Identifying the Maximum Concentration of Results in Bilateral Sports Competitions

Round 1

Reviewer 1 Report

This paper shows that the HHI is not a stable measure of competitive balance if the number of points in a championship is not fixed. In particular, the complete cascade distribution does not accurately represent the upper-level of competitive imbalance. The authors derive what the distribution is under this situation and what represents the extreme end of competitive imbalance.

Some further clarification, especially with regards to terminology, would help readers less familiar with European football. By "points in a championship," I take it to mean that, 3 points are awarded for a win, 1 point is awarded for a tie, and no points are awarded for a loss. The issue seems to be that if 3 points are awarded for a win rather than 2, then the distribution is unstable. Some clarification on why 3 points is a problem but not 2 would be helpful. 

Also some clarification about what it means when "the total number of points of a championship is constant." This be a familiar concept to European football fans, but less so to fans of other sports. A brief overview of how a championship is awarded under this system would be helpful.

An example of measuring competitive balance with the traditional HHI vs the authors' HHI would also be helpful in illustrating the authors' point regarding the problem with the HHI under this situation. 

Author Response

Dear reviewer,

As authors of the manuscript ID mathematics-1213463, titled: “Identifying the maximum concentration of results in bilateral sports competitions”, we want to thank you for your comments and suggestions, which have greatly helped us improve the quality, readability and scientific rigor of our paper.

The changes to the manuscript are reported in the uploaded document in the order they were requested. However, we must note that some of the line numbers referred to by the reviewer may have changed as a result of the changes suggested by reviewers.

All these changes have been carried out taking into account the requirements of the journal. We hope we have fully addressed your concerns. As authors of the paper, we would like to thank you for your suggestions to make improve the quality of manuscript.

Sincerely,

LIST OF CHANGES MADE TO THE MANUSCRIPT

We have taken all the reviewers' suggestions into consideration. Particularly:

1. We have completed the Abstract incorporating the suggestion of one referee.
2. The Introduction has been amended to incorporate:
   • More information on the concept of concentration.
   • More information on the concept of bilaterality.
   • A reference to the existence of other competitive balance metrics (wins) in addition to the points achieved.
   • An extension of the reference to the Uncertainty of Outcome Hypothesis.
   • Redrafting of some paragraphs.
   • An example of measuring competitive balance with the traditional HHI_max (Complete cascade distribution) versus the correct HHI_max (Truncated cascade distribution).
3. The Section Materials and Methods has been amended to incorporate:
   • More detailed information on scoring patterns and the meaning of "the total number of points of a championship is constant."
   • We detail that the scoring system of the European football leagues does not fulfil the stability condition.
   • We stress that several measures of competitive balance should be redone.
4. In Section Discussion and Conclusions:
   • We stress that the reconstruction of results is a wrong choice and give an example for the five major European football leagues.

In addition, the document has been reviewed for English language, grammar and punctuation by a professional to improve the proofreading test.

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Manuscript ID Mathematics-1213463

“Identifying the maximum concentration of results in bilateral sports competitions”

 Summary of Paper

The objective of this paper is to construct a perfectly unbalanced distribution that satisfies the stability condition which is needed to generate the maximum value of an HHI, a commonly used measure of market concentration. The perfectly unbalanced distribution is called the truncated cascade distribution. The distribution is then applied to scoring systems used in the major European football leagues to demonstrate that the maximum concentration level (as measured by HHI) can be obtained. The authors show that complete cascade distribution commonly used in the sports economics literature to evaluate competitive balance does not always satisfy the stability condition and, therefore, does not generate the maximum concentration level. In this case, competitive balance is underestimated.

Assessment of Paper

The paper makes an interesting methodological contribution to the extensive competitive balance literature in sports economics. However, it took a long time to get to why I should care about the properties of the complete cascade distribution or why getting a perfectly unbalanced distribution that satisfies the stability condition is of interest. This is finally discussed on page 10 of a 13 page manuscript. The relevance of the research question and the importance of the contribution to the competitive balance literature should be stated in the introduction and in the abstract. I have some suggestions for minor revisions to improve the readability and clarity of the paper.

