Non-Cooperative Representations of Cooperative Games

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Authors,

Thank you for your submission. The manuscript presents a thoughtful and rigorous approach to bridging cooperative and non-cooperative game theory. To further improve its clarity and completeness, I offer the following suggestions:

1. Acronyms and Abbreviations: Please define all acronyms (e.g., NE for Nash Equilibrium, PFF for Partition Function Form) upon first use in both the abstract and main text. An acronyms and abbreviations list, including for formulas, is recommended.
   
   
2. Language and Proofing: The manuscript would benefit from careful English proofreading to address minor typos and improve sentence clarity.
   
   
3. Literature: Please consider citing more relevant work from the past five years to enhance the contemporary context of the study.
   
   
4. Focus and Structure: The presentation can be streamlined to focus more clearly on the proposed representation method and the theoretical contributions.
   
   
5. Conclusion Section: A concise conclusion section is recommended to summarize the findings and suggest directions for future research.

Best regards,

Author Response

1. Acronyms and Abbreviations: Please define all acronyms (e.g., NE for Nash Equilibrium, PFF for Partition Function Form) upon first use in both the abstract and main text. An acronyms and abbreviations list, including for formulas, is recommended.

An abbreviations list and list of mathematical notation commonly used in the manuscript are now added prior to the references.

2. Language and Proofing: The manuscript would benefit from careful English proofreading to address minor typos and improve sentence clarity.

The manuscript has been thoroughly scanned for typos and the appropriate edits have been made. Certain sentences have been changed for clarity.

3. Literature: Please consider citing more relevant work from the past five years to enhance the contemporary context of the study.

More recent work has been cited as part of additional Discussion and Conclusion sections in the manuscript.

4. Focus and Structure: The presentation can be streamlined to focus more clearly on the proposed representation method and the theoretical contributions.

A Discussion section has been added to provide more clarity to the context and importance of the theoretical contributions to the paper.

5. Conclusion Section: A concise conclusion section is recommended to summarize the findings and suggest directions for future research.

A Conclusion section has been added which summarizes the main results of the paper and suggests areas of further research.

Reviewer 2 Report

Comments and Suggestions for Authors

1. The main theorems (strictly super-additive PF games ↔ Nash etc.) were already available in a public pre-print more than a year ago. If this manuscript is an extension of that work, you should say so explicitly, cite the earlier version, and make clear what is genuinely new.

2. Lemma 1 requires every coalition to possess a strictly dominant joint strategy. Outside of degenerate or constant-sum environments this is extremely restrictive and should be defended.

3. The assumption of cost-free, externally enforceable binding agreements (lines 56-64)  rules out the very commitment problems that motivate many cooperative-to-non-cooperative translations; discuss realism and possible extensions to self-enforcing arrangements.

4. Theorem 1 is stated for arbitrary n but the threshold construction for θ is only demonstrated for 2- and 3-player cases (pp. 6-8). A rigorous argument that the maximum in (8) is finite and computable for general n is still missing.

5. Several inequalities in the proof (e.g. lines 268-274, 285-296) hinge on un-stated bounds; please include explicit derivations.

6. You claim that weakly super-additive CF games are represented under Nash (Theorem 2) but the argument relies on pay-off additivity that fails once positive externalities are allowed. Provide either a counter-example or a clarified proof.

 

Author Response

 

1. The main theorems (strictly super-additive PF games ↔ Nash etc.) were already available in a public pre-print more than a year ago. If this manuscript is an extension of that work, you should say so explicitly, cite the earlier version, and make clear what is genuinely new.

 

This manuscript edited for publication is supposed to be a refinement of the preprint based on feedback I have previously received from colleagues during seminars and reading groups. No new theorems have been introduced in this paper. Instead, the presentation of the results have changed considerably since the preprint.

 

2. Lemma 1 requires every coalition to possess a strictly dominant joint strategy. Outside of degenerate or constant-sum environments this is extremely restrictive and should be defended.

 

I detail some of the implications of the construction in the newly added Discussions section of the manuscript. 

 

3. The assumption of cost-free, externally enforceable binding agreements (lines 56-64)  rules out the very commitment problems that motivate many cooperative-to-non-cooperative translations; discuss realism and possible extensions to self-enforcing arrangements.

 

A treatment of this issue is now handled in the Discussions section. It should be noted that cost-free agreements to form coalitions (or equivalently costless enforcement) are a common assumption in the literature, see for example Ray and Vohra (1997). Although unrealistic, any costs incurred in coalition formation can be subsumed by the value of the coalition. So, this assumption is no more than a renormalization of the worth function and does not affect the proofs presented in the paper.

