Review Reports - The Bateson Game: A Model of Strategic Ambiguity, Frame Uncertainty, and Pathological Learning

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The literature review in the introduction section is relatively brief. This may make it difficult for readers to fully understand the relationship between Bateson games and existing research. What contributions have you made based on previous work? Please clarify your innovative points.
Can the model be extended to multiple frames?
Does convergence occur after removing the meta-communication penalty? Are oscillations more severe in the multi-frame setting?
Can supplementary experiments be added?
The penalty function is set as an exogenous constant. How does the sender implement this penalty?
What social efficiency losses result from ambiguity?
Every minute detail should be explained more thoroughly. Please supplement the article's definitions and theorems, and refine the theoretical proof process.

Comments for author File: Comments.pdf

Author Response

Concern: The literature review is relatively brief. Please clarify your innovative points.

Our Response: We agree with the reviewer that the original introduction did not sufficiently situate the Bateson Game within the existing literature. The revised manuscript addresses this in two key ways:

Expanded Literature Review (Section 1.1): We have significantly expanded Section 1.1, "Strategic Ambiguity: Beyond Types to Frames." This section now explicitly contrasts our concept of frame uncertainty with traditional models of type uncertainty (e.g., Spence, Cho & Kreps) and more recent work on Knightian uncertainty and global games. We clarify that while existing models focus on ambiguity about a player's characteristics or the state of the world, the Bateson Game models ambiguity about the fundamental rules of the interaction itself. As we state, "The Receiver's problem shifts from one of inference ('What type of player are you?') to one of hermeneutics ('What game are we playing?')".
Clarified Innovative Contributions (Section 1.3): We have revised Section 1.3, "Contribution and Outline," to state our innovative points more clearly. The primary contribution is the formalization of a new class of signaling game driven by frame uncertainty. The key innovation, which distinguishes this work, is the demonstration that this structure leads to two distinct pathological learning outcomes—the "ambiguity trap" and the "certainty trap"—contingent on the Sender's incentive structure. This dual-outcome dynamic, proven theoretically and demonstrated empirically, is a novel contribution to the literature on boundedly rational learning in games.

Concern: Can the model be extended to multiple frames?

Our Response: Thank you for this important question. The model is indeed designed for this generalization.

Formal Definition: In Section 2.1, "Game Primitives," the set of interpretive frames is formally defined as F=f_1,f_2,...,f_k, with k≥2. While our computational analysis uses the simplest non-trivial case of

k=2 for clarity and illustrative power, the core axioms and theoretical proofs are constructed to hold for any finite number of frames.

Future Research: We acknowledge that the dynamics of a multi-frame game are a rich area for future work. In the revised manuscript, we have added Section 5.4, "Limitations and Future Research," where we explicitly propose this as a key extension: "Multi-Frame and Continuous Frame Spaces: Exploring the dynamics when the Receiver faces a large or continuous spectrum of possible interpretations".

Concern: Does convergence occur after removing the meta-communication penalty? Are oscillations more severe in the multi-frame setting?

Our Response: This is an excellent two-part question that probes the core mechanics of the game.

Removing the Meta-Communication Penalty (Axiom 2): If the meta-communication penalty were removed, the game would cease to be a Bateson Game. The "Question" action would become a costless and dominant strategy for the Receiver. In the first round, a rational Receiver would simply ask to clarify the frame, and the ambiguity would be resolved. The pathological learning dynamics we describe are entirely dependent on this penalty, which formalizes Bateson's "tertiary injunction" by preventing the Receiver from escaping the contradictory frame. Without the penalty, the trap dissolves, and convergence to a correct belief would be immediate.
Oscillations in a Multi-Frame Setting: This is a fascinating empirical question. While our current simulations are limited to two frames, we hypothesize that oscillations would not only persist but could become more complex, and potentially more severe, in a multi-frame setting. The Sender would have a richer set of options for exploitation, potentially leading the Receiver's beliefs on a more chaotic or complex path through the belief space. We have added this to our agenda for future research, as noted in the new Section 5.4.

