Making the most of potential: potential games and genotypic convergence

We consider genotypic convergence of populations and show that under fixed fitness asexual and haploid sexual populations attain monomorphic convergence (even under genetic linkage between loci) to basins of attraction with locally exponential convergence rates; the same convergence obtains in single locus diploid sexual reproduction but to polymorphic populations. Furthermore, we show that there is a unified theory underlying these convergences: all of them can be interpreted as instantiations of players in a potential game implementing a multiplicative weights updating algorithm to converge to equilibrium, making use of the Baum–Eagon Theorem. To analyse varying environments, we introduce the concept of ‘virtual convergence’, under which, even if fixation is not attained, the population nevertheless achieves the fitness growth rate it would have had under convergence to an optimal genotype. Virtual convergence is attained by asexual, haploid sexual and multi-locus diploid reproducing populations, even if environments vary arbitrarily. We also study conditions for true monomorphic convergence in asexually reproducing populations in varying environments.

dynamics of a (potential) game, and by applying insights from game theory (regret minimization) and by using the Baum Eagon theorem. The main results are as follows: (-) If the fitness of genotypes is fixed (i.e., there is a constant environment), and if reproduction is asexual, haploid, or single-locus diploid, the authors prove that the dynamics will always converge to a fixed point (which is shown to be a Nash equilibrium). (-) In all other cases, the dynamics converges "virtually". This means that the dynamics itself may not converge (which is intuitively clear); however, the reproducing population asymptotically attains the same mean growth rate that it would have obtained, had it been comprised by the expost optimal genotype. General evaluation: The article is absolutely impressive. It combines techniques from different fields (population genetics, algorithms, game theory) to propose an elegant framework for how to analyse convergence of evolving populations. The introduction is extremely well written and provides a good summary of what the authors aim to do (especially the table in Figure 1 is very helpful). In the following, I only have a few smaller comments that the authors may take into account to further improve the paper.
Major Comment: (-) Maybe the biggest advantage of the paper, its multidisciplinarity, also entails a certain risk. There are probably not too many scholars that have sufficient knowledge in both population genetics and (algorithmic) game theory to appreciate all of the papers' findings. I do believe that the authors do a very good job in explaining the concepts they apply. However, in a few cases, some further exposition would have been useful. For example, I'd appreciate if the authors could explain in more detail, perhaps in Section 1.2, what virtual convergence is good for. What are the precise conceptual insights it offers? Can an it be used as a tool to prove further statements of interest? Similarly, what is the intuition behind a potential function? How does it help to prove things? Moreover, I'd appreciate an explicit statement about which results are already known (although the existing proof may be based on different techniques). For example, I'd assume that all nonvirtual convergence results are already known, is this correct? Given the many variables (and the relations between them), it might be useful to have a table that contrasts the variables (and interpretations) in game theory and the variables (and interpretations) in population genetics. Finally, I'd appreciate if there was a short conclusions section that (once more) summarises the results and puts them into the perspective of the previous literature.

