# Publications

### 2019

S.A. Bloks

Are Referendums and Parliamentary Elections Reconcilable? The Implications of Three Voting Paradoxes. Journal Article

In: Moral Philosophy and Politics, vol. 6, no. 2, pp. 281-311, 2019.

Abstract | BibTeX | Tags: Democracy, Elections, Majority voting, Referendums | Links:

@article{S.A.Bloks2019,

title = {Are Referendums and Parliamentary Elections Reconcilable? The Implications of Three Voting Paradoxes.},

author = {S.A. Bloks},

url = {http://suzannebloks.com/wp-content/uploads/2020/12/Bloks_Referendums_and_Parliamentary-Elections-MOPP2019.pdf},

doi = {https://doi.org/10.1515/mopp-2018-0055},

year = {2019},

date = {2019-11-17},

journal = {Moral Philosophy and Politics},

volume = {6},

number = {2},

pages = {281-311},

abstract = {In representative democracies, referendum voting and parliamentary elections provide two fundamentally different methods for determining the majority opinion. We use three mathematical paradoxes – so-called majority voting paradoxes – to show that referendum voting can reverse the outcome of a parliamentary election, even if the same group of voters have expressed the same preferences on the issues considered in the referendums and the parliamentary election. This insight about the systemic contrarieties between referendum voting and parliamentary elections sheds a new light on the debate about the supplementary value of referendums in representative democracies. Using this insight, we will suggest legal conditions for the implementation of referendums in representative democracies that can pre-empt the conflict between the two methods for determining the majority opinion.},

keywords = {Democracy, Elections, Majority voting, Referendums},

pubstate = {published},

tppubtype = {article}

}

In representative democracies, referendum voting and parliamentary elections provide two fundamentally different methods for determining the majority opinion. We use three mathematical paradoxes – so-called majority voting paradoxes – to show that referendum voting can reverse the outcome of a parliamentary election, even if the same group of voters have expressed the same preferences on the issues considered in the referendums and the parliamentary election. This insight about the systemic contrarieties between referendum voting and parliamentary elections sheds a new light on the debate about the supplementary value of referendums in representative democracies. Using this insight, we will suggest legal conditions for the implementation of referendums in representative democracies that can pre-empt the conflict between the two methods for determining the majority opinion.

### 2018

S.A. Bloks

Referendums in Representative Democracies. Law and the Mathematics of Voting. Masters Thesis

Utrecht University, 2018, (Supervised by R. Nehmelman and J.H. Gerards).

Abstract | BibTeX | Tags: Democracy, Elections, Majority voting, Referendums | Links:

@mastersthesis{Bloks2018d,

title = {Referendums in Representative Democracies. Law and the Mathematics of Voting.},

author = {S.A. Bloks},

url = {http://suzannebloks.com/wp-content/uploads/2020/12/Bloks_Referendums-in-Representative-Democracies_Dissertation-LLM.pdf},

year = {2018},

date = {2018-11-08},

school = {Utrecht University},

abstract = {European representative democracies are currently facing the challenges of rising populist powers and a decline in political legitimacy, which is accompanied by an increasing demand for more direct forms of democracy. The most widely discussed direct democratic instrument is the instrument of referendums, but does supplementing representative democracies with referendums have the desired effects?

The thesis deals with the question of the supplementary values of referendums in representative democracies. Opinions are divided on this topic. A variety of arguments is used by proponents and opponents of referendums about citizen’s knowledge and capacity, citizen’s power and representative’s power and prestige. Proponents and opponents do not seem to agree on the premises of these arguments and, furthermore, on how to weigh these arguments against each other. Hence, it is a currently unsettled question whether, and if yes, how referendums could and should be combined with representative democracy.

This question is addressed in the thesis from a multidisciplinary angle: a contribution to the legal-political debate about referendums in representative democracies is made by combining the legal-political arguments with mathematical insights. The mathematical insights show that the arguments on both sides of the debate about the supplementary value of referendums should be nuanced and, hence, the insights can help to find common ground between proponents and opponents of referendums in representative democracies.

