A Disproportionality Bias in the Bureau of the Regional Assembly of Madrid
Round 1
Reviewer 1 Report
Summary
The authors examine the voting behavior of the parliamentary groups to appoint the governing Bureau (President, Vice-President, and Secretaries) in the Madrid Regional Assembly for the elections running from 1983 to 2021. They compare actual allocations to those obtained under the Nash Equilibrium (NE) and d´Hondt (d´H) allocation rules and claim that the allocation that would obtain in d'Hondt allocation depends on the number of voting stages under which the President, Vice-President, and Secretaries are elected. Their results show that while the NE was only followed in 4 of the 12 Legislatures for Vice-Presidents and argue that this gives an indication that parliamentary groups would gain seats in the Bureau were they to modifying their strategies.
Major comments
As presently written, the article is too condensed and difficult to follow. This can be seen below, there are many parts in which the article is not clear. Moreover, the authors make a lot of emphasis about a sequence of elections, and yet do not use subgame perfect Nash equilibrium in their analysis. In addition, the authors assume that the reader is familiar with the political landscape in Spain and in Madrid which is not the case for most readers of this journal.
The results as currently presented are too condensed. The authors do not take the time to guide the reader as to how to understand the different tables and the information contained in the tables both in the main text an in the appendix. The lack of explanation of the results makes it hard to follow what they are doing. The authors are forcing the reader to work too hard to understand what they're doing. Many would have stopped reading this paper earlier on.
One of the major points that makes this manuscript incomplete is that the authors never describe the political parties or their alignments or how they can relate to each other when it comes to coalitions. Coalitions are really important in this paper. The authors give the impression that they expect the reader to be extremely familiar with the political situation in Madrid and any other parts of Spain. Similarly in other parts of the paper.
On page 2, line 46 it says that Vice-Presidents are simultaneously elected. In line 63 it says that there is a first vice-president. This means that more detail is needed in the description of the rules.
Basic concepts: the Nash equilibrium
In Table 1 the acronyms of the parties or groups are given but the parties are never defined nor what their political inclinations are, nor what kind of alliances or coalitions can be formed. Brothers clearly state that in their model what matters is the number of seats. Yet one would expect that the type of coalitions/alliances that might arise will depend on the political ideologies of the parties.
There is a disconnect between the formal definition of the Nash equilibrium and immediately plunging into the distribution of seats in tables 1 and 2 in page 4.
In the paragraph starting in line 137 in Page 4 the authors state that a result is an Nash equilibrium and yet the reader has no clue how the others concluded this. A more detailed analysis of this needs to be shown.
In the paragraph on line 143 of page 4 the authors talk about the order in which the vice presidents are elected as being relevant and that the allocation was not optimal because certain parties could have formed coalitions. Again, we don't know why these parties would form coalitions. It seems from what the author is saying that all parties would care about are gaining add vice president or secretary. Yet it would seem that their policy positions may be crucial in forming those alliances.
On page 5 in line 158 the authors claim that 97 seats would have been needed. It is not clear where this 97 is coming from.
D'Hondt distribution and other proportional rules
Many tables are included in this section with little or no description to guide the reader as to what the reader should learn from these tables.
In this section the authors talk a lot about certain groups of parties splitting their votes differently, but we don't know if these parties would have been willing to do so.
In order to understand the formula given in line 284 on page 9 it would be good to have some examples for the E_i and N_i and to interpret these results before applying this formula in the tables.
On page 9 line 285 on page 9 why is the legislature divided into two groups? What do these parties have in common that they can be grouped into a single group?
Line 298 on page 9: why is the 0.95 multiplied by 7? Clearly it gives a percentage, but the number should be interpreted.
Theoretical analysis
Page 10 line 316, hypothesis 1, what does it mean for groups to be candidates?
Page 10 line 323, hypothesis 3, what does it mean that ties are settled evenly? Did you mean but each died candidate gets an equal probability being chosen? This question also applies two line 327.
Page 10 line 339, what does the assumption of loyal and in order mean?
Page 10 line 347 and 348 how is the expected number of seats found?
The way the paragraph at the top of page 11 is written it's not clear.