Concern: Can supplementary experiments be added?

Our Response: This was a critical suggestion, and we have thoroughly addressed it by making the computational analysis a central feature of the revised paper. The original manuscript only described a single, hypothetical simulation. The new manuscript presents two distinct, fully realized experiments that empirically demonstrate the paper's core theoretical claim: that the game can produce two different pathological traps depending on the Sender's incentives.

Experiment 1: The Certainty Trap (Section 4.2): We designed a payoff matrix where the Sender has a dominant incentive to always choose Frame 2. The simulation results, shown in the new Figures 1 and 2, demonstrate that the Receiver's belief rapidly converges to a stable but incorrect certainty that the frame is Frame 2. This confirms the "certainty trap" dynamic.
Experiment 2: The Ambiguity Trap (Section 4.3): We then designed a second payoff matrix with cyclical incentives, where the Sender is always rewarded for contradicting the Receiver's belief. The results, shown in the new Figures 3 and 4, provide a stark contrast. The Receiver's belief is locked in a state of perpetual, high-frequency oscillation, and their actions are split 50/50, demonstrating strategic paralysis. This confirms the "ambiguity trap" dynamic.

These supplementary experiments are no longer just illustrative; they are a core part of our evidence, demonstrating the richness of the model and providing empirical grounding for our refined theoretical claims.

Concern: The penalty function is set as an exogenous constant. How does the sender implement this penalty?

Our Response: We thank the reviewer for pushing for more detail on this crucial mechanism. While the penalty is modeled as an exogenous cost for theoretical tractability, we have expanded the discussion in the revised paper to provide concrete, real-world examples of how this penalty is implemented.

Application Sections (5.2 and 5.3): In the application sections, we now explain how the penalty manifests. In the context of AI Alignment (Section 5.2), the penalty is the high cost of verification: "Techniques like SHAP or LIME are computationally expensive, time-consuming, and their outputs can be complex... This high cost of verification serves as a powerful deterrent to routine oversight". In

Institutional Gaslighting (Section 5.3), the penalty is social: "The employee might be labeled as 'not a team player,' 'overthinking things,' or 'lacking commitment.' This penalty enforces the ambiguity".

Future Research (Section 5.4): We also acknowledge the limitation of an exogenous penalty and propose endogenizing it as a key area for future research. In Section 5.4, we suggest a model where "the cost of questioning could be a strategic move by the Sender, who chooses how severely to punish clarification".

Concern: What social efficiency losses result from ambiguity?

Our Response: This is an excellent point. The original paper did not adequately quantify the welfare implications. We have added a new section, Section 5.1: The Structure of Exploitation and Social Welfare, to explicitly address this.

Defining Social Welfare Loss: We introduce the concept of Social Welfare Loss from Ambiguity (SWLA), defined as "the difference between the sum of payoffs in a first-best scenario... and the sum of average payoffs realized in the Bateson Game".
Analyzing Loss in Both Traps: We analyze this loss for both scenarios demonstrated in our new simulations. In the certainty trap, the joint payoff happens to be efficient for that specific parametrization, but we argue the pathology lies in the Receiver's complete loss of agency. In the

ambiguity trap, the loss is severe and quantifiable: "the Receiver consistently gets -10 and the Sender gets +10, for a joint payoff of 0. This represents a significant welfare loss compared to the maximum of 11". This new section directly answers the reviewer's question using the empirical results from our supplementary experiments.

Concern: Every minute detail should be explained more thoroughly. Please supplement the article's definitions and theorems, and refine the theoretical proof process.

Our Response: We have taken this comment to heart and have thoroughly revised the paper for greater clarity, detail, and rigor.

Expanded Definitions (Section 2): The entire formal model in Section 2 has been expanded with more detailed explanations of the game primitives, the sequence of play, and the meaning of each axiom.
Refined Theoretical Proofs (Section 3): The theoretical core of the paper in Section 3 has been completely reworked. The proof of PBE failure (Section 3.1) is now more detailed and explicit. Most importantly, the central theorem has been refined to account for the discovery of the two distinct traps. The new proof in

Section 3.2 ("Pathological Learning Dynamics: Ambiguity and Certainty Traps") is more rigorous and comprehensive. It is now broken into two distinct cases, formally proving the conditions that lead to either the ambiguity trap (cyclical incentives) or the certainty trap (aligned incentives or a dominant strategy).