The thesis consists of two articles. The first article, titled ‘Are referendums and parliamentary elections reconcilable’, examines the role of referendums in representative democracies in general. It looks at parliamentary systems in representative democracies and asks whether referendums and parliamentary elections can and should be combined. By drawing on three mathematical paradoxes, the article shows that referendums and parliamentary elections represent two different voting systems that could be contradictory. Because of the systemic contrarieties between referendums and parliamentary elections, the article argues that referendums should only be introduced in representative democratic systems with parliamentary elections under certain strict legal conditions. This article is published in the Journal for Moral Philosophy and Politics.

The second article, titled ‘De Wet raadgevend referendum afschaffen of verbeteren?’ (Abolishing or improving the Dutch Consultative referendum law?), focusses on referendums in the Netherlands. In particular, it considers the decision of the Dutch government to abolish the Dutch Consultative referendum law. In the article, it is argued that the government’s arguments for abolishing the law do not justify this decision and instead ask for improving the law. Using mathematical insights about the effects of quorum requirements in referendums, suggestions are given for changing the quorum requirement in the law. These changes could solve the problems of the Consultative referendum law that the Dutch government raised. This article is published in het Nederlands Juristenblad (NJB).

The research of this dissertation was awarded the Dare to Cross Over prize 2017 by Utrecht University, which is a prize for interdisciplinary graduate research.},

note = {Supervised by R. Nehmelman and J.H. Gerards},

keywords = {Democracy, Elections, Majority voting, Referendums},

pubstate = {published},

tppubtype = {mastersthesis}

}

European representative democracies are currently facing the challenges of rising populist powers and a decline in political legitimacy, which is accompanied by an increasing demand for more direct forms of democracy. The most widely discussed direct democratic instrument is the instrument of referendums, but does supplementing representative democracies with referendums have the desired effects?

The thesis deals with the question of the supplementary values of referendums in representative democracies. Opinions are divided on this topic. A variety of arguments is used by proponents and opponents of referendums about citizen’s knowledge and capacity, citizen’s power and representative’s power and prestige. Proponents and opponents do not seem to agree on the premises of these arguments and, furthermore, on how to weigh these arguments against each other. Hence, it is a currently unsettled question whether, and if yes, how referendums could and should be combined with representative democracy.

This question is addressed in the thesis from a multidisciplinary angle: a contribution to the legal-political debate about referendums in representative democracies is made by combining the legal-political arguments with mathematical insights. The mathematical insights show that the arguments on both sides of the debate about the supplementary value of referendums should be nuanced and, hence, the insights can help to find common ground between proponents and opponents of referendums in representative democracies.

The thesis consists of two articles. The first article, titled ‘Are referendums and parliamentary elections reconcilable’, examines the role of referendums in representative democracies in general. It looks at parliamentary systems in representative democracies and asks whether referendums and parliamentary elections can and should be combined. By drawing on three mathematical paradoxes, the article shows that referendums and parliamentary elections represent two different voting systems that could be contradictory. Because of the systemic contrarieties between referendums and parliamentary elections, the article argues that referendums should only be introduced in representative democratic systems with parliamentary elections under certain strict legal conditions. This article is published in the Journal for Moral Philosophy and Politics.

The second article, titled ‘De Wet raadgevend referendum afschaffen of verbeteren?’ (Abolishing or improving the Dutch Consultative referendum law?), focusses on referendums in the Netherlands. In particular, it considers the decision of the Dutch government to abolish the Dutch Consultative referendum law. In the article, it is argued that the government’s arguments for abolishing the law do not justify this decision and instead ask for improving the law. Using mathematical insights about the effects of quorum requirements in referendums, suggestions are given for changing the quorum requirement in the law. These changes could solve the problems of the Consultative referendum law that the Dutch government raised. This article is published in het Nederlands Juristenblad (NJB).

The research of this dissertation was awarded the Dare to Cross Over prize 2017 by Utrecht University, which is a prize for interdisciplinary graduate research.

The thesis deals with the question of the supplementary values of referendums in representative democracies. Opinions are divided on this topic. A variety of arguments is used by proponents and opponents of referendums about citizen’s knowledge and capacity, citizen’s power and representative’s power and prestige. Proponents and opponents do not seem to agree on the premises of these arguments and, furthermore, on how to weigh these arguments against each other. Hence, it is a currently unsettled question whether, and if yes, how referendums could and should be combined with representative democracy.