On page 11 line at 361, hypothesis 6 is hard to understand. Something seems missing from the statement of the hypothesis. Although the explanation below makes it clearer. Perhaps rewriting the hypothesis might make it clear too.
The way this section is written it doesn't look like a theoretical section. it is just a list of hypotheses. Little detail is given us to where this hypothesis come from.
Methodology
Note that on page 12 line 412 the authors are talking about a winning coalition. There has been no discussion of coalitions in the manuscript at this point even though conditions have been mentioned several times. Given that coalitions are an important element of the discussion, more attention should be devoted to explaining how coalitions form, how they vote, how party disciplines apply across coalition members.
On page 13 line 432, the authors talk about two groups even though there were many parties in the legislature. We still don't know what two groups would be important or why some parties would form a coalition government. Why would a group with 92 seats split their vote equally 46-46, possibility but no explanation is given for this. In the same paragraph, why would it only be required do you have 105 seats? Different calculations are also shown in this paragraph, but it is not clear to the reader where these calculations are coming from. The discussion is too condensed.
Coalition versus parties
On page 13 line 452 the authors are talking about information published in the news media, yet no citation of the news media is given here or the content of this media reports. The authors talk about two different coalitions considered in 2019 but do not explain what these coalitions are or what they mean for the analysis.
On page 13 line 454 the authors suddenly talk about the limitation of four members per political party, yet this constraint has not been incorporated into the analysis before. The authors have not shown how this constraint would affect their results.
It is not clear what the theory is in this “Theoretical” section, it is not clear how the analysis will be undertaken and yet we're heading into analyzing the results after this.
Analysis of the results
On page 13 line 463 the authors talk about that Section 3 demonstrates a relationship between the Nash Equilibrium and the d’Hondt distribution, I did not see a demonstration of this relationship in that section. I did see comments related the two, but I did not see a proof or demonstration of the relationship.
Statements are made that certain allocation of seats satisfy the Nash criteria, but no proof is given. As a minimum the authors should take an example and show/prove that no one party could benefit from deviating from the election outcome. Just showing the two distributions is not a proof.
In table 9 on page 14 there is a list of elections in different years and then there are columns showing weather in Nash equilibrium, the d’Hondt Vice-p and the Global d’Hondt within with coalitions and the columns and rows have a yes or no. Just stating that this is what happen is not enough to prove things so the analysis here is quite incomplete and certainly does not demonstrate anything. Since these elections were included in the Appendix a reference should be made to them here.
On page 15 line 502, the authors describe that in certain years the elections were not a Nash equilibrium outcome, yet no explanation is given for these circumstances.
In the next few paragraphs in this section the authors regularly comment but the results are in Nash equilibrium but again no proof is given of this. The author should discuss in detail the results for two or three elections than what is currently been done in the paper, then they could say that there are elections with similar characteristics.
The authors make the point of talking about sequence of elections and should be applying and relating it to the subgame perfect Nash equilibrium strategies.
Minor comments
Grammar problems:
- There are grammar problems, sometimes it is hard to understand what the authors are saying.
- The sentence in line 65to 68 on page 3 is really hard to understand.
- The paragraph In Page 3 line 79 is really hard to understand.
- What does the excess for coalition or party in line 103 on Page 3 mean?
There are several instances where Spanish words were left in manuscript. See for example, the equation given in line 197 where the word “si” was left in the equation when the translation of the document was made.
There are many instances in tables where numbers with decimals have a comma rather than a period to indicate the decimal. While comas are used in Spanish for decimals, periods are used in English.
The authors should make sure that all tables are in the same page, not split between pages.
In Line 216 on page 6 is there an extra 10 that shouldn't be there?
In Line 241 on page 7 is there an extra 1?
The last paragraph on page 11 refers to some literature with no citations, add citations so the reader can understand what articles you're referring to.
On page 12 line 406 should it be 4 divisions?
On page 15 the paragraph below table 10 is repeated twice.
It is better to say appendix than annex in English.
Author Response
Please see the attachment.
Author Response File: 
Author Response.docx
Reviewer 2 Report
This paper looks to apply game theory to the applied context of the Madrid regional assembly. This could be an interesting contribution that would be suitable for Games. But I have several concerns about how the paper is currently written. These can be summarised:
- The motivation and contribution of the paper is not sufficiently clear. It would be nice to see how the paper contributes to the large literature on political choice and/or behavioural game theoretic modelling of political choice? At the moment it reads more like a mathematical exercise with no real purpose. For instance, why is the Madrid Assembly an interesting example to study? What new can we learn from the analysis relative to the literature?