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript is proposed a conceptually new model of a signaling game – “Bateson’s Game”, in which strategic ambiguity does not arise from information asymmetry, but from frame ambiguity and meta-communication penalties. The author successfully formalizes this idea through a strictly defined game structure and demonstrates its dynamics through simulations and theoretical results on lack of convergence.

The work is well written, clearly structured, and meets academic standards. It is of interest to researchers in the fields of game theory, behavioral economics, institutional design, and AI communication. However, minor corrections are needed to improve interpretive clarity and broaden the discussion around applications.

The manuscript shows a clear motivation based on Bateson's ideas about double messages and makes a good distinction from classical signaling games. I would recommend including a short illustrative example (even a hypothetical one) to give a more intuitive sense of the problem.

The structure of the game (players, messages, frames, actions, utilities) is formalized clearly, but a visual diagram (game tree or table) can be included to show the interaction between actions and frames.

It is shown that under the given conditions there is no convergence in the recipient's strategies under limited training - an original and important result, but a brief intuition behind the proof could be added in the text to make it accessible to readers without a strong mathematical background.

The simulations confirmed the theoretical expectations – there is a fluctuation in beliefs, question punishment and strategic freezing, but it would be useful to show a figure or graph of the evolution of beliefs. Also, to better explain how the utility structure is chosen and to discuss what would happen under different parameters (e.g. smaller penalty for “questions”).

The applications mentioned (AI alignment, institutional ambiguity) are interesting, but too abstract. I suggest that the author develop one of them in more detail or propose a hypothetical implementation in a real environment (e.g., algorithmic moderation or judicial procedure).

Relevant sources are used (Aumann, Skyrms, Gintis). It is possible to enrich it with newer publications after 2020 in the field of "AI interpretability".

The manuscript must be submitted in the MDPI template.

Author Response

Concern: "I would recommend including a short illustrative example (even a hypothetical one) to give a more intuitive sense of the problem."

Our Response: We agree completely that a concrete example is essential for making the abstract model more accessible. The original manuscript was indeed too abstract in its introduction.

Action Taken: We have added a detailed, intuitive example in the revised manuscript. In Section 5.3, "Application: Institutional Opacity and Gaslighting," we now present a clear, real-world scenario:

"Consider an employee (Receiver) in a company with a dysfunctional culture (Sender). The official corporate communication espouses values of 'innovation and risk-taking' (Frame 1). However, the actual reward structure, observed through promotions and project approvals, punishes failure and rewards conservative, low-risk behavior (Frame 2).... An attempt to resolve this contradiction by 'Questioning' the policy... is met with a penalty. The employee might be labeled as 'not a team player,' 'overthinking things,' or 'lacking commitment.'"

This example directly illustrates the conflicting frames, the meta-communication penalty, and the resulting "double bind" in a way that is immediately relatable, addressing the reviewer's concern for a more intuitive entry point.

Concern: "a visual diagram (game tree or table) can be included to show the interaction between actions and frames."

Our Response: This is an excellent suggestion for improving clarity. The original paper lacked a visual representation of the payoff structure.

Action Taken: The revised manuscript now includes two explicit payoff matrices in the new Section 4, "Computational Analysis."

Table 1 details the payoff structure used to simulate the "Certainty Trap."
Table 2 details the payoff structure used to simulate the "Ambiguity Trap."

These tables provide a clear and immediate visual reference for the utilities associated with each action-frame pair, making the strategic tensions described by the axioms much easier to understand.

Concern: "a brief intuition behind the proof could be added in the text to make it accessible to readers without a strong mathematical background."

Our Response: We thank the reviewer for this important point. The original proof was overly formal and lacked an intuitive summary.

