Communication in Weighted Networks: A Game Theoretic Approach
Round 1
Reviewer 1 Report
The paper is related to the transferable utility cooperative graph-restricted games. It introduces weights to the properties of the graph to derive a novel allocation rule.
Some comments:
I am used to every equation in the paper being enumerated. Maybe this is not relevant to the publishing policy of the journal.
The description of related work is limited and based on a rather small number of old references. I would expect more new studies on related topics.
There are several references in lines 84 to 89 of the characterization from other authors. It would be beneficial to describe it in more detail and compare them with the one from the authors. Maybe this can be fused with the “Remarks”.
There is a typo at the beginning of line 119.
It is difficult to conclude if the proposed method would be beneficial.
Author Response
Dear Editor and Reviewers,
We thank you for your valuable comments and suggestions concerning our manuscript entitled “Communication in weighted networks: a game theoretic approach”. Based on these comments and suggestions, we improved our paper. All the changes that we have included in the paper are now in red color. Below we summarize our revisions and responses to your comments.
Best regards,
Elena C. Gavilán, Conrado M. Manuel and Daniel Martín
Response to the comments of Reviewer #1:
Comment 1: I am used to every equation in the paper being enumerated. Maybe this is not relevant to the publishing policy of the journal.
Author response:
Unfortunately the style of the journal is to number only the equations referred to in the text. In any case, we have no objection to numbering all the equations.
Comment 2: The description of related work is limited and based on a rather small number of old references. I would expect more new studies on related topics.
Author response: We have introduced 16 new references which are generally very recent. Several of them are at the bottom of page 1 and the remaining ones in the body of text referring to the characterizations of the Myerson, position and mixed values on page 5.
Comment 3: There are several references in lines 84 to 89 of the characterization from other authors. It would be beneficial to describe it in more detail and compare them with the one from the authors. Maybe this can be fused with the “Remarks”.
Author response:
We have described it in more detail and in addition we have introduced the last paragraph of Section 2.
Comment 4: There is a typo at the beginning of line 119.
Author response:
Corrected.
Comment 5: It is difficult to conclude if the proposed method would be beneficial.
Author response: We have included a new paragraph in the middle of page 2 in addition to the one mentioned at the end of Section 2.
Response to the comments of Reviewer #2:
Comment : Generally speaking, I don’t see great impact of this work. Though the title speaks of a game theoretic approach, I don’t see any strategic behaviors of the agents from the analysis. A lot of symbols and definitions need to be clearly explained. The writting should be much improved.
Author response:
We have included a new paragraph in the middle of page 2 and another one at the end of Section 2.
Comment 1: In the abstract, it would be better not to use citation like [1], [2], .... For example, I suggest using author names and year.
Author response:
Unfortunately this is the style of the journal. As the reviewer says, if we can decide we prefer the references with the author and year.
Comment 2: Line 11: What is a TU-game? Though it will be introduced at line 58, but here readers may have no idea about it at the very beginning.
Author response: We have included a paragraph at the beginning in the introduction section.
Comment 3: In Sect.1 (e.g., Line 15), the authors mentioned Shapley value without any definition and explanation. It’d better to roughly introduce its main idea here. Though in Lines 64–65 the Shapley value and the weighted Shapley value are defined, I recommend the authors to explain its meaning clearer.
Author response: We have introduced in the opening paragraph the idea behind the Shapley value. We have also introduced additional explanations on the Shapley value and the weighted Shapley value before Section 2.2.
Comment 4: Line 18: What’s the position value? How does it correspond to the ”allocation rule”?
Author response: Paragraph -1 in the first page has been rewritten trying to make clearer what the position value is. Earlier in the abstract and in the first paragraph it was intended to make it clear that an allocation rule is a point solution or value.
Comment 5: Line 62-63: What’s the uS here? It hasn’t been defined yet here. Also, I think Δv(S) is not well defined, either.
Author response: We have defined the unanimity games and the Harsanyi dividends.
Comment 6: Line 83: vγ (S) -. Why use ’S’ here?
Author response: It is necessary to use ’S’ to know what is the value of each coalition, which depends on its connected components.
Comment 7: Line 83: As η denotes a subset of γ, N/η may not be well defined and unclear for the readers.
Author response: We have defined it in the paragraph -1 of the Section 2.2.
Comment 8: The Myerson value, μ, ([2], [3]) and the position value, π, [5] and [6] allocate payoffs for players in a communication situation. -. This sentence is not clear and it may have a grammar error.
Author response: We have corrected the sentence.
Comment 9: To me, I am not very used to see citations at the starts of sentences. Authors’ names are usually mentioned conventionally.
Author response: We have eliminated the citations at the starts of sentences.
Comment 10: wv,γ is not defined.
Author response: The game is defined in paragraph 4 of Section 2.3.