- The game needs to be formally set out. At the moment the reader basically has to construct the game themselves based on the rules set out on page 2. I would consider it essential to formally set out the game - the players, strategy set and payoff functions. It is particularly important to set out the payoff functions in detail. Without this it is impossible to calculate the Nash equilibrium or D Hondt distribution. Of course, we do not know what the payoff functions are in reality and so any assumptions should be justified and alternatives compared.
- The analysis could be more clearly structured. Again, if the motivation of the paper was clearer then it would help the reader through the analysis. What are the key results that come out of this analysis and how do they fit with the literature?
- The literature section needs to be considerably extended to acknowledge the large literature on strategic voting in assemblies and parliaments.
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
While the paper improved with the revision, it is still very hard to read. The revised manuscript seems incomplete, there are parts of the paper that are hard to follow and parts of the paper that lack details.
I agree with the comment made by the other referee that the paper, even in the revised form, reads like a mathematical exercise. The authors do not specify how they contribute to the large literature in political science and social choice or review that literature. adding some additional citations is not enough to put the paper within the context of the larger broader literature.
Moreover, as emphasize all throughout the paper, the authors may goal is to “analyse the consequences of voting in stages,” yet they use the Nash equilibrium derived for simultaneous games and claim that the analysis of sequential games is left to future work. This is also clearly stated when the authors formulate hypothesis 1 later on in the manuscript. While something can be learned from applying the Nash equilibrium of a simultaneous game in this context, the game as presented in this manuscript is by nature sequential. The Equilibrium concept to be used is that of subgame perfection and so the analysis as currently carried out is incomplete. The authors examine the stages of sequential voting. As can be seen in the (wrongly numbered section 4.1, should be 3.1) the authors discuss the number of voting stages.
The authors claim to have carried out media analysis to understand discrepancy of voting between the votes obtained by candidate and by the group she belongs to. No evidence is provided of the media analysis done or the media sources. No information is given on the discussion of the positions of the candidates on page 13. The lack of details does not allow the reader to follow how they carried out this type of analysis. The manuscript in that sense is incomplete. The analysis has a lot of empirical work but get the detail of the empirical work carried out is incomplete. Moreover, the authors claim both in the paper and in the response to my earlier comments That groups only care about the number of votes obtained and the coalitions that form do not depend in any way in their positions and give an example of extreme right wing and extreme left-wing parties that formed a coalition. But no discussion is given as to whether this is a common occurrence or just one exception in their sample. There is no argument presented in the paper to justify that parties care only about the number of posts won and that their ideological positions play role in forming the coalitions. A simple statement by the author saying that this is the case is not enough.
There are many questions that I asked in my previous comments that were answered in the file with responses to my questions but were not addressed in the manuscript. The reason I asked those questions is because it was not clear in the text what it meant and so needed clarification in the text. In addition, the authors sometimes use words there are not regularly used in English. The authors should try to look for words that better reflect what they're trying to convey. For example, breaking ties evenly is not regularly used in English, as the authors explain is that the tie is breaking randomly. A similar comment applies to the word loyal it needs to be defined in their context.
The authors insisted so much that the proof of Nash equilibrium, that both the other referee and I were asking for, are already published in another paper, that I read the paper by Perez and De la Cruz published in 2014 in Social Choice and Welfare on the “Implementation of Jefferson-d’Hondt rule in the formation of a parliamentary committee.”
Just because a section summarizes the work of others does not mean that you do not need to provide more detail. You cannot assume or force the reader to go and read the previous paper in order to understand the content of the current manuscript. Enough detail has to be given in this manuscript to understand what is happening in your context and in the empirical application that you are undertaking.
In essence, a lot of work needs to be done to sell this manuscript and the authors have not spend the time to do so.
Minor comments:
The word evenly in hypothesis 3 is still unclear. It is not a word that is regularly used in the literature.
Author Response
Please see the attachment.
Regards.
Author Response File: Author Response.docx