Action Taken: We have substantially revised Section 3.2, "Pathological Learning Dynamics: Ambiguity and Certainty Traps," to include a clear, intuitive explanation of the proof's logic. We now introduce the proof by stating:

"The stability of the Receiver's belief depends on whether this choice by the Sender generates a payoff prediction error for the Receiver."

We then walk through the two cases (Ambiguity Trap and Certainty Trap) in plain language. The core intuition is now presented clearly: any stable belief leads to a predictable action, which the Sender will exploit. This exploitation either creates a "surprise" (a prediction error) that destabilizes the belief, forcing oscillation, or it confirms the belief, locking it into a stable trap. This makes the logic of the proof accessible to a broader audience.

Concern: "it would be useful to show a figure or graph of the evolution of beliefs. Also, to better explain how the utility structure is chosen and to discuss what would happen under different parameters."

Our Response: This was a critical suggestion that has led to the most significant improvement in the paper. The original manuscript only described the simulation results textually.

Action Taken (Figures): The revised manuscript now features a comprehensive Section 4, "Computational Analysis," which includes four figures that directly visualize the simulation results.

Figures 1 and 2 show the belief evolution and action frequencies for the "Certainty Trap," demonstrating rapid convergence to a stable, incorrect belief.
Figures 3 and 4 show the belief evolution and action frequencies for the "Ambiguity Trap," providing a stark visual of persistent, high-frequency oscillation and strategic paralysis.

Action Taken (Utility Structure & Parameters): We now explicitly explain how the utility structures in Tables 1 and 2 were chosen to create either dominant or cyclical incentives for the Sender, directly leading to the different traps. Furthermore, we have added a point to

Section 5.4 ("Limitations and Future Research") committing to a "Comprehensive Simulation Analysis" to explore how different parameters (learning rates, rationality) affect which trap emerges.

Concern: "The applications mentioned (AI alignment, institutional ambiguity) are interesting, but too abstract. I suggest that the author develop one of them in more detail..."

Our Response: We agree that a more detailed application significantly enhances the paper's impact.

Action Taken: We have substantially expanded Section 5.2, "Application: Deceptive Alignment in Artificial Intelligence." This section now provides a detailed mapping of each game element to a specific aspect of the AI safety problem. For example, we now explicitly define the Meta-Communication Penalty as the real-world cost and complexity of using AI interpretability tools like SHAP or LIME, citing relevant literature. We also connect this application to our discovery of the two traps, noting the chilling possibility that an AI might learn that the most effective manipulation is to guide a user into a "certainty trap" of confident misinterpretation.

Concern: "It is possible to enrich [the bibliography] with newer publications after 2020 in the field of 'AI interpretability'."

Our Response: We thank the reviewer for this suggestion to keep our discussion current.

Action Taken: We have updated the bibliography with several recent and relevant publications. In the AI interpretability section, we now cite sources such as Kim (2022) on human-AI relationships and Rudin (2019) on the limits of black-box explanations. We have also added other recent works, such as

Neeman et al. (2023) and Ui (2023), to ensure our discussion is situated within the contemporary literature.

Concern: "The manuscript must be submitted in the MDPI template."

Our Response: This has been addressed.

Action Taken: The entire manuscript has been reformatted using the official MDPI LaTeX template. The revised document now adheres to all of the journal's formatting and style guidelines.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper is a “follow up” of an article “The Bateson Game: Strategic Entrapment, Frame Ambiguity, and the Logic of Double Binds”, which is already published (online) and available at https://zenodo.org/records/15331010 (also referred in this paper at the bottom of page 6, as a source article).

I am therefore not sure whether the current paper can be accepted as a new submission. I think this does not stand alone, unless of course, the materials from the original paper (such as the theoretical equilibrium) are included here.

The main theorem presented here in Section 3.1 is not clear at all as the background materials are missing. There is hardly any discussion of this result (theorem).

Section 3.4 mentions the simulation results however not presented here, instead are referred back to the original paper. There is no discuss here either.