Comments   

1. Abstract: The abstract is vague. It is not clear exactly what you are doing or how it moves the competitive balance literature forward. What do you mean by concentration? After reading the paper, I know you mean concentration in terms of market share or market concentration but this could be clearer.
2. Page 1, line 27: What do you mean by “maximum market concentration”? You eventually get to it later in the paper but the reader should know right away.
3. Page 2, lines 59-62: This needs to be written more clearly. I am not sure what you are trying to convey about differences in the competitive process for sports relative to other markets.
4. Page 2, lines 75-81: This is a narrow view of competitive balance. I believe there are many papers in this extensive literature that do not use points to develop measures of competitive balance. A commonly used metric is wins.
5. Page 2, lines 85-86: There is an extensive literature in sports economics on the uncertainty of outcome hypothesis. The literature has advanced in recent years such that Rottenberg’s hypothesis and its implications for attendance and television audiences has been challenged using reference dependent preferences. Further, the extensive empirical literature evaluating the uncertainty of outcome is mixed. The conclusion stated on lines 85-86 is not unambiguously supported in this literature. This should be addressed.
6. Page 3, lines 105-106: This sentence is unclear.
7. Page 3, line 111: What do you specifically mean by the bilateral nature of the encounters in sports competitions? Is it that only two teams can meet in any one competitive encounter?
8. Page 3, lines 136-140: You should say this sooner in the paper. Now I know what you mean by a bilateral encounter.
9. Page 3, lines 142-146: Why do we care about the calculation of a perfectly unbalanced distribution in the context of competitive balance?
10. Page 4, lines 147-152: What are some examples of when this is used? An example of what you mean in this paragraph would be helpful.
11. Page 10: line 365: What is UCL? I am pretty sure it is the United Champions League but I do not think it has been defined or mentioned before now.
12. Page 10, lines 374-377: I suggest saying this much sooner in the paper. It belongs in the abstract and the introduction because this is the real contribution of the paper.
13. Page 10, lines 374-377: This is an interesting conclusion but now I would like to know your thoughts on what the implications of underestimating competitive balance are. How much of an underestimation exists in the literature? Does it make a significant difference in the interpretation and implications of existing results? Should all of the papers using the complete cascade distribution be redone?
14. Page 10, lines 380-383: The sentences are unclear.
15. Page 11, lines 387-393: Where is this coming from? What do the numbers mean – 8 in the case of baseball, 30 in the case of American football, etc?

Author Response

Dear reviewer,

As authors of the manuscript ID mathematics-1213463, titled: “Identifying the maximum concentration of results in bilateral sports competitions”, we want to thank you for your comments and suggestions, which have greatly helped us improve the quality, readability and scientific rigor of our paper.

The changes to the manuscript are reported in the uploaded document in the order they were requested. However, we must note that some of the line numbers referred to by the reviewer may have changed as a result of the changes suggested by reviewers.

All these changes have been carried out taking into account the requirements of the journal. We hope we have fully addressed your concerns. As authors of the paper, we would like to thank you for your suggestions to make improve the quality of manuscript.

Sincerely,

LIST OF CHANGES MADE TO THE MANUSCRIPT

We have taken all the reviewers' suggestions into consideration. Particularly:

1. We have completed the Abstract incorporating the suggestion of one referee.
2. The Introduction has been amended to incorporate:
   • More information on the concept of concentration.
   • More information on the concept of bilaterality.
   • A reference to the existence of other competitive balance metrics (wins) in addition to the points achieved.
   • An extension of the reference to the Uncertainty of Outcome Hypothesis.
   • Redrafting of some paragraphs.
   • An example of measuring competitive balance with the traditional HHI_max (Complete cascade distribution) versus the correct HHI_max (Truncated cascade distribution).
3. The Section Materials and Methods has been amended to incorporate:
   • More detailed information on scoring patterns and the meaning of "the total number of points of a championship is constant."
   • We detail that the scoring system of the European football leagues does not fulfil the stability condition.
   • We stress that several measures of competitive balance should be redone.
4. In Section Discussion and Conclusions:
   • We stress that the reconstruction of results is a wrong choice and give an example for the five major European football leagues.

In addition, the document has been reviewed for English language, grammar and punctuation by a professional to improve the proofreading test.

Please see the attachment

Author Response File: Author Response.pdf