Comment 11: Line 119: acoonection set-graph -. a connection set-graph
Author response: Corrected.
Comment 12: Line 129: Why wuS,γ (T,η)) is at most 1? For example, for more than one C ∈ T/η such that S ⊆C aren’t this value more than 1 possibly?
Author response: As the connected components are disjoint, the same set ’S’ cannot be contained in two components and therefore a value greater than 1 is not possible.
Comment 13: Line 160: the symbol ’l’ should be corrected.
Author response: Corrected.
Comment 14: I don’t see there is much difference between Theorem 2, Corollary 3, Theorem 4 and Corollary 4. The authors need to better clarify their results in Section 4.
Author response: The theorems refer to the characterization of the weighted mixed value. The corollaries to the particular case where all weights are unitary or equal, which leads to the mixed value defined by Feltkamp and van den Nouweland. Then, the corollaries are new characterizations of the mixed value. We have introduced the citation of the mixed value in the statement of the corollaries to try to make it clear that it is a characterization of a pre-existing value which follows in a simple way from the one introduced in the corresponding theorem.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The authors introduced weighted mixed value allocation on nodes and links of a network to well representing asymmetries of players and links that can be different types of information transmission. They also present characterizations of such kind of value allocation using five different properties.
Generally speaking, I don't see great impact of this work. Though the title speaks of a game theoretic approach, I don't see any strategic behaviors of the agents from the analysis. A lot of symbols and definitions need to be clearly explained. The writting should be much improved.
Put the issues specifically as below:
1. In the abstract, it would be better not to use citation like [1], [2], .... For example, I suggest using author names and year.
2. Line 11: What is a TU-game? Though it will be introduced at line 58, but here readers may have no idea about it at the very beginning.
3. In Sect.1 (e.g., Line 15), the authors mentioned Shapley value without any definition and explanation. It'd better to roughly introduce its main idea here. Though in Lines 64--65 the Shapley value and the weighted Shapley value are defined, I recommend the authors to explain its meaning clearer.
4. Line 18: What's the position value? How does it correspond to the "allocation rule"?
5. Line 62-63: What's the uS here? It hasn't been defined yet here. Also, I think \Delta_v (S) is not well defined, either.
6. Line 83: v^{\gamma}(S) -> Why use 'S' here?
7. Line 83: As \eta denotes a subset of \gamma, N/\eta may not be well defined and unclear for the readers.
8. Line 83: The Myerson value, \mu, ([2], [3]) and the position value, \pi, [5] and [6] allocate payoffs for players in a communication situation. -> This sentence is not clear and it may have a grammar error.
9. Line 84: To me, I am not very used to see citations at the starts of sentences. Authors' names are usually mentioned conventionally.
10. Line 84: w_{v,\gamma} is not defined.
11. Line 119: acoonection set-graph -> a connection set-graph
12. Line 129: Why w_{u_S, \gamma}(\{T, \eta\})) is at most 1? For example, for more than one C\in T/\eta such that S\subseteq C, aren't this value more than 1 possibly?
13. Line 160: the symbol 'l' should be corrected.
14. I don't see there is much difference between Theorem 2, Corollary 3, Theorem 4 and Corollary 4. The authors need to better clarify their results in Section 4.
Author Response
Dear Editor and Reviewers,
We thank you for your valuable comments and suggestions concerning our manuscript entitled “Communication in weighted networks: a game theoretic approach”. Based on these comments and suggestions, we improved our paper. All the changes that we have included in the paper are now in red color. Below we summarize our revisions and responses to your comments.
Best regards,
Elena C. Gavilán, Conrado M. Manuel and Daniel Martín
Response to the comments of Reviewer #1:
Comment 1: I am used to every equation in the paper being enumerated. Maybe this is not relevant to the publishing policy of the journal.
Author response:
Unfortunately the style of the journal is to number only the equations referred to in the text. In any case, we have no objection to numbering all the equations.
Comment 2: The description of related work is limited and based on a rather small number of old references. I would expect more new studies on related topics.
Author response: We have introduced 16 new references which are generally very recent. Several of them are at the bottom of page 1 and the remaining ones in the body of text referring to the characterizations of the Myerson, position and mixed values on page 5.
Comment 3: There are several references in lines 84 to 89 of the characterization from other authors. It would be beneficial to describe it in more detail and compare them with the one from the authors. Maybe this can be fused with the “Remarks”.
Author response:
We have described it in more detail and in addition we have introduced the last paragraph of Section 2.
Comment 4: There is a typo at the beginning of line 119.
Author response:
Corrected.
Comment 5: It is difficult to conclude if the proposed method would be beneficial.
Author response: We have included a new paragraph in the middle of page 2 in addition to the one mentioned at the end of Section 2.
