Optimal Dynamic Control of Proxy War Arms Support
Round 1
Reviewer 1 Report
The article is devoted to the proxy war of the blues with the reds, where the blues use the territory and the army of the greens to harm the reds supporting greens with weapons. Assuming the passive behavior of the reds in choosing a strategy and their limited resources, the author builds a model for the optimal planning of the supply of weapons to the greens from the blues, taking into account the dynamics over time. The mathematical model is based on a simple optimal control problem, which is explicitly solved in quadratures using the Pontryagin maximum principle. The resulting explicit solution is visualized as graphs of the optimal supply function for various problem parameters. The mathematical model itself is very simple, at the level of an exercise for students. Its interpretation may be of interest as an example of a simplified simulation of a real situation.
Minor remarks.
1. In formula (7) on page 7, it should be ( x=0 or x=K) instead of ( y=0 or y=K).
2. Analitical calculations are too detailed and could be shorted.
3. Software listings in the appendix are trivial and don't give useful information. They are not needed to be presented.
Author Response
Dear Reviewer 1,
Thank you for your comments. Please see the attachment.
Best regards,
The Author
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The authors analyze a differential game where a coalition of countries has a proxy war with a country that is trying to invade a third country that wants to belong to the coalition.
The paper's contributions rely on describing, in a formal way, the implications of the current war in Europe. Although the paper introduces an interesting approach to analyzing war conflicts between countries, its structure and presentation need to improve. Below, I list some of my comments.
1. Introduction is not well-organized in the sense that it is not clear the relationship between the paper and the literature. I suggest rewriting the introduction by emphasizing in it the problem statement, analysis, contributions, and the paper's relationship with the literature. Also, I recommend creating a second section, named Theoretical Framework, where authors discuss the generalities of the optimal control theory.
2. Concerning the literature discussion, the authors focus on books and seminal papers. In this sense, the authors need to revise other papers, such as
Zhang, T., Li, C., Ma, D., Wang, X., & Li, C. (2021). An optimal task management and control scheme for military operations with dynamic game strategy. Aerospace Science and Technology, 115, 106815
Zhang, Y., Zhang, P., Wang, X., Song, F., Li, C., & Hao, J. (2022). An open loop Stackelberg solution to optimal strategy for UAV pursuit-evasion game. Aerospace Science and Technology, 129, 107840.
Kim, G. J., Kim, B., & Kim, J. (2019). Equilibrium in a war of attrition with an option to fight decisively. Operations Research Letters, 47(4), 326-330.
3. Materials and Methods are confusing. The context description is long and verbose, while the mathematical model appears disconnected from the previous part. Particularly, the authors should improve the presentation of the state variable, strategies space, objective function, and system dynamics. For example, it is not clear if the authors pretend to analyze a zero-sum or a pursuit-evasion game. Does the state variable refer to facility location? Do players want to minimize the cost of transporting military resources? Why do authors describe location by considering only one variable? Is it not convenient to introduce coordinates x(t) and y(t)?
In general, I suggest introducing the game-theoretical model in a more traditional way (1. players, 2. actions, 3. game rules, 4. payoffs)
4. In the equilibrium analysis, I do not understand the meaning of partial functions. Also, the authors do not clearly discuss the difference between the static problem and the dynamic optimization problem that the three players face; the introduction of the static analysis is difficult to follow. Moreover, as far as I understand, the central analysis focuses on a deterministic problem. Why do authors talk about expected losses and gains?
5. Results presentation can be improved by introducing them as Propositions or Theorems. The algebraic manipulations can be included in an Appendix. Also, numerical examples and game variations need to be rewritten. What are the implications of following equilibrium strategies? Which are the main assumptions of the interaction? I think that authors can polish the presentation of the results by establishing relationships between exogenous parameters and their implications on war outcomes.
6. There is an abuse of graphics. Also, their relationship with the main model is not clear. For example, is there a parameter that relates to nuclear weapons?
7. The use of English can be improved. Particularly verb tenses. Some ideas are written in simple future instead of the simple present (which is the correct form)
Author Response
Dear Reviewer 2,
Thank you for your comments. Please see the attachment.
Best regards,
The Author
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Thanks for addressing my questions and clarifying my doubts in the previous review stage. Particularly, in their answer, the authors emphasize that their study does not pretend to establish a differential game, which was my first understanding since the theoretical framework reviews game-theoretical models in detail, and the problem description emphasizes the existence of three agents, which may be strategic.
Having in mind that the paper focuses on deriving an optimal control strategy concerning the location of the war front, my comments are the following:
1. The introduction immediately starts with the theoretical framework discussion, which is not the best way to present the paper. Including introductory paragraphs that state and motivate the paper's objective is necessary; if such paragraphs summarize the differences and similarities with other papers, the paper's objective and contribution become clear. Some of the previous ideas are included in the abstract. I consider that the paper may be enhanced by simplifying the abstract and explaining the paper's contribution and methodology in more detail in the introduction.
2. Concerning the use of English, the authors use simple future to present and solve the model. Since the paper does not validate the model empirically, it is better to write verbs in the simple present tense to discuss the model.
3. The theoretical framework discusses game theoretical models concerning proxy wars. Moreover, the authors consider dynamic game theory as a keyword of their study. The previous points confuse the readers since they can expect a game theoretical model/analysis. However, the paper points in another direction. Specifically, in the present study, not all agents are strategic in the sense of game theoretical models, which is why I suggested specifying the model's assumptions concerning the model features in my previous review. Typically, in Game Theory, proxy wars are modeled as principal-agent relationships where the resources of the first one impact the payoffs of the second one, and the actions of the second one impact the benefits of the first one. Later, the game solution searches for strategies under which agents have no incentives to deviate. In the paper, there is only one decision variable (x) that represents blue's decision-making. Other agents' actions are linked to one agent's decision-making since the war front is modeled as a point in a line whose maximum length is constant. The previous assumption makes it possible to dismiss strategic analysis from two agents during the model solution. Yes, the paper discusses the previous fact in the Materials and Methods section, but such an explanation is split into three subsections. In my opinion, as I suggested in my previous review, it would be better for the paper to describe the problem while introducing the notation at the same time. Otherwise, the introduction of notations f1 , f2, and f2 may suggest that we miss something in the game description. The previous confusion increases when we look at the previous functions, and they depend on x and other variables; this is not clear since there is an empty space in the functions' argument (equation 38). Given the model's assumptions, this is not the case, and it is possible the direct implementation of the optimal control analysis. Maybe, it is a typo related to the file's compilation and not a miss-specification from the model basics.
Also, my previous suggestion may be unnecessary (given that the other reviewer has no problems with the model), but I think that commenting on such a point in the introduction is important since the proxy war is analyzed by simplifying the interaction between the agents. In other words, the paper provides a non-principal-agent analysis of a proxy war, which represents a contribution.
5. Minor details. From line 346, the document displays empty squares in some equations, which may generate confusion concerning the notation and the variables. See my previous comment.
Author Response
Dear Reviewer 2,
Thank you for your well motivated comments and suggestions. I have now followed them and adjusted the manuscript. I attach a document with my detailed description of my responses to your point.
Best regards,
The Author
Author Response File: Author Response.pdf