Minor Comments:
(-) Last paragraph Section 1.1: "relatively small increased" -> "relatively small increases" (-) First paragraph Section 2.5: The index for {w_g^t} should be "g\in \Gamma" instead of "\omega \in \Gamma" (-) Section 3: Could you please add some references to the relevant literature on regret minimisation, and to the multiplicative weights algorithm? (-) Section 3.2: "l_i is greater (respectively less than) than..." --omit the first "than" (-) Proposition 8: Does this Proposition need an additional assumption? For example, in its present form, the Proposition seems wrong for pure Nash equilibria that are weakly dominated -for example for the 2x2 symmetric payoff matrix where the first player's payoff is 1 1 (first row) and 1 2 (second row). Here, the first strategy is a Nash equilibrium, but it's not locally stable under replicator dynamics.
In the proof, the authors seem to assume the Nash equilibrium is strict. (-) Section 6.1.1: "are generically monotonically increase" -> "increasing" Decision letter (RSOS-210309.R0) We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Dr Edhan
The Editors assigned to your paper RSOS-210309 "Making the Most of Potential: Potential Games and Genotypic Convergence" have now received comments from reviewers and would like you to revise the paper in accordance with the reviewer comments and any comments from the Editors. Please note this decision does not guarantee eventual acceptance.
We invite you to respond to the comments supplied below and revise your manuscript. Below the referees' and Editors' comments (where applicable) we provide additional requirements. Final acceptance of your manuscript is dependent on these requirements being met. We provide guidance below to help you prepare your revision.
We do not generally allow multiple rounds of revision so we urge you to make every effort to fully address all of the comments at this stage. If deemed necessary by the Editors, your manuscript will be sent back to one or more of the original reviewers for assessment. If the original reviewers are not available, we may invite new reviewers.
Please submit your revised manuscript and required files (see below) no later than 21 days from today's (ie 04-May-2021) date. Note: the ScholarOne system will 'lock' if submission of the revision is attempted 21 or more days after the deadline. If you do not think you will be able to meet this deadline please contact the editorial office immediately.
Please note article processing charges apply to papers accepted for publication in Royal Society Open Science (https://royalsocietypublishing.org/rsos/charges). Charges will also apply to papers transferred to the journal from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (https://royalsocietypublishing.org/rsos/chemistry). Fee waivers are available but must be requested when you submit your revision (https://royalsocietypublishing.org/rsos/waivers). Comments to the Author(s) Making the Most of Potential: Potential Games and Genotypic Convergence I think this is a very interesting work and can be considered for publication after the following concerns are addressed: 1. The references can be given in Figure 1 corresponding to each result. In addition, should it be a   Table 1. 6. The discussion should be elaborated and Conclusion Section should be given too. 7. on a side note, there seems to be some overlap with the results from: https://doi.org/10.1002/advs.202001995 https://doi.org/10.1038/srep34889 If so, do discuss these two papers in the introduction /background to motivate your work further.
Reviewer: 2 Comments to the Author(s) Summary: In this paper, the authors study evolutionary convergence for different kinds of reproduction (asexual, haploid, diploid). They do so by interpreting the dynamics as the evolutionary dynamics of a (potential) game, and by applying insights from game theory (regret minimization) and by using the Baum Eagon theorem. The main results are as follows: (-) If the fitness of genotypes is fixed (i.e., there is a constant environment), and if reproduction is asexual, haploid, or single-locus diploid, the authors prove that the dynamics will always converge to a fixed point (which is shown to be a Nash equilibrium).
(-) In all other cases, the dynamics converges "virtually". This means that the dynamics itself may not converge (which is intuitively clear); however, the reproducing population asymptotically attains the same mean growth rate that it would have obtained, had it been comprised by the expost optimal genotype.

General evaluation:
The article is absolutely impressive. It combines techniques from different fields (population genetics, algorithms, game theory) to propose an elegant framework for how to analyse convergence of evolving populations. The introduction is extremely well written and provides a good summary of what the authors aim to do (especially the table in Figure 1 is very helpful). In the following, I only have a few smaller comments that the authors may take into account to further improve the paper.
Major Comment: (-) Maybe the biggest advantage of the paper, its multidisciplinarity, also entails a certain risk. There are probably not too many scholars that have sufficient knowledge in both population genetics and (algorithmic) game theory to appreciate all of the papers' findings. I do believe that the authors do a very good job in explaining the concepts they apply. However, in a few cases, some further exposition would have been useful. For example, I'd appreciate if the authors could explain in more detail, perhaps in Section 1.2, what virtual convergence is good for. What are the precise conceptual insights it offers? Can an it be used as a tool to prove further statements of interest? Similarly, what is the intuition behind a potential function? How does it help to prove things? Moreover, I'd appreciate an explicit statement about which results are already known (although the existing proof may be based on different techniques). For example, I'd assume that all nonvirtual convergence results are already known, is this correct? Given the many variables (and the relations between them), it might be useful to have a table that contrasts the variables (and interpretations) in game theory and the variables (and interpretations) in population genetics. Finally, I'd appreciate if there was a short conclusions section that (once more) summarises the results and puts them into the perspective of the previous literature.