 

As for external enforcement, again this is a common motivation for defining the worth function. But one should not confuse this with the motivation for a particular solution concept to a cooperative game. To take an obvious example, many variants of the core that have been proposed as solutions to partition function form games rely on expectations as to how the partition of the player set looks following a deviation from the grand coalition. Clearly, this suggests that from the perspective of the players, any agreement to form the grand coalition is not truly binding. However, this does not detract from the idea that the value of the grand coalition can be understood to be what players can feasibly achieve if they have a binding agreement towards acting in the collective best interests.

 

It is worth emphasizing that my paper is agnostic as to which solution concepts for cooperative games are reasonable or realistic. It only seeks to demonstrate that many cooperative games can be given non-cooperative foundations that are more general than the construction in von Neumann and Morgenstern’s pioneering work (1944). I firmly believe that the former research agenda, while important, is beyond the scope of this paper.

 

4. Theorem 1 is stated for arbitrary n but the threshold construction for θ is only demonstrated for 2- and 3-player cases (pp. 6-8). A rigorous argument that the maximum in (8) is finite and computable for general n is still missing.

 

As the editor has noted, the previous examples for the 2- and 3- player cases are purely for illustration. A proof of the general case follows directly after the statement of Theorem 1. The argument that the maximum in (8) is finite is given directly below equation (8): “Note that  exists as the maximum is taken over a finite set; the size of the set is bounded by the number of distinct pure strategy profiles in Γ(N, v, θ)”. This in turn is quite clearly finite since the construction involves a finite player set and finite strategy sets.

 

5. Several inequalities in the proof (e.g. lines 268-274, 285-296) hinge on un-stated bounds; please include explicit derivations.

 

Parts of the proof of Theorem 1 have been rewritten for greater clarity.

 

6. You claim that weakly super-additive CF games are represented under Nash (Theorem 2) but the argument relies on pay-off additivity that fails once positive externalities are allowed. Provide either a counter-example or a clarified proof.

 

The proof for Theorem 2 holds as it is. The whole point is that characteristic function form games do not accommodate externalities. An externality occurs when the value of a coalition is impacted by the behavior of external players when they decide to form or dissolve coalitions of their own. Equivalently, an externality occurs when the value of a coalition changes depending on the partition of the player set. But the partition of the player set is even never specified in a characteristic function. In fact, I use this point in the proof of Theorem 2: see lines 317-8.

Reviewer 3 Report

Comments and Suggestions for Authors

The paper is devoted to undoubtedly relevant issues concerning the interrelations of the theory of cooperative and non-cooperative games. For this reason, it will be of undoubted interest to specialists in this field.

Unfortunately, after reading this paper, one is left with the impression of an incomplete study. One gets the feeling that the authors are in a hurry to publish an intermediate result that they consider important.

The work does not contain the sections “Discussion” and “Conclusions”.

The list of sources (References) is extremely laconic and fragmentary, which makes it difficult to position this study in the general pool of game-theoretical works.

The article would look more interesting and significant if it reflected the issues of the relationship between solutions to strategic games (Nash equilibrium, etc.), on the one hand, and cooperative games, on the other (Shapley vector, Core, Nucleolus, etc.)

The term "representation for cooperative (or non-cooperative) games" is quite local (even for specific game-theoretical literature). It is desirable to include additional explanations in the text of the work regarding its theoretical and practical meaning.

Author Response

 

1. The paper is devoted to undoubtedly relevant issues concerning the interrelations of the theory of cooperative and non-cooperative games. For this reason, it will be of undoubted interest to specialists in this field. Unfortunately, after reading this paper, one is left with the impression of an incomplete study. One gets the feeling that the authors are in a hurry to publish an intermediate result that they consider important.

 

I can understand this criticism insofar as the manuscript submitted aimed to provide the minimal literature review necessary to understand the motivations behind the research question. That combined with the fact that there is no Discussion or Conclusion section will likely make a reader feel this work is incomplete. It should be noted that many of the choices in structuring the paper this way was a result of recommendations and criticisms I had regarding a previous draft that detailed more heavily the intellectual history associated with tying cooperative and non-cooperative game theory together.  I have since re-introduced some of these ideas and comments in the newly added Discussion and Conclusion sections.

 

2. The work does not contain the sections “Discussion” and “Conclusions”.

 

Addressed. See above.

 

3. The list of sources (References) is extremely laconic and fragmentary, which makes it difficult to position this study in the general pool of game-theoretical works.

