The superdominance relation, the positional winner, and more missing links between Borda and Condorcet
| Date | 01 January 2019 |
| Author | Raúl Pérez-Fernández,Bernard De Baets |
| DOI | 10.1177/0951629818809417 |
| Published date | 01 January 2019 |
| Subject Matter | Articles |
Article
Journal of Theoretical Politics
2019, Vol.31(1) 46–65
ÓThe Author(s) 2018
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DOI: 10.1177/0951629818809417
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The superdominance
relation, the positional
winner, and more missing
links between Borda and
Condorcet
Rau
´lPe
´rez-Ferna
´ndez
Department of Data Analysis and MathematicalModelling, Ghent University, Belgium
Bernard De Baets
Department of Data Analysis and MathematicalModelling, Ghent University, Belgium
Abstract
The field of social choice dates back to the eighteenth century, when Borda and Condorcet
started a never-ending discussion about the use of either positional or pairwise information.
Three centuries later, after countless axiomatic characterizations of voting rules, impossibilitythe-
orems and many other study subjects, researchers still debate whether positional information is
really sensitive to manipulation or pairwise informationdisregards the transitivity of voters’ prefer-
ences. In a previous paper, we introduced the notions ofsupercovering relation and pairwise win-
ner,which resulted in a meeting point for both points of view of the theory of social choice. In this
paper, we continue this direction and propose the notions of superdominance relation and posi-
tional winner that will prove to be the alter egos of the supercovering relation and the pairwise
winner when positional information (rather than pairwise information) is considered. Moreover,
we analyse a new interesting choiceset: the unsuperdominated set.
Keywords
Positional winner; scorix; superdominance relation; votrix;unsuperdominated set
Corresponding author:
Rau
´lPe
´rez-Ferna
´ndez, KERMIT, Department of Data Analysis and MathematicalModelling, Ghent University,
Coupure links 653, 9000 Gent, Belgium.
Email: raul.perezfernandez@ugent.be
1. Introduction
Back in the eighteenth century, Borda (1781) and Condorcet (1785) proposed two
points of view of the theory of social choice based on the use of positional informa-
tion and pairwise information, respectively. Although the two approaches may
comply with each other in most cases, one could easily find examples in which they
differ significantly. For instance, consider the example in which 101 voters are
asked to rank four candidates – say a,b,cand d– resulting in 51 voters expressing
the ranking abcdand 50 voters expressing the ranking bcda.
According to the philosophy advocated by Borda, candidate bshould be consid-
ered the best candidate since almost half of the number of voters consider bto be
the best candidate, while the other voters consider bto be the second best candi-
date. Candidate ais here ‘penalized’ for being considered the worst candidate by
almost half of the number of voters. However, candidate ais considered better than
any other candidate by more than half of the number of voters, and, thus, accord-
ing to the philosophy advocated by Condorcet, candidate ashould be considered
the best candidate.
In a previous paper (Pe
´rez-Ferna
´ndez and De Baets, 2018), we advocated that
this disagreement arises because Condorcet’s proposal overlooks the transitivity of
the rankings of the voters. Thus, we introduced a new way of comparing candi-
dates based on pairwise information – the supercovering relation – and proved this
relation to be a meeting point for the works of Borda and Condorcet. A new type
of winner emerged, the pairwise winner, which is a candidate that supercovers all
other candidates and assures both the Borda winner and the Condorcet winner to
exist, and more importantly, to agree. In this paper, we introduce another way of
comparing candidates based on positional information – the superdominance rela-
tion – that will also be proven to be a meeting point for the works of Borda and
Condorcet. Similarly, another type of winner will emerge, the positional winner (a
candidate that superdominates all other candidates), assuring again both the Borda
winner and the Condorcet winner to exist and to agree.
Finally, we will propose a new choice set – the unsuperdominated set – that,
like the unsupercovered set (Pe
´rez-Ferna
´ndez and De Baets, 2018) and unlike
any choice set contained in the Smith set (Good, 1971; Smith, 1973)
1
,doesnot
need to reduce to the Condorcet winner in the case where it exists. Although this
might seem controversial at first (and might go against the major part of the
most recent research in the field), all these sets contained in the Smith set might
be at least questionable for some examples (such as the one in the first para-
graph of this introduction), even though they have actually been criticized for
being rather large in the case where the Condorcet winner does not exist (Dutta,
1988).
The rest of the paper is organized as follows. We introduce the superdominance
relation in Section 2. This superdominance relation is used to define the positional
winner in Section 3 and the unsuperdominated set in Section 4. We round up with
some conclusions and a brief discussion on open problems in Section 5.
Pe
´rez-Ferna
´ndez and De Baets 47
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