Minor Comments:
(-) Last paragraph Section 1.1: "relatively small increased" -> "relatively small increases" (-) First paragraph Section 2.5: The index for {w_g^t} should be "g\in \Gamma" instead of "\omega \in \Gamma" (-) Section 3: Could you please add some references to the relevant literature on regret minimisation, and to the multiplicative weights algorithm? (-) Section 3.2: "l_i is greater (respectively less than) than..." --omit the first "than" (-) Proposition 8: Does this Proposition need an additional assumption? For example, in its present form, the Proposition seems wrong for pure Nash equilibria that are weakly dominated -for example for the 2x2 symmetric payoff matrix where the first player's payoff is 1 1 (first row) and 1 2 (second row). Here, the first strategy is a Nash equilibrium, but it's not locally stable under replicator dynamics. In the proof, the authors seem to assume the Nash equilibrium is strict. (-) Section 6.1.1: "are generically monotonically increase" -> "increasing" ===PREPARING YOUR MANUSCRIPT=== Your revised paper should include the changes requested by the referees and Editors of your manuscript. You should provide two versions of this manuscript and both versions must be provided in an editable format: one version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); a 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them. This version will be used for typesetting if your manuscript is accepted. Please ensure that any equations included in the paper are editable text and not embedded images.
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Author's Response to Decision Letter for (RSOS-210309.R0)
See Appendix A.

Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
Please proofread the manuscript carefullyand check that the references are cited properly, before uploading. It is extremely difficult to read the revised manuscript as there is literally no point to point responses except for some very brief comments in the cover letter. the authors have also uploaded 4 sets of revised manuscript totaling 200 over pages , rendering the reading very difficult.

Comments to the Author(s)
The authors have taken all my major suggestions into account. In particular, (i) They explain now in more detail how the framework of potential games (and potential functions) is useful. (ii) They explain in more detail how their results add to the existing literature, and (iii) They now provide a summarizing Conclusions section. I appreciate all these changes. In my opinion, already the first version of the manuscript was very good, and hence I support publication.
Just two minor comments: (-) I find this sentence a bit weird: "When the environment is fixed, they [reproductive processes] will converge to the stability of local optima as represented by Nash equilibria". I'd rather write "... they will converge to local optima represented by Nash equilibria" (-) On the same page, there is a "tuhs" that should be replaced by "thus"