 

Many additional references are now provided as part of the Discussion section. They aim to provide some context for the antecedents of this research question. This was part of some material in an earlier draft that detailed the development of cooperative games in much greater depth.

 

4. The article would look more interesting and significant if it reflected the issues of the relationship between solutions to strategic games (Nash equilibrium, etc.), on the one hand, and cooperative games, on the other (Shapley vector, Core, Nucleolus, etc.)

 

I agree that that relating solution concepts across non-cooperative and cooperative games is an interesting research agenda. The Nash Program (Serrano, 2021) is precisely this: it tries to provide a rigorous micro-foundation for different cooperative solution concepts. But this is quite distinct from the purposes of my paper. My paper argues for a rigorous micro-foundation for the cooperative game itself. It is agnostic as to how one ought to solve the resulting cooperative game.

Von Neumann and Morgenstern (1944) define characteristic function form games as resulting from zero-sum games where the players can form coalitions and play against their complement. They defined the value or worth of that coalition to be their maximin payoff. My research attempts to generalize this notion in three different ways: it asks what can be said of cooperative games with externalities, i.e. partition function form games; it asks what happens if the non-cooperative representation was general-sum instead of zero-sum; and it asks what happens if the worth function was not defined by maximin payoffs but instead Nash equilibrium payoffs and generalizations and refinements of. All of this is clarified further in the Discussion section.

 

Incidentally, I do show in the Discussion that using the right representation is important to derive cooperative solutions that make sense economically. But again, cooperative solutions are not the primary focus of this research.

 

5. The term "representation for cooperative (or non-cooperative) games" is quite local (even for specific game-theoretical literature). It is desirable to include additional explanations in the text of the work regarding its theoretical and practical meaning.

 

Yes, unfortunately, there is no existing terminology (to my knowledge) for the idea that one can derive a cooperative game from a non-cooperative game.  I settled on the term “representation” in an attempt to address this. A cooperative game represents a non-cooperative game if it summarizes the payoff outcomes from coalition formation and a non-cooperative game represents a cooperative game if it details the strategy sets needed to generate the value of each coalition in a way that is consistent with equilibrium. It is implied by von Neumann and Morgenstern’s work (1944) that all characteristic function form games are represented by a general-sum game where the solution concept is maximin. But they never use this vocabulary. I have added comments on this in the Discussion section.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Everything was fixed.

Author Response

1. N/A: No suggestions for revisions were made by the referee.

Reviewer 3 Report

Comments and Suggestions for Authors

I have read the author's response to the article and its corrected version, with the degree of detail that the deadlines allow. I consider it necessary to note that the authors have done significant work to improve the quality of the text. At the same time, I must admit that this has not fundamentally increased the level of scientific significance.

As far as I could understand, the work can be attributed to the direction associated with the construction of characteristic functions of cooperative games. A certain difficulty for the reader's correct understanding of this idea is introduced by the term "representation of the game" (cooperative or non-cooperative) actively used by the author. Perhaps, when preparing the final version of the article, it is necessary to explain this more clearly.

Obviously, the work in this direction would be of interest in identifying some substantive dependencies between the concepts of solving strategic and cooperative games. However, making such a requirement is a sign of unrealistic maximalism.

I think that it is possible to limit ourselves to a request to more clearly emphasize in the introductory part of the article the objective limitations of the approach considered by the authors and the problems that it leaves unresolved.

If these comments are taken into account, then I believe that the work can be published.

Author Response

1. A certain difficulty for the reader's correct understanding of this idea is introduced by the term "representation of the game" (cooperative or non-cooperative) actively used by the author. Perhaps, when preparing the final version of the article, it is necessary to explain this more clearly.

This paper motivates the idea of a non-cooperation representation of a cooperative game in the introduction by describing how von Neumann and Morgenstern (1944) originally constructed cooperative games from zero-sum games. A more general definition of a non-cooperative representation has subsequently been added to this section: see lines 56-60. A more mathematically formal definition is given by Definition 4 (pg. 5) in the Preliminaries section.

 

2. I think that it is possible to limit ourselves to a request to more clearly emphasize in the introductory part of the article the objective limitations of the approach considered by the authors and the problems that it leaves unresolved.

The introduction now ends with a brief comment on the restrictions imposed by the approach in the paper. Since the Discussion section handles this topic in much broader detail, a more thorough treatment of the assumptions and limitations is left out of the introduction. The reader is instead redirected to the Discussion and Conclusion sections of the paper.