This question is addressed in the thesis from a multidisciplinary angle: a contribution to the legal-political debate about referendums in representative democracies is made by combining the legal-political arguments with mathematical insights. The mathematical insights show that the arguments on both sides of the debate about the supplementary value of referendums should be nuanced and, hence, the insights can help to find common ground between proponents and opponents of referendums in representative democracies.

The thesis consists of two articles. The first article, titled ‘Are referendums and parliamentary elections reconcilable’, examines the role of referendums in representative democracies in general. It looks at parliamentary systems in representative democracies and asks whether referendums and parliamentary elections can and should be combined. By drawing on three mathematical paradoxes, the article shows that referendums and parliamentary elections represent two different voting systems that could be contradictory. Because of the systemic contrarieties between referendums and parliamentary elections, the article argues that referendums should only be introduced in representative democratic systems with parliamentary elections under certain strict legal conditions. This article is published in the Journal for Moral Philosophy and Politics.

The second article, titled ‘De Wet raadgevend referendum afschaffen of verbeteren?’ (Abolishing or improving the Dutch Consultative referendum law?), focusses on referendums in the Netherlands. In particular, it considers the decision of the Dutch government to abolish the Dutch Consultative referendum law. In the article, it is argued that the government’s arguments for abolishing the law do not justify this decision and instead ask for improving the law. Using mathematical insights about the effects of quorum requirements in referendums, suggestions are given for changing the quorum requirement in the law. These changes could solve the problems of the Consultative referendum law that the Dutch government raised. This article is published in het Nederlands Juristenblad (NJB).

The research of this dissertation was awarded the Dare to Cross Over prize 2017 by Utrecht University, which is a prize for interdisciplinary graduate research.

S.A. Bloks

Condorcet Winning Sets Masters Thesis

London School of Economics and Political Science, 2018, (Supervised by B. von Stengel).

Abstract | BibTeX | Tags: Majority voting | Links:

@mastersthesis{S.A.Bloks2018,

title = {Condorcet Winning Sets},

author = {S.A. Bloks},

url = {http://suzannebloks.com/wp-content/uploads/2020/12/Bloks_2018_Condorcet-Winning-Sets_Dissertation-LSE.pdf},

year = {2018},

date = {2018-04-24},

school = {London School of Economics and Political Science},

abstract = {In voting, a voting rule has to determine the winning candidate on the basis of the voters’ preferences over the set of candidates, which together constitute a preference profile. According to Condorcet’s majority rule, the winning candidate is the candidate that is pairwise preferred to all other candidates by a majority of voters. When candidate a is preferred to candidate b by a majority of voters, we say that a “beats” b and we call this the majority relation over a and b. However, Condorcet’s majority rule is not always able to choose a single winner. Therefore, we study Condorcet winning sets. A set Y is a Condorcet winning set if for every candidate z not in Y a majority of voters prefer some candidate in Y to z. Then, the Condorcet dimension of a preference profile is the minimum size of a Condorcet winning set on the preference profile.

In this research, we approach an open question posed by Elkind et al. (2015): do there exist preference profiles whose Condorcet dimension exceeds 3? Preference profiles have been found with minimum Condorcet winning sets of size 2 and 3, but it remains unanswered whether preference profiles can be constructed with minimum Condorcet winning sets of size 4 or higher.

We will approach this question by using the (known) relation between Condorcet winning sets on preference profiles and dominating sets in tournaments. If there are no ties (which is always the case with an odd number of voters), the majority relation defines a tournament on the set of candidates. This is a relation R so that either xRy or yRx for any two candidates x;y. Then, a dominating set on a tournament is a set of candidates S such that for every candidate z not in S there is a candidate y in S for which y beats z. The dominating dimension of a tournament is the minimum size of a dominating set on the tournament.

