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PCSY-D-24-00006

Emergence of an unpredictable evolution in a spatial prisoner’s dilemma via a player’s multiple perspectives

PLOS Complex Systems

Dear Dr. Sakiyama,

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Aming Li, Ph.D.

Academic Editor

PLOS Complex Systems

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Does this manuscript meet PLOS Complex Systems’s publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Partly

Reviewer #2: No

Reviewer #3: Yes

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: N/A

Reviewer #3: No

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3. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: No

Reviewer #2: No

Reviewer #3: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS Complex Systems does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: No

Reviewer #2: Yes

Reviewer #3: No

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5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: In this manuscript, the authors propose two spatial prisoner's dilemma models, where individuals change the neighborhood definition in accordance with their strategies and sometimes select the rule for this change using probability or local information. Their simulation results show that the models can generate characteristic population patterns that may be linked to self-organized criticality. After reading this work, I have some following comments or questions.

(1) After reading the model description, it is unclear how the individuals change the neighborhood. Why can individuals change the neighborhood? The authors should present some more clear description.

(2) In order to improve the readability, I suggest the authors present the mathematical equations or parameters in the model description. It is not very clear to see how individuals obtain the payoff and update the neighborhood according to the present version of the manuscript.

(3) Regarding the SLI model, it is not clear why the updating probability of neighborhood is defined as that form (see line 116). The authors should provide some justification.

(4) In the manuscript, the authors do computer simulations and mention that a total of 10000 time steps are performed in the iteration. I wonder whether the system can reach the stationary state after 10000 time steps. Does the step number is enough for each parameter set?

(5) In this work, the population size is set to 10000. Indeed the population size plays an important role in evolutionary outcomes. I wonder whether the results are still valid for larger population size.

(6) There are some previous works on evolutionary game dynamics, which are related with this study. For example, see Applied Mathematics and Computation 328 (2018) 162-170, Mathematical Models and Methods in Applied Sciences 29 (2019) 2127-2149, and Journal of the Royal Society Interface 20 (2023) 20220653. I think that the authors can mention these works when they revise the manuscript.

Reviewer #2: The manuscript titled "Emergence of Unpredictable Evolution in a Spatial Prisoner’s Dilemma via Multiple Player Perspectives" presents an exploration into the dynamics of the spatial prisoner's dilemma (SPD). The spatial prisoner's dilemma models agents interacting on a grid. This study aims to advance our understanding of evolutionary game theory by simulating two proposed models within SPD. To update the number of neighborhoods, one model employs a fixed probability weight, while the other uses a local kernel to calculate the ratio related to maximum and minimum scores among neighborhood. The models are well-designed but might be oversimplified for evolutionary dynamics.

Upon review, I am convinced that the emergence of critical transitions in evolutionary game theory represents an interesting pattern that can be investigated through simulations. However, the writing requires improvement. The model parameters are not clearly defined (for example, what does parameter b in the payoff matrix represent?) This paper does not qualify for publication unless the following major and minor points are addressed.

Major Points:

1. The dynamic computational process for updating the neighbors is clearly outlined, albeit very simple and abstract. Why is a fixed probability used? How is it chosen? Why is b set to 1.9? What significance does this parameter have? You have not mentioned any definition. How is the payoff matrix interpreted with different b and p values? To represent the larger payoff when a defector encounters a cooperator, why not test other values of b? The manuscript will not be convincing unless a sensitivity analysis with different parameters is performed.

2. What is the main implication of these two models? How do the two models define and contribute to our understanding of prisoner’s dilemma? What practical meaning arise from having a critical phase transition in this evolutionary game theory? How does it relate to the prisoner’s dilemma? What additional contributions do these various and complexly defined spatial structures, i.e., neighborhoods, provide?

3. The authors use simulations for all computations, but where is the code?

Minor Comments:

1. In line 31, it is unclear what SOC refers to. It needs to be defined at its first appearance.

2. In lines 39-41, what are the signs and ranges of different types of payoffs? How are they simulated? Including more details and providing the codes would be beneficial.

3. Regarding space and agents, how many trials were simulated? Why use periodic boundaries? Why is the initial defector density set at 0.5? What would be the effects of using different initial conditions? I believe these conditions are very important for critical phase transitions. More details are needed about the configuration.

4. In the title of Fig 3, "looser" may be corrected to "loser."

Reviewer #3: This paper proposed two spatial game models (PS and SLI) to investigate to the emergence of unstable or unpredictable evolution in spatial population. And some interesting phenomena were observed, such as the power-law tailed distribution of time interval. However, this paper is organized and written poorly, which should be carefully revised.

I invite the authors consider the problems as follows:

1.what kind of PD game is used in this paper, the weak PD or donation game? The authors should clearly state the meaning about the parameter b.

2.The simulation is done on a two-dimensional lattice in this paper. Is it a square lattice or Moore lattice? They should check the robustness of their results on other topologies, such as random graph.

3.I suggest the authors should depict the game model through flowsheet.

4.They observe some interesting phenomena about SPD, but they don’t deeply analyze them. Some crucial mechanisms about the evolution of unpredictable behavior should be carefully summarized. Some heavy tailed distributions about time interval have been observed in previous investigations, such as, [Chaos 29, 103103, 2019], which may help to analyze the related mechanisms.

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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Emergence of an unpredictable evolution in a spatial prisoner’s dilemma via a player’s multiple perspectives

PCSY-D-24-00006R1

Dear Ms. Sakiyama,

We are pleased to inform you that your manuscript 'Emergence of an unpredictable evolution in a spatial prisoner’s dilemma via a player’s multiple perspectives' has been provisionally accepted for publication in PLOS Complex Systems.

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

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

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

Best regards,

Aming Li, Ph.D.

Academic Editor

PLOS Complex Systems

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Reviewer Comments (if any, and for reference):

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

Reviewer #3: All comments have been addressed

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2. Does this manuscript meet PLOS Complex Systems's publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: N/A

Reviewer #3: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS Complex Systems does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The authors have addressed my comments accordingly, and I would like to recommend the publication of the work in the journal.

Reviewer #2: Thank you for your comprehensive responses and the revisions made to the manuscript. Below are further comments and clarifications required based on your responses:

You have expanded on the implications of these models for understanding the evolutionary dynamics in SPD. However, the practical significance and potential applications of identifying a critical phase transition remain vague. Could you elaborate on how these findings could influence real-world strategies or policies? Additionally, it might be beneficial to discuss if there are any potential limitations or conditions under which these models may not hold.

Additionally, there seems to be a typographical error "PSD" in line 39. Did you mean "SPD"?

Please consider these points to further refine your manuscript and enhance its contribution to the field.

Reviewer #3: This paper has been revised correctly, and the acceptance is considerable.

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7. 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.

Do you want your identity to be public for this peer review? If you choose “no”, your identity will remain anonymous but your review may still be made public.

For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

Reviewer #3: Yes: Zhihai Rong

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