As a reader, I am thus completely confused!! What’s the purpose of this paper? The author(s) must include a big chunk of the original paper here to make any sense of the contribution. Otherwise, this short note is not an independent work.

I am happy with the simulations as a contribution; however, it has to be presented properly.

The abstract and the Introduction of this paper are thereby completely misleading, as it does not do what it claims to!

The conclusion is also short and does not provide any proper discussion.

The author(s) must decide whether or not to present it as a separate paper or not; if yes, then we need to see more details.

To sum up, I do think this paper has some substance (simulations) but it has to be presented properly before it can be a standalone paper, let alone be accepted for publication!!

Author Response

Concern: "This paper is a 'follow up'... I am therefore not sure whether the current paper can be accepted as a new submission. I think this does not stand alone, unless... the materials from the original paper (such as the theoretical equilibrium) are included here."

Our Response: We wholeheartedly agree with the reviewer's assessment of the previous draft. It was not a complete paper. The current, revised manuscript has been written to be a fully self-contained article.

Action Taken: All essential background and theoretical material has been integrated directly into the new manuscript.

Section 2 ("The Bateson Game: A Formal Model") now contains the complete formal definition of the game, its primitives, and a detailed explanation of the three core axioms.
Section 3 ("Equilibrium Failure and Learning Dynamics") now includes the full, rigorous proof of the non-existence of a separating Perfect Bayesian Equilibrium (Section 3.1).

The paper no longer requires any external reference to be understood.

Concern: "The main theorem presented here in Section 3.1 is not clear at all as the background materials are missing. There is hardly any discussion of this result (theorem)."

Our Response: The reviewer is correct. The original theorem was presented without sufficient context or proof.

Action Taken: The theoretical core of the paper has been completely rewritten and expanded in the new Section 3. The main theorem on learning dynamics has been significantly refined and moved to Section 3.2 ("Pathological Learning Dynamics: Ambiguity and Certainty Traps").

The new theorem is more nuanced, now formally distinguishing between the conditions that lead to the "ambiguity trap" (perpetual oscillation) and the "certainty trap" (stable, false belief).
The proof is now presented in full, with a step-by-step logical argument that is broken down into two distinct cases, making it far more rigorous and clear.
We have also added intuitive explanations to accompany the formal proof, ensuring the logic is accessible.

Concern: "Section 3.4 mentions the simulation results however not presented here, instead are referred back to the original paper. There is no discussion here either."

Our Response: This was the most significant flaw in the original draft, and we have completely rectified it. The simulations are no longer an external reference but are now a central, empirical pillar of the paper.

Action Taken: We have created a new, extensive Section 4 ("Computational Analysis") that presents and analyzes two distinct simulation experiments in full detail.

Full Presentation of Results: We now include the specific payoff matrices (Table 1 and Table 2) and present the results directly in the paper with four new figures.
Detailed Discussion: Each simulation is followed by a detailed analysis section (4.2.1 and 4.3.1) that interprets the figures and explains the underlying dynamics.

For clarity, here are the results that are now fully integrated and discussed in the paper:

Experiment 1: The Certainty Trap The simulation shows the Receiver's belief converging to a stable, incorrect certainty (Figure 1), leading to a single, fixed strategy (Figure 2).

Experiment 2: The Ambiguity Trap The second simulation shows the Receiver's belief locked in perpetual oscillation (Figure 3), leading to strategic paralysis (Figure 4).

The paper now stands on the substance of these properly presented simulations, just as the reviewer advised.

Concern: "The abstract and the Introduction of this paper are thereby completely misleading, as it does not do what it claims to!"

Our Response: We agree that the abstract and introduction of the previous draft were misleading because the paper did not contain the promised content.

Action Taken: The Abstract and Introduction have been completely rewritten to accurately reflect the new, self-contained structure of the paper.

The new Abstract now explicitly states that we "present a computational analysis contrasting these outcomes [the two traps], demonstrating empirically how different parametrizations lead to either trap".
The new Introduction (Section 1.3) provides a detailed outline that the paper now follows precisely, ensuring that all claims made at the outset are fully delivered within the manuscript itself.