Decision letter (RSOS-210309.R1)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Dr Edhan
On behalf of the Editors, we are pleased to inform you that your Manuscript RSOS-210309.R1 "Making the Most of Potential: Potential Games and Genotypic Convergence" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referees' reports. Please find the referees' comments along with any feedback from the Editors below my signature.
We invite you to respond to the comments and revise your manuscript. Below the referees' and Editors' comments (where applicable) we provide additional requirements. Final acceptance of your manuscript is dependent on these requirements being met. We provide guidance below to help you prepare your revision.
Please submit your revised manuscript and required files (see below) no later than 7 days from today's (ie 23-Jul-2021) date. Note: the ScholarOne system will 'lock' if submission of the revision is attempted 7 or more days after the deadline. If you do not think you will be able to meet this deadline please contact the editorial office immediately.
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Thank you for submitting your manuscript to Royal Society Open Science and we look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch.
Kind regards, Royal Society Open Science Editorial Office Royal Society Open Science openscience@royalsociety.org on behalf of Dr Derek Abbott (Associate Editor) and Mark Chaplain (Subject Editor) openscience@royalsociety.org Reviewer comments to Author: Reviewer: 1 Comments to the Author(s) Please proofread the manuscript carefullyand check that the references are cited properly, before uploading. It is extremely difficult to read the revised manuscript as there is literally no point to point responses except for some very brief comments in the cover letter. the authors have also uploaded 4 sets of revised manuscript totaling 200 over pages , rendering the reading very difficult.
Reviewer: 2 Comments to the Author(s) The authors have taken all my major suggestions into account. In particular, (i) They explain now in more detail how the framework of potential games (and potential functions) is useful. (ii) They explain in more detail how their results add to the existing literature, and (iii) They now provide a summarizing Conclusions section. I appreciate all these changes. In my opinion, already the first version of the manuscript was very good, and hence I support publication.
Just two minor comments: (-) I find this sentence a bit weird: "When the environment is fixed, they [reproductive processes] will converge to the stability of local optima as represented by Nash equilibria". I'd rather write "... they will converge to local optima represented by Nash equilibria" (-) On the same page, there is a "tuhs" that should be replaced by "thus" ===PREPARING YOUR MANUSCRIPT=== Your revised paper should include the changes requested by the referees and Editors of your manuscript. You should provide two versions of this manuscript and both versions must be provided in an editable format: one version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); a 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them. This version will be used for typesetting. Please ensure that any equations included in the paper are editable text and not embedded images.
Please ensure that you include an acknowledgements' section before your reference list/bibliography. This should acknowledge anyone who assisted with your work, but does not qualify as an author per the guidelines at https://royalsociety.org/journals/ethicspolicies/openness/.
While not essential, it will speed up the preparation of your manuscript proof if you format your references/bibliography in Vancouver style (please see https://royalsociety.org/journals/authors/author-guidelines/#formatting). You should include DOIs for as many of the references as possible.
If you have been asked to revise the written English in your submission as a condition of publication, you must do so, and you are expected to provide evidence that you have received language editing support. The journal would prefer that you use a professional language editing service and provide a certificate of editing, but a signed letter from a colleague who is a native speaker of English is acceptable. Note the journal has arranged a number of discounts for authors using professional language editing services (https://royalsociety.org/journals/authors/benefits/language-editing/).

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Decision letter (RSOS-210309.R2)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.
Dear Dr Edhan, I am pleased to inform you that your manuscript entitled "Making the Most of Potential: Potential Games and Genotypic Convergence" is now accepted for publication in Royal Society Open Science.
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On behalf of the Editors of Royal Society Open Science, thank you for your support of the journal and we look forward to your continued contributions to Royal Society Open Science.

May 2021
To: The Editor of Royal Society Open Science, We thank you for the opportunity to revise again our manuscript, Making the Most of Potential, Potential Games and Genotypic Convergence, for resubmission to the journal Royal Society Open Science.
Here is a summary of the changes that we implemented: • Figure 1 has been renamed Table 1.
• Table 1 now includes references to the theorems in the manuscript.
• In several places we now explain explicitly our use of notation such as d + g or d + i , in response to a question by reviewer 1. • Reviewer 1 asked why we set w t g and w t h to be e 1/3 and e 1/2 on page 21. The answer is that this is just one example of a process of temporally varying environments that does not converge to a single genotype. It so happens that the example we chose involves terms of the form e 1/3 and e 1/2 , to illustrate what may happen. There is nothing beyond that implied in the example.
• Reviewer 1 also asks about the examples in what is now Table 3 of the manuscript. We wish to emphasise that what appears in Table 3 is only an illustrative numerical 'toy' example, to aid us in making a point. It represents a possible hypothetical population that occasionally experiences 'rainy years' and at times 'drought years'. There is no intention here to state a theorem or imply any direct applicability of the models of the manuscript. • We have added more discussion in the introduction and a conclusions section, in response to a comment of Reviewer 1. • Reviewer 1 mentioned some overlaps between the subjects covered in our work and those of Cheong et al., 2016, andBabajanyan et al., 2020. We have added a literature review section that now includes explicit reference to those research articles. • Reviewer 2 asked if it might be possible to add more detail in Section 1.2 regarding the concept we introduce of virtual convergence, including conceptual insights. We have done so. • Similarly, Reviewer 2 asked about the intuition behind a potential function. More on this concept has therefore been added in Section 1.1. • We have added a literature review section in Section 1.4. This section includes explicit delineation of the results in our manuscript