Dear Prof. Baek,

Thank you very much for submitting your manuscript "Friendly-rivalry solution to the iterated n-person public-goods game" for consideration at PLOS Computational Biology.

As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. In light of the reviews (below this email), we would like to invite the resubmission of a significantly-revised version that takes into account the reviewers' comments.

The paper has now been revised by three of our reviewers, who have made a number of criticisms that are sufficiently adverse as to suggest that a major revision is due before a final decision about publication can be made. We therefore ask you to address all comments by the reviewers. However, we ask you to pay particular attention to:

- the technical aspects raised by reviewer 3. This might require to show that the results are robust beyond what is now presented in the MS.

- We would also like that similarities and differences with respect to the paper arXiv:2004.00261 were clearly stated. PCB only published highly original contributions, and we appreciate that you have studied the same strategy for the PD game, which apparently leads to very similar findings and even the discussion presented.

We cannot make any decision about publication until we have seen the revised manuscript and your response to the reviewers' comments. Your revised manuscript is also likely to be sent to reviewers for further evaluation.

When you are ready to resubmit, please upload the following:

[1] A letter containing a detailed list of your responses to the review comments and a description of the changes you have made in the manuscript. Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out.

[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).

Important additional instructions are given below your reviewer comments.

Please prepare and submit your revised manuscript within 60 days. If you anticipate any delay, please let us know the expected resubmission date by replying to this email. Please note that revised manuscripts received after the 60-day due date may require evaluation and peer review similar to newly submitted manuscripts.

Thank you again for your submission. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.

Sincerely,

Yamir Moreno

Guest Editor

PLOS Computational Biology

Stefano Allesina

Deputy Editor

PLOS Computational Biology

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The paper has now been revised by three of our reviewers, who have made a number of criticisms that are sufficiently adverse as to suggest that a major revision is due before a final decision about publication can be made. We therefore ask you to address all comments by the reviewers. However, we ask you to pay particular attention to:

- the technical aspects raised by reviewer 3. This might require to show that the results are robust beyond what is now presented in the MS.

- We would also like that similarities and differences with respect to the paper arXiv:2004.00261 were clearly stated. PCB only published highly original contributions, and we appreciate that you have studied the same strategy for the PD game, which apparently leads to very similar findings and even the discussion presented.

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: I found the study "Friendly-rivalry solution to the iterated n-person public-goods game" by Murase and Baek very interesting, if not important. I believe that the study should be published subject to addressing the issues listed below.

NON-NEGOTIABLE

• There are many mathematical symbols. A table of symbols is absolutely needed.

MAJOR

L209-225... I really struggled trying to understand the argument about efficiency. If I had more time, maybe I would fully grasp it, but as it is, the available information did not really convince me. I strongly suggest that the authors consider rephrasing the argument for the benefit of their readers.

L265-272... This whole paragraph sounds as if the authors were saying: "We checked our claims and you should trust us!" It is utterly unclear whether the checks were performed on paper or computationally. If the former is the case, then the authors should detail their calculations in an appendix or supporting text. If the latter is the case, then a reference to the precise part of the code should be made available.

Fig. 3... What do gray dots represent? What does blueish color coding stand for?

MINOR

L139... Eq. (1) does not really define a matrix.

L191, 192... I'm not sure if I got 'former' and 'latter' right at this particular spot in the manuscript. Perhaps the authors should rephrase.

L266... CAPRI-2 rather than CAPRI-n.

L366... The authors should cite OACIS in the Methods section.

Reviewer #2: This manuscript presents an interesting study on the social dilemma of cooperation by an approach based on the iterated n-person public-goods Game.

The manuscript is well presented. I found correct mathematics as well as the results obtained both via simulations and formal reasoning. However, I have some concerns about both the justification of the model and the discussion.

The authors extend the concept of Nash equilibrium to long-term payoffs. This extension has been adopted in recent literature, for example, as authors cited, in the context of zero-determinant strategies.

The authors define a set of conditions for a particular collaborative strategy (CAPRI). If I understand the model correctly, the condition for the Nash equilibrium, for the particular case of a symmetric strategic profile where all the agents share the strategy $\\Omega$, is reformulated as:

"It must be guaranteed that none of the co-players can obtain higher long-term payoffs against $\\Omega$ regardless of their strategies and initial states when e = 0."

Furthermore, the required memory length of players adopting such a strategy is m = 2n − 1. I find this concept far from the original Nash Equilibrium, and, at least, would expect a deep discussion by the authors on this (or perhaps to rename it). Nevertheless, the authors study the system for small values of n, which, one the one hand, make the memory requirement realistic while, on the other hand, bring this approach very close to that of pairwise interactions which has been addressed by the authors in Ref. 22.

The memory considered (2n − 1) is very long for large sets of co-players, furthermore when the authors state "not far from human behavior." Does it mean that, in an n-agents interaction, an agent may have a memory of 2n-1 previous interactions of everybody? I think that the authors should contextualize their model and justify it according to its particular requirements on memory length to differentiate it from that of pairwise interactions.

Reviewer #3: See attached PDF

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Have all data underlying the figures and results presented in the manuscript been provided?

Large-scale datasets should be made available via a public repository as described in the PLOS Computational Biology data availability policy, and numerical data that underlies graphs or summary statistics should be provided in spreadsheet form as supporting information.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

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Reviewer #1: Yes: Marko Jusup

Reviewer #2: No

Reviewer #3: No

Figure Files:

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Attachments

Attachment

Submitted filename: Review_PLOS_CAPRI.pdf

Dear Prof. Baek,

We are pleased to inform you that your manuscript 'Friendly-rivalry solution to the iterated n-person public-goods game' has been provisionally accepted for publication in PLOS Computational Biology.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow up email. A member of our team will be in touch with a set of requests.

Please note that your manuscript will not be scheduled for publication until you have made the required changes, so a swift response is appreciated.

IMPORTANT: The editorial review process is now complete. PLOS will only permit corrections to spelling, formatting or significant scientific errors from this point onwards. Requests for major changes, or any which affect the scientific understanding of your work, will cause delays to the publication date of your manuscript.

Should you, your institution's press office or the journal office choose to press release your paper, you will automatically be opted out of early publication. We ask that you notify us now if you or your institution is planning to press release the article. All press must be co-ordinated with PLOS.

Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Computational Biology. 

Best regards,

Yamir Moreno

Guest Editor

PLOS Computational Biology

Stefano Allesina

Deputy Editor

PLOS Computational Biology

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Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: I'm satisfied with the authors' answers to my comments.

Reviewer #2: I appreciate the efforts made by the authors to improve the manuscript as well as the answers to the reviewers' suggestions.

Reviewers' concerns have revealed some weaknesses in the model. In my opinion, in addition to the improvements made in the current version and the necessary corrections, the authors have contextualized these weak points by making the limitations explicit.

The revised version of this paper has improved significantly the original and, in my opinion, it can be accepted for publication.

Reviewer #3: I thank the authors for their careful response to the queries I raised. Although I would be interested to see a further analysis of the role of errors in determining the evolutionary dynamics of the strategies presented in this paper, I agree that the changes made in #11 and #12 are sufficient to address the issues I raised, and I'm therefore happy to recommend accepting the paper

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Have all data underlying the figures and results presented in the manuscript been provided?

Large-scale datasets should be made available via a public repository as described in the PLOS Computational Biology data availability policy, and numerical data that underlies graphs or summary statistics should be provided in spreadsheet form as supporting information.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

Reviewer #3: Yes: Alexander J. Stewart