Concern: "The conclusion is also short and does not provide any proper discussion."

Our Response: The reviewer is correct; the original conclusion was underdeveloped.

Action Taken: The Conclusion (Section 6) has been significantly expanded. It now provides a proper synthesis of the paper's refined theoretical claims and the new empirical findings from the simulations. It discusses the key insight that "manipulation can be engineered and durable, and that it may be most effective when it generates certainty rather than confusion," a conclusion that is only possible because of the new, fully-presented simulation results.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

report attached

Comments for author File: Comments.pdf

Author Response

Concern: "The formal development is imprecise and incomplete... lacks a motivation... uncertain about how the game studied differs from standard games... does not connect its results to the literature... does not appear to do an equilibrium analysis." Our Response: We agree entirely that the original draft failed on these crucial dimensions. The revised manuscript has been rebuilt from the ground up to be formally precise, well-motivated, and situated within the literature. This includes: a complete formal definition in Section 2 ("The Bateson Game: A Formal Model") of the strategic environment, including the game primitives, the sequence of play, and the three defining axioms; a new Section 1.1 ("Strategic Ambiguity: Beyond Types to Frames") dedicated to motivating the study and differentiating frame uncertainty from type uncertainty and Knightian uncertainty; a standard equilibrium analysis in Section 3.1 ("The Impossibility of a Stable Frame-Revealing Equilibrium") with a formal proof that no separating Perfect Bayesian Equilibrium (PBE) can exist; and a clear definition of the dynamic adjustment process in Section 4.1 ("Simulation Environment and Agent Model"), specifying the Sender's behavior and the Receiver's error-correction updating rule.
Concern: Imprecise technical-sounding terms. Our Response: We agree. The revised paper now introduces all terms within a clear context. "Bateson Game" is formally defined by its axioms in Section 2.2. "Frame uncertainty" is defined and contrasted with other forms of ambiguity in Section 1.1. "Boundedly rational learning" is specified as an error-correction model in Section 4.1.
Concern: The definition of M (the message set) is incomplete. Our Response: This has been corrected. In Section 2.1, the message set is now formally defined as .
Concern: "Do you want utility functions defined on A x I... Otherwise, payoffs do not depend on m." Our Response: Thank you for catching this critical imprecision. The reviewer is correct that payoffs do not depend directly on the message m. In the revised model (Section 2.1), utility functions are precisely defined as and . Payoffs depend on the action and the true frame. The message m is a signal the Sender uses to try and influence the Receiver's belief about the frame, but it does not directly enter the utility function.
Concern: Unneeded notation; Section 2 should specify the actual game, strategies, and solution concept. Our Response: We have completely rewritten Section 2 to be a comprehensive and precise definition of the game. It now includes the game primitives, sequence of play, axioms, and a discussion of strategies and beliefs (Section 2.3). We address the solution concept in Section 3.1 with a full PBE analysis.
Concern: "I am not sure what is... It is a function of history." Our Response: The reviewer's intuition is correct, and our original definition was poor. In the revised manuscript (Section 2.3), we clarify that is the Receiver's belief distribution over the set of frames F at time t, which is updated based on the history of play.
Concern: "If R can observe payoffs, why can’t he figure out the true frame? [He] 'inverts' to find ." Our Response: This is a crucial point that we failed to explain properly. The ability to "invert" the payoff requires the Receiver to know the full payoff matrix, which we do not assume. We have clarified the informational assumptions of our boundedly rational learner. The Receiver observes their own realized payoff for the action they took, but they do not know the complete payoff function for all actions and frames. Learning is not a process of logical deduction but of error correction. If the Receiver expects +10 but gets -10, they know their belief was wrong, but they cannot uniquely deduce the true frame from that single data point. Their belief update is a gradual adjustment based on this "surprise," as modeled by the Rescorla-Wagner-style rule now detailed in Section 4.1.
Concern: "I would not use the term oscillation... Incomplete learning seems like a better term." Our Response: This is a thoughtful semantic point. We have adopted the reviewer's suggestion in spirit by reframing our results. While we retain the term "oscillation" for the "ambiguity trap" because the visual evidence from our new simulations is so strikingly oscillatory, we now categorize both outcomes under the broader heading of "Pathological Learning Dynamics" (Section 3.2). We explicitly contrast the oscillatory "ambiguity trap" with the convergent "certainty trap," demonstrating that "incomplete learning" can manifest in multiple ways. This makes our terminology more precise and our findings richer.
Concern: Theorem labeling. Our Response: This has been corrected. All theorems, propositions, tables, and figures are now properly numbered and cross-referenced throughout the manuscript.
Concern: The description of S's and R's behavior is not defined. Our Response: The reviewer is absolutely correct. This was a major omission. We have added Section 4.1 ("Simulation Environment and Agent Model") which provides a precise, formal description of the agents' behavior in the simulation. The Sender (S) is a myopic best-responder, strategically selecting the true frame in each round t to maximize their immediate payoff, contingent on the Receiver's action . The Receiver (R) is a boundedly rational agent using an error-correction learning rule to update beliefs based on prediction errors, and a softmax function for action selection.
Concern: "The paper needs to present simulation results and discuss them in detail." Our Response: We agree completely. The original draft's reference to an external link was inappropriate. The revised manuscript now makes the simulation results a central pillar of the paper's contribution. The new Section 4 ("Computational Analysis") is dedicated entirely to this. We present two distinct experiments, complete with payoff matrices, and provide four figures visualizing the results, which are then discussed in detail. Experiment 1, The Certainty Trap, demonstrates stable convergence to a false belief. Experiment 2, The Ambiguity Trap, demonstrates persistent, high-frequency oscillation and strategic paralysis. These results, now fully integrated, provide the empirical grounding that was missing and directly support our refined theoretical claims.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