The concept of dominating sets in tournaments is related to but stronger than the concept of Condorcet winning sets on preference profiles. The difference lies in the underlying notion of collective dominance. In a Condorcet winning set Y, for every candidate z not in Y there is some candidate in Y that is preferred to z by a majority of voters, but that candidate may depend on the voter. By contrast, in a dominating set S, for every candidate z not in S there is a candidate in S that beats z, i.e. there is one candidate y in S that is preferred to z by the majority of voters. This means that every dominating set on a tournament is a Condorcet winning set on the corresponding preference profile, but not vice versa. We will use this relation to make advancements towards constructing preference profiles with Condorcet dimension 4 or higher.

Firstly, we will use results by Erdöos (1963) and Szekeres and Szekeres (1965) concerning dominating sets in tournaments to obtain bounds on the minimum number of candidates that is needed for preference profiles with a given Condorcet dimension.

Secondly, we will analyse constructions of preference profiles with Condorcet dimension 2 and 3, in particular, Condorcet cycles and Kronecker preference profiles. The latter have been constructed by Elkind et al. (2015). We will show that the minimum-sized Condorcet winning sets on these preference profiles are also dominating sets in the corresponding tournament. Also, these preference profiles have a cyclic structure that is necessary for obtaining a higher Condorcet dimension (as we will prove with a theorem by Dasgupta and Maskin (2008)), but we will show that these cyclic structures cannot be extended in order to obtain a preference profile with Condorcet dimension 4 or higher.

Lastly, we will construct so-called quadratic residue tournaments with dominating dimension 3 and 4, which have first been constructed by Szekeres and Szekeres (1965), and provide a novel way of constructing preference profiles that realise quadratic residue tournaments of dominating dimension 3 and 4. The preference profile that represents the quadratic residue tournament with dominating dimension 3 is a new preference profile with Condorcet dimension 3. As the preference profile corresponding to the tournament with dominating dimension 4 does not have Condorcet dimension 4, we will suggest further research questions that our new results raise.},

type = {MSc Dissertation},

note = {Supervised by B. von Stengel},

keywords = {Majority voting},

pubstate = {published},

tppubtype = {mastersthesis}

}

In voting, a voting rule has to determine the winning candidate on the basis of the voters’ preferences over the set of candidates, which together constitute a preference profile. According to Condorcet’s majority rule, the winning candidate is the candidate that is pairwise preferred to all other candidates by a majority of voters. When candidate a is preferred to candidate b by a majority of voters, we say that a “beats” b and we call this the majority relation over a and b. However, Condorcet’s majority rule is not always able to choose a single winner. Therefore, we study Condorcet winning sets. A set Y is a Condorcet winning set if for every candidate z not in Y a majority of voters prefer some candidate in Y to z. Then, the Condorcet dimension of a preference profile is the minimum size of a Condorcet winning set on the preference profile.

In this research, we approach an open question posed by Elkind et al. (2015): do there exist preference profiles whose Condorcet dimension exceeds 3? Preference profiles have been found with minimum Condorcet winning sets of size 2 and 3, but it remains unanswered whether preference profiles can be constructed with minimum Condorcet winning sets of size 4 or higher.

We will approach this question by using the (known) relation between Condorcet winning sets on preference profiles and dominating sets in tournaments. If there are no ties (which is always the case with an odd number of voters), the majority relation defines a tournament on the set of candidates. This is a relation R so that either xRy or yRx for any two candidates x;y. Then, a dominating set on a tournament is a set of candidates S such that for every candidate z not in S there is a candidate y in S for which y beats z. The dominating dimension of a tournament is the minimum size of a dominating set on the tournament.

The concept of dominating sets in tournaments is related to but stronger than the concept of Condorcet winning sets on preference profiles. The difference lies in the underlying notion of collective dominance. In a Condorcet winning set Y, for every candidate z not in Y there is some candidate in Y that is preferred to z by a majority of voters, but that candidate may depend on the voter. By contrast, in a dominating set S, for every candidate z not in S there is a candidate in S that beats z, i.e. there is one candidate y in S that is preferred to z by the majority of voters. This means that every dominating set on a tournament is a Condorcet winning set on the corresponding preference profile, but not vice versa. We will use this relation to make advancements towards constructing preference profiles with Condorcet dimension 4 or higher.