Response to the comments of Reviewer #2:
Comment : Generally speaking, I don’t see great impact of this work. Though the title speaks of a game theoretic approach, I don’t see any strategic behaviors of the agents from the analysis. A lot of symbols and definitions need to be clearly explained. The writting should be much improved.
Author response:
We have included a new paragraph in the middle of page 2 and another one at the end of Section 2.
Comment 1: In the abstract, it would be better not to use citation like [1], [2], .... For example, I suggest using author names and year.
Author response:
Unfortunately this is the style of the journal. As the reviewer says, if we can decide we prefer the references with the author and year.
Comment 2: Line 11: What is a TU-game? Though it will be introduced at line 58, but here readers may have no idea about it at the very beginning.
Author response: We have included a paragraph at the beginning in the introduction section.
Comment 3: In Sect.1 (e.g., Line 15), the authors mentioned Shapley value without any definition and explanation. It’d better to roughly introduce its main idea here. Though in Lines 64–65 the Shapley value and the weighted Shapley value are defined, I recommend the authors to explain its meaning clearer.
Author response: We have introduced in the opening paragraph the idea behind the Shapley value. We have also introduced additional explanations on the Shapley value and the weighted Shapley value before Section 2.2.
Comment 4: Line 18: What’s the position value? How does it correspond to the ”allocation rule”?
Author response: Paragraph -1 in the first page has been rewritten trying to make clearer what the position value is. Earlier in the abstract and in the first paragraph it was intended to make it clear that an allocation rule is a point solution or value.
Comment 5: Line 62-63: What’s the uS here? It hasn’t been defined yet here. Also, I think Δv(S) is not well defined, either.
Author response: We have defined the unanimity games and the Harsanyi dividends.
Comment 6: Line 83: vγ (S) -. Why use ’S’ here?
Author response: It is necessary to use ’S’ to know what is the value of each coalition, which depends on its connected components.
Comment 7: Line 83: As η denotes a subset of γ, N/η may not be well defined and unclear for the readers.
Author response: We have defined it in the paragraph -1 of the Section 2.2.
Comment 8: The Myerson value, μ, ([2], [3]) and the position value, π, [5] and [6] allocate payoffs for players in a communication situation. -. This sentence is not clear and it may have a grammar error.
Author response: We have corrected the sentence.
Comment 9: To me, I am not very used to see citations at the starts of sentences. Authors’ names are usually mentioned conventionally.
Author response: We have eliminated the citations at the starts of sentences.
Comment 10: wv,γ is not defined.
Author response: The game is defined in paragraph 4 of Section 2.3.
Comment 11: Line 119: acoonection set-graph -. a connection set-graph
Author response: Corrected.
Comment 12: Line 129: Why wuS,γ (T,η)) is at most 1? For example, for more than one C ∈ T/η such that S ⊆C aren’t this value more than 1 possibly?
Author response: As the connected components are disjoint, the same set ’S’ cannot be contained in two components and therefore a value greater than 1 is not possible.
Comment 13: Line 160: the symbol ’l’ should be corrected.
Author response: Corrected.
Comment 14: I don’t see there is much difference between Theorem 2, Corollary 3, Theorem 4 and Corollary 4. The authors need to better clarify their results in Section 4.
Author response: The theorems refer to the characterization of the weighted mixed value. The corollaries to the particular case where all weights are unitary or equal, which leads to the mixed value defined by Feltkamp and van den Nouweland. Then, the corollaries are new characterizations of the mixed value. We have introduced the citation of the mixed value in the statement of the corollaries to try to make it clear that it is a characterization of a pre-existing value which follows in a simple way from the one introduced in the corresponding theorem.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
The authors have made significant refinement of their manuscript, though there are still some concerns listed below.
Line 28: [2][3] -> [2,3].
Line 38: [8], [9], [10], [11], [12] and [13] -> [8-13]. This can be achieved using \usepackage[sort&compress,numbers]{natbib}.
I would say this sentence can be rephrased as "Most of recent works have obtained characterizations in different settings [8-13]."
Line 39 & 123: I suggest use another name instead of calling it the "third" game/value.
Line 91 & 95: An n-person TU-game -> the 'n' should be italicized because it's a variable. Please check it through the entire manuscript.
In Line 95, \Delta_v(S) = \sum_{T\subseteq S} (-1)^{s-t}v(T) -> I don't see what the s and t are for here. Is s = |S|? Also, it's also unclear about the lower case s in Sh_i(N,v).
Line 125: It seems that Sh_{\ell}(\gamma, r_{\gamma}^v) is not well-defined.
So far we only have Sh_{i} defined on the players yet we haven't defined Sh_{\ell} for the edges.
Line 165: ...to obtain new characterization -> ...to obtain new characterizations.
Author Response
