On the Concept of Equilibrium in Sanctions and Countersanctions in a Differential Game

Round 1

Reviewer 1 Report (New Reviewer)

 The paper deals with noncooperative differential games and only with quadratic payoff functions. A strict form of active equilibria is studied instead of Nash equilibria. The main result, giving explicit expressions for equilibrium strategies is unclear. Does the game Γ in line 713 coincide with the game Γ in line 709? What do entries (2.2), (2.3) in line 713 mean? For this reason, the paper should be rejected. Besides, illustrative examples are desired.

I also note a number of misprints and inaccuracies.

·         The claim θ>0 in line 206 is redundant since the claim 0 ≤ t₀< θ is given in line 208.

·         The claim that u_i are constants in line 206 contradicts the definition of u_i in line 213.

·         x is missed in line 221.

·         Time discounting of payments should be added in line 228 since the introduction says about economic problems.

·         The lines 252 – 259 duplicate the lines 244 – 251.

·         The designation Û in line 261 is not explained.

·         It must be said about the left side of formula (4) instead of formula (4) in line 266.

·         J₂ should be instead of J₁ in line 290.

·         Lines 396, 397 do not belong to the proof of Lemma 3.

·         The game Γ_ν (line 456) is not presented in the paper. Probably, the game Γ_v is meant.

·         Statements in lines 574 – 577 are not clear.

·         Many formulas are not aligned properly. Other formulas are given too big or too small.

Author Response

Dear collegue!

Thank`s fou your professional and detal review! We Have made chages according to most of your comments. 

Best regards!

Author Response File: Author Response.docx

Reviewer 2 Report (New Reviewer)

The review is attached

Comments for author File: Comments.docx

Author Response

Dear colleague!

Thank you for your professional and detailed feedback on our study! We have made changes in accordance with your comments.

Best regards!

Author Response File: Author Response.docx

Reviewer 3 Report (Previous Reviewer 2)

The authors addressed all the issues listed in the report of the former submission.. There are no additional comments. Consequently, I am proposing the publication of the paper in the current form in the journal.

Best regards!

Author Response

Dear colleague,

Thank you for your appreciation of our research!

Best wishes to you!

Round 2

Reviewer 1 Report (New Reviewer)

The reviewer’s work is a big deal. Inattention to this work makes a bad impression when the authors answer hastily and do not take into account some of the comments.

                Unfortunately, the authors did not take into account my main comment. The following remains unclear to me. Does the game Γ referred to in line 709 (line 703 in the revised version) coincide with the game Γ referred to in line 713 (line 707 in the revised version)? If it is the same game, why it is introduced twice in the statement of the theorem? If the games are different, why the both games designated by Γ? Why the lines 703 – 706 highlighted in yellow? In my opinion, these lines coincide with lines 709 – 712 in the previous version.

                The authors did not reply on my comment about time discounting of payments.

                My remark that statements in lines 574 – 577 (lines 568 – 571 in the revised version) are not clear had not been considered. It seems that something is wrong with English grammar in this place.

                I recommend to reject the paper.

Author Response

Dear colleague!

Thanks for your comments! They improve the quality of our research.

Responses to your comments:

1. Does the game Γ referred to in line 709 (line 703 in the revised version) coincide with the game Γ referred to in line 713 (line 707 in the revised version)? If it is the same game, why it is introduced twice in the statement of the theorem? If the games are different, why the both games designated by Γ? Why the lines 703 – 706 highlighted in yellow? In my opinion, these lines coincide with lines 709 – 712 in the previous version.

 - Yes, they are coincided. The definition of game G is introduced on line 703. The changes have been made in the text and are highlighted in color.

2. The authors did not reply on my comment about time discounting of payments.

- The issue of time discounting of payments hasn`t been discussed in this article. But it may be a subject for future research. You're right! Thanks!

3. My remark that statements in lines 574 – 577 (lines 568 – 571 in the revised version) are not clear had not been considered. It seems that something is wrong with English grammar in this place.

- The statements have been corrected in lines 568-571. Changes have been made in the text and are highlighted in color.

Best regards!

Author Response File: Author Response.docx

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

The submitted paper is a considerable contobution to scholarship. In addition, the work plays an important role in the contemporary discussion. The mathematical part is fine. The central shortcoming of the paper is the lack of the literature review. The authors have to add at least one-page literature review whereby they have to mention both mathematical and political science literature. Refer also to https://link.springer.com/article/10.1057/s41287-019-00221-7

Try to reduce mathematical part by presenting some formulas in the appendix.

Author Response

Dear colleague!

Thank you for the high-quality and professional review!

The literature review in the form you have written about is not possible due to the fact that for our information only two authors of this publication are engaged in the creation and development of the mathematical theory of sanctions and counter-sanctions at this stage. Political science research is not part of the author's professional sphere of interest, but thank you for the possible direction of future research. Mathematical constructions are an integral part of the theoretical proof mentioned in the article and cannot be placed in a separate appendix.

Best regards!

Reviewer 2 Report

Dear Authors,

The topic of the manuscript is interesting, but there are some issues that should be considered in order to improve the quality of the paper.

First of all, the authors made great effort to present the idea, keeping the logical flow throughout the manuscript. The manuscript really has a great potential.

As far as important and necessary major improvements, the comments are as follows:

• Please state clearly in the abstract who can benefit from your work
• Methods section should be explained in more details
• The discussion should be more thoroughly elaborated, making a connection with previous research in the field. Furthermore, you should be more precise in explaining who and how can benefit from the results presented in the manuscript
• Please add in the conclusion limitations of your study and some specific propositions for further research which can be derived from your work.

Best regards!

Author Response

Dear colleague!

Thank you for your professional and detailed review of the article!

In the article we present a new methodology for solving the problem of the stability of equilibria of sanctions and countersanctions. The coefficient criteria under which there is an equilibrium of sanctions and counter-sanctions in the game and there is no generally accepted Nash equilibrium. The results of the research presented in the article belong to the fundamental ones and are aimed at forming the development of a new mathematical concept of sanctions and countersanctions, i.e., they are an increment of knowledge to the theory of active equilibria. They are useful to those who are developing the theory and strategy of war collision, matters of competition. At the present stage of research, it makes no sense to talk about specific proposals for applied research so far.

Best regards!

Reviewer 3 Report

The contribution of this manuscript is not clear. Besides the novelty claimed in Section 3, it is not evidenced firstly because the references are too old. The newer references are from the authors, which summed with the 39% similarity index from Turnitin indicates self-plagiarism, or at least, inadequate self-citation. Another big problem is poor presentation. No Figs. and Tabs. and single-paragraph discussion and conclusion are too bad for a scientific paper. I am sorry.

Author Response

The article presents a new methodology for solving the problem of the stability of equilibria of sanctions and countersanctions. The coefficient criteria are established, under which there is an equilibrium of sanctions and countersanctions in the game and there is no generally accepted Nash equilibrium. Proof mathematical constructions in this case do not involve figures and tabs. The creation and development of the mathematical theory of sanctions and counter-sanctions at this stage is only engaged by the authors of the article and, probably, this is why the assumption of "self-plagiarism or inadequate self-citation" arose. Unfortunately, there are no other publications on this direction in the theory of active equilibria. The new research results presented in the article are fundamental and are aimed at the formation and development of the author's mathematical concept of sanctions and counter-sanctions.

Best regards!