Firstly, we will use results by Erdöos (1963) and Szekeres and Szekeres (1965) concerning dominating sets in tournaments to obtain bounds on the minimum number of candidates that is needed for preference profiles with a given Condorcet dimension.

Secondly, we will analyse constructions of preference profiles with Condorcet dimension 2 and 3, in particular, Condorcet cycles and Kronecker preference profiles. The latter have been constructed by Elkind et al. (2015). We will show that the minimum-sized Condorcet winning sets on these preference profiles are also dominating sets in the corresponding tournament. Also, these preference profiles have a cyclic structure that is necessary for obtaining a higher Condorcet dimension (as we will prove with a theorem by Dasgupta and Maskin (2008)), but we will show that these cyclic structures cannot be extended in order to obtain a preference profile with Condorcet dimension 4 or higher.

Lastly, we will construct so-called quadratic residue tournaments with dominating dimension 3 and 4, which have first been constructed by Szekeres and Szekeres (1965), and provide a novel way of constructing preference profiles that realise quadratic residue tournaments of dominating dimension 3 and 4. The preference profile that represents the quadratic residue tournament with dominating dimension 3 is a new preference profile with Condorcet dimension 3. As the preference profile corresponding to the tournament with dominating dimension 4 does not have Condorcet dimension 4, we will suggest further research questions that our new results raise.

In this research, we approach an open question posed by Elkind et al. (2015): do there exist preference profiles whose Condorcet dimension exceeds 3? Preference profiles have been found with minimum Condorcet winning sets of size 2 and 3, but it remains unanswered whether preference profiles can be constructed with minimum Condorcet winning sets of size 4 or higher.

We will approach this question by using the (known) relation between Condorcet winning sets on preference profiles and dominating sets in tournaments. If there are no ties (which is always the case with an odd number of voters), the majority relation defines a tournament on the set of candidates. This is a relation R so that either xRy or yRx for any two candidates x;y. Then, a dominating set on a tournament is a set of candidates S such that for every candidate z not in S there is a candidate y in S for which y beats z. The dominating dimension of a tournament is the minimum size of a dominating set on the tournament.

The concept of dominating sets in tournaments is related to but stronger than the concept of Condorcet winning sets on preference profiles. The difference lies in the underlying notion of collective dominance. In a Condorcet winning set Y, for every candidate z not in Y there is some candidate in Y that is preferred to z by a majority of voters, but that candidate may depend on the voter. By contrast, in a dominating set S, for every candidate z not in S there is a candidate in S that beats z, i.e. there is one candidate y in S that is preferred to z by the majority of voters. This means that every dominating set on a tournament is a Condorcet winning set on the corresponding preference profile, but not vice versa. We will use this relation to make advancements towards constructing preference profiles with Condorcet dimension 4 or higher.

Firstly, we will use results by Erdöos (1963) and Szekeres and Szekeres (1965) concerning dominating sets in tournaments to obtain bounds on the minimum number of candidates that is needed for preference profiles with a given Condorcet dimension.

Secondly, we will analyse constructions of preference profiles with Condorcet dimension 2 and 3, in particular, Condorcet cycles and Kronecker preference profiles. The latter have been constructed by Elkind et al. (2015). We will show that the minimum-sized Condorcet winning sets on these preference profiles are also dominating sets in the corresponding tournament. Also, these preference profiles have a cyclic structure that is necessary for obtaining a higher Condorcet dimension (as we will prove with a theorem by Dasgupta and Maskin (2008)), but we will show that these cyclic structures cannot be extended in order to obtain a preference profile with Condorcet dimension 4 or higher.

Lastly, we will construct so-called quadratic residue tournaments with dominating dimension 3 and 4, which have first been constructed by Szekeres and Szekeres (1965), and provide a novel way of constructing preference profiles that realise quadratic residue tournaments of dominating dimension 3 and 4. The preference profile that represents the quadratic residue tournament with dominating dimension 3 is a new preference profile with Condorcet dimension 3. As the preference profile corresponding to the tournament with dominating dimension 4 does not have Condorcet dimension 4, we will suggest further research questions that our new results raise.