This revised version does provide all the details however it is now (too) close to the originally published paper; incidentally, the author does not even site the paper in this version (which may amount to “self-plagiarism”!).

I am therefore not sure whether which parts of this revised version should be (re-)published here again. The current paper must tell the readers and clearly indicate the “new” contributions that were not present in the original published paper.

The model (Section 2) is very well presented, except the very last part (Remark 2.8) – this remark should be clarified – perhaps more details is required here.

I am confused by the term “Clarification-Resolving Equilibrium” as in Definition 3.2 and the statement of the main result (Theorem 3.3). Does it say that there does not exist any Clarification-Resolving Equilibrium? There should be some comments and discussions on this result (and by the same token, after the next two results, that is, Theorems 4.4 and 5.5). All these contributions should have discussed more in the paper.

I am pleased to report that the author has taken care of all the previous comments and thus the revised version has improved a lot in terms of presentation. However, I wonder what happened to the simulations presented in the earlier version? This revised paper seems to be a completely different theoretical paper!!

To sum up, I do think this paper has improved a lot and can be published. However, the author should a. be honest about the new contributions and b. present all the results of the of this standalone paper more carefully for the readers to understand.

Author Response

Relationship to Prior Work, Novelty, and Potential Self-Plagiarism

As I mentioned in my earlier report (on the previous version), this paper is a “follow up” of an article “The Bateson Game: Strategic Entrapment, Frame Ambiguity, and the Logic of Double Binds”, which is already published (online) and available at https://zenodo.org/records/15331010. This revised version does provide all the details however it is now (too) close to the originally published paper; incidentally, the author does not even site the paper in this version (which may amount to “self-plagiarism”!). I am therefore not sure whether which parts of this revised version should be (re-)published here again. The current paper must tell the readers and clearly indicate the “new” contributions that were not present in the original published paper.

Response:

I apologize for the oversight in not explicitly citing my own preprint in the previous submission and for the lack of clarity regarding the novelty of this manuscript. I understand the seriousness of this concern and have taken immediate steps to rectify it.

