equilibrium is…","keywords":["Bayesian persuasion","information design","large language models","equilibrium finding","extensive-form games","prompt optimization","natural language dialogue"],"falsifier":"Compare the receiver LLM's chosen actions, over many signals in the three benchmark problems, with the Bayes best response to the posterior induced by the sender's actual mixed strategy; if the deviation rate is non-negligible, the obedience constraints are misestimated and the claimed $\\varepsilon$-approximate Bayes correlated equilibrium does not hold for these agents.","tokens_in":56828,"feed_emoji":"💬","tokens_out":10427,"duration_ms":83004,"temperature":0.7,"pith_summary":"Bayesian persuasion, the study of how a sender can steer a receiver by committing to an information-revealing strategy, has mostly been solved only in toy games with small discrete state and signal spaces. This paper claims that by re-casting the problem as a mediator-augmented extensive-form game in which both players are LLMs and strategies are prompts, the same equilibrium logic can be applied to real natural-language dialogues such as writing recommendation letters, presenting courtroom evidence, and announcing police patrols. The proposed verbalized Bayesian persuasion (VBP) framework is claimed to converge to a $\\varepsilon$-approximate Bayes correlated equilibrium in static settings and a $\\varepsilon$-approximate Bayes-Nash equilibrium in multistage settings, and the experiments reproduce the theoretical sender payoffs of classic Bayesian persuasion while discovering new persuasive strategies in language. A sympathetic reader would care because, if the claim holds, persuasion in everyday communication becomes amenable to game-theoretic guarantees rather than ad hoc prompting.","feed_headline":"Plain-language persuasion games now solve to approximate equilibrium","feed_subtitle":"Sender and receiver LLMs bargain in natural language, reaching approximate equilibrium in dialogue.","key_machinery":"The central object is the verbalized mediator-augmented extensive-form game, a reformulation of Bayesian persuasion in which the sender acts as a mediator who commits to a signaling scheme and the receiver best-responds, with states, signals, and actions rendered as text rather than numbers. The solver that carries the argument is the prompt-space response oracle (Prompt-PSRO), which restricts each player's strategy space to a growing pool of prompt categories and contents, computes a meta-game equilibrium over that pool, and uses LLM-based optimizers, namely OPRO in static settings and FunSearch in multistage settings, as approximate best-response oracles. Three supporting mechanisms make the reduction work: the commitment assumption is verbalized by writing the sender's style and its probability into the receiver's prompt, obedience constraints are estimated by sampling and added as a penalty, and information obfuscation uses self-reflection to help aligned LLMs produce strategically vague signals. The theoretical load is carried by rewriting the obedience-constrained maximization as a bilinear saddle-point problem, which connects the verbalized game to the known polynomial-time equilibrium results for mediator-augmented games.","core_discovery":"The paper's central claim is that a Bayesian persuasion problem can be faithfully represented as a verbalized mediator-augmented extensive-form game, in which the state, the sender's signal, the receiver's action, and the commitment assumption are all expressed as text, and that this game can be solved by a generalized equilibrium-finding algorithm that alternates between LLM-based approximate best response and a meta-game solver. Strategies are not optimized in the LLM's weight space; they are optimized in prompt space, where a signaling scheme is a probability distribution over writing-style prompts such as tone, detail level, and emphasis. The paper further claims that the sender's commitment can be verbalized by including the sender's style and its probability in the receiver's prompt, that obedience constraints can be estimated by sampling and penalized through reward shaping, and that a step-wise variant using conditional prompt functions extends the method to multistage games. Proposition 5.1 states that the result is a $\\varepsilon$-approximate Bayes correlated equilibrium in static BP and a $\\varepsilon$-approximate Bayes-Nash equilibrium in multistage BP, and the experiments on three classic problems are presented as evidence that the framework reproduces known equilibria and transfers to richer language settings.","pith_inferences":["If the receiver LLM's Bayesian updating is imperfect, the equilibrium guarantee describes best response to the simulated receiver, not to the human it stands in for; a direct test against human receivers would be needed before treating VBP as a tool for real-world persuasion.","The oscillation between honesty and deception observed with aligned models, which vanishes with an unaligned model, suggests alignment acts as an extra payoff perturbation; one could test this by varying the receiver's stated normative