The Incidence of Some Voting Paradoxes Under Domain Restrictions
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The Incidence of Some Voting Paradoxes Under Domain Restrictions Hannu Nurmi1
© The Author(s) 2020
Abstract Voting paradoxes have played an important role in the theory of voting. They typically say very little about the circumstances in which they are particularly likely or unlikely to occur. They are basically existence findings. In this article we study some well known voting paradoxes under the assