Lipschitz Continuity Results for a Class of Parametric Variational Inequalities and Applications to Network Games

Round 1

Reviewer 1 Report

Methodological Rigor

1. The paper mentions the use of Lagrange multipliers techniques. While this is a well-established method, the paper could benefit from a more detailed exposition of why this particular technique is most suitable for the problem at hand. Are there alternative methods that were considered and discarded? If so, why?
2. The paper discusses an algorithm based on the Lipschitz estimate. It would be beneficial to include a complexity analysis of the algorithm. This would provide a clearer understanding of its efficiency and limitations.

Theoretical Contributions

1. Uniqueness of Nash Equilibrium: The paper delves into Network Games and mentions the unique Nash equilibrium. However, it is unclear how the Lipschitz continuity results contribute to our understanding of this equilibrium. A section that synthesizes these two aspects would be highly beneficial.
2. Generalized Nash Equilibrium Problems: The paper also mentions Generalized Nash equilibrium problems. A more in-depth discussion or a separate section dedicated to how the Lipschitz continuity results apply to these generalized problems would add value.

Empirical Validation

Possibility of Case Studies or Real-world Applications: The paper could benefit from a dedicated section that explores the potential applicability of the Lipschitz continuity results in real-world contexts, particularly within the domain of network games. This exploration could outline hypothetical scenarios where the theoretical contributions could be impactful, thereby offering a conceptual bridge between the paper's theoretical rigor and practical utility.

 

Author Response

see attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments to Raciti et al

 

 

 

Summary

This mathematics study deals with parameter dependent variational inequalities leading to a solution as a function of the parameter. Specifically, the study improves existing estimates of the global Lipschitz constant for the continuity on the parameter assuming customary regularity conditions. Furthermore, the authors distinguish the case when it is possible to express the function at the core of the definition of the problem as a sum of a parameter dependent part and a part depending on a variable in Euclidean space. Moreover, the situation where the admissible subset is independent of the parameter also receives separate treatment. Subsequently the study provides an algorithm for approximating the solution of the inequality over the parameter interval from the values at a finite number of parameter points. As applications, there are numerical experiments on parametric games with the purpose of finding Nash equilibria, where the new Lipschitz constant estimations outperform previous estimates.

General comments

The authors have written an elegant and mathematically convincing treatise offering novel results on a topic certainly suitable for the Algorithms journal. Additionally, the sectional structure of the manuscript is logical, albeit I somewhat miss a Conclusion section that also seem to be customary among the articles published in Algorithms. A Conclusion section would in a compact form summarize the main findings of the paper and hint to future research. In all, there could more flesh on the mathematical skeleton in form of short comments connecting the flow of definitions and theorems with the intended applications. This is however only a minor remark; in general, it is an impressive work.

The English language of the manuscript is excellent. That is also true for the setup of the mathematical equations, the figures, and the tables.

Specific comments

Line 86: What do you mean by continuity of the data?

Theorem 1: Does the superscript T denote transposition? It would be more consistent to use the same inner product notation as in Equation (1) at the beginning of the chapter.

Theorem 2: The theorem follows the previous one so abruptly that the reader first may believe it is a generalization of that. It would be helpful with a soft transition (just a couple of lines) letting us know the next few theorems and corollaries will be your mathematical main results.

Proof of Theorem 2 first line: It is a bit presumptuous to state that we know we can express x1 and x2 as the projections given. Maybe it is not necessary to give a proof, but you could at least mention from where it follows.

Line 176: At least in the first part of the double comparative, it should be normal word order (predicate last). Furthermore, I think that, in contrast to many other languages, in English the word order also in the second part should be normal. (In that case, maybe you will need to replace the second appearance of is with becomes for stylistic reasons.)  

Section 4.1. Basics of Network games: The first part on graph theory is very elucidative and helpful, whereas the part on concepts of game theory is a bit too compact. You introduce the notation to distinguish the action of player i before an action notation in general. By the way, why is graph theory in lowercase and Game Theory in uppercase?

Line 244: …has been…

Comments for author File: Comments.pdf

Author Response

see attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have substantially improved the manuscript. While there are minor reservations regarding the novelty of the work, it is deemed to be of sufficient quality for publication in its current form.