preferences and measuring the equilibrium lie rate.","The same mediator-augmented interface could be pointed at multi-receiver mechanism design: since the formulation already supports multiple players, VBP-style prompt search may solve information design problems where one sender recommends actions to many receivers, a direction the paper lists as future work."],"forward_implications":["If VBP works as claimed, Bayesian persuasion can be run on arbitrary conversational text rather than binary states and discrete signals, so recommendation letters, legal arguments, and public announcements can be analyzed as strategic information design problems.","The framework yields a convergence guarantee: in static settings the output is a $\\varepsilon$-approximate Bayes correlated equilibrium, so a sender that deploys the found meta-strategy cannot gain more than $\\varepsilon$ by deviating, with an analogous guarantee in multistage settings.","The verbalized commitment assumption gives a practical implementation of the defining feature that separates Bayesian persuasion from cheap talk: the receiver's prompt contains the sender's strategy and its probability.","Multistage persuasion with a long-lived receiver becomes tractable through conditional prompt functions, and the observed honesty-deception oscillations suggest bargaining-like dynamics rather than a fixed unilateral commitment.","The method is designed to generalize across dialogue domains without retraining the LLM, since the experiments require only LLM inference plus prompt optimization, not fine-tuning."],"supporting_citations":[{"why":"Defines the Bayesian persuasion model and the commitment assumption that the paper verbalizes.","marker":"[Kamenica & Gentzkow, 2011]"},{"why":"Supplies the mediator-augmented game formulation and the equilibrium computation theory behind Proposition 5.1.","marker":"[Zhang & Sandholm, 2022]"},{"why":"Contributes the prompt-space response oracle, the LLM strategy-optimization loop that VBP extends.","marker":"[Gemp et al., 2024]"},{"why":"Provides the MARL baseline and the extended obedience constraint that VBP estimates by sampling.","marker":"[Lin et al., 2023]"},{"why":"Gives the Bayes correlated equilibrium concept that static VBP is claimed to approximate.","marker":"[Bergemann & Morris, 2016]"},{"why":"Provides the OPRO optimizer used to generate categorical prompt strategies in static settings.","marker":"[Yang et al., 2024]"},{"why":"Provides the FunSearch program-search method used to discover conditional prompt functions in multistage settings.","marker":"[Romera-Paredes et al., 2024]"}],"fun_headline_variants":["Verbalized Bayesian persuasion: LLMs reach near-equilibrium in dialogue games","Talking persuasion: LLM-based Bayesian games solve to approximate equilibrium","Natural-language persuasion games now solved via LLM equilibrium search","LLM-powered verbalized Bayesian persuasion reaches approximate equilibrium","Dialogue games for persuasion: verbalized Bayesian model finds equilibria"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a receiver LLM instructed to follow the Bayesian decision rule actually does so, and that a small set of prompt categories such as tone and detail level spans enough of the signaling space for near-optimal persuasion.","fun_headline_variants_meta":{"raw":{"variants":["Verbalized Bayesian persuasion: LLMs reach near-equilibrium in dialogue games","Talking persuasion: LLM-based Bayesian games solve to approximate equilibrium","Natural-language persuasion games now solved via LLM equilibrium search","LLM-powered verbalized Bayesian persuasion reaches approximate equilibrium","Dialogue games for persuasion: verbalized Bayesian model finds equilibria"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001199,"raw_usage":{"total_tokens":4946,"prompt_tokens":949,"completion_tokens":3997,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":3909}},"tokens_in":565,"tokens_out":3997,"duration_ms":23533,"temperature":1.0,"reasoning_tokens":3909,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T14:53:11.145324+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare the receiver LLM's chosen actions, over many signals in the three benchmark problems, with the Bayes best response to the posterior induced by the sender's actual mixed strategy; if the deviation rate is non-negligible, the obedience constraints are misestimated and the claimed $\\varepsilon$-approximate Bayes correlated equilibrium does not hold for these agents.","supporting_citations":[{"cited_title":"and Sandholm, T","cited_arxiv_id":null,"evidence_quote":"Supplies the mediator-augmented game formulation and the equilibrium computation theory behind Proposition 5.1."},{"cited_title":"Information design in multi-agent reinforcement learning","cited_arxiv_id":null,"evidence_quote":"Provides the MARL baseline and the extended obedience constraint that VBP estimates by sampling."},{"cited_title":"V., Zhou, D., and Chen, X","cited_arxiv_id":null,"evidence_quote":"Provides the OPRO optimizer used to generate categorical prompt strategies in static settings."}],"review_version":1}