Citation Added: I have now explicitly cited the Zenodo preprint as Reference [7] (Page 13, Lines 468-469) and introduced it in the text (Page 2, Lines 75-79).
Clarification of Novelty: I have thoroughly revised the Introduction, specifically Section 1.3 (Page 2, Line 75 to Page 3, Line 94), to clearly delineate the relationship between this submission and the foundational preprint. I detail the novel contributions of this manuscript, which include:

The reformulation of the analysis using Perfect Bayesian Equilibrium (PBE), the standard for signaling games, rather than BNE.
The introduction of a novel theoretical distinction between the "Ambiguity Trap" and the "Certainty Trap" (Theorem 1), offering a deeper characterization of learning dynamics.
A new computational analysis designed to validate these distinct dynamic outcomes.

Clarification on Versions: I also wish to clarify the confusion regarding the similarity between the current submission and the version available on Zenodo. The Zenodo repository hosts an evolving working paper. The research conducted during this revision period led to significant updates in the manuscript, which were subsequently reflected in the Zenodo repository. I confirm that the current submission to Games represents a distinct, finalized, and significantly expanded contribution, as detailed above.

Clarification of Remark 2.8

The model (Section 2) is very well presented, except the very last part (Remark 2.8) – this remark should be clarified – perhaps more details is required here.

Response:

I appreciate the positive feedback on the model presentation. In the revised structure of the paper, the content previously in Remark 2.8 has been expanded and clarified in the new Remark 1 (The Nature of Frame Uncertainty) (Page 4, Lines 152-163).

This revised remark now provides a detailed explanation of the distinction between traditional state uncertainty and the interpretive mapping uncertainty (frame uncertainty) central to the Bateson Game, addressing the ambiguity noted by the reviewer.

Clarification-Resolving Equilibrium and Discussion of Results

Response:

I apologize for the ambiguity in the previous version's theoretical structure. The reorganization of the paper was largely driven by the need to address this specific confusion.

Clarification-Resolving Equilibrium (Formerly Def 3.2 and Theorem 3.3): I have explicitly addressed the concept of the "Clarification-Resolving Equilibrium" (Page 5, Lines 204-208). Furthermore, the new Corollary 1 (Page 5, Line 209-223) directly addresses the reviewer's question. I prove that if the Meta-Communication Penalty (Axiom 2) is strict, the 'Question' action is never played in PBE. Therefore, a Clarification-Resolving Equilibrium indeed does not exist under these standard conditions.
Synthesis of Main Results (Formerly Theorems 3.3, 4.4, 5.5): The core insights previously distributed across these theorems have been synthesized and refined into Proposition 1 (Equilibrium Failure) and Theorem 1 (Pathological Learning Dynamics: Ambiguity and Certainty Traps).
Discussions Added: As requested, I have added substantial discussion sections immediately following each main result to interpret their significance:

Discussion of Proposition 1 (Page 5, Lines 196-203).
Discussion of Corollary 1 (Page 6, Lines 224-227).
Discussion of Theorem 1 (Page 7, Lines 275-289).

I believe this revised framework and the added discussions provide a much clearer exposition of the theoretical results.

Missing Simulations

Response:

The reviewer correctly noted the absence of simulations in the previous revision, which had focused purely on the analytical proofs available at that time. With the development of the new, clearer theoretical framework during this revision cycle (specifically, the distinction between the Ambiguity Trap and the Certainty Trap in Theorem 1), I felt it was essential to demonstrate these distinct outcomes empirically.

I have therefore introduced a new Computational Analysis (Section 4). I justify this addition (Page 8, Lines 302-306), noting that these simulations are specifically designed to validate and illustrate the novel theoretical findings of Theorem 1.

Summary

To sum up, I do think this paper has improved a lot and can be published. However, the author should a. be honest about the new contributions and b. present all the results of this standalone paper more carefully for the readers to understand.

Response:

I thank the reviewer for their guidance and encouragement. As detailed above, I have taken significant steps to delineate the novelty of this work (Comment 1) and have reorganized the theoretical framework and expanded the discussion around all main results (Comments 2 and 3) to ensure clarity and impact for the readers.

Author Response File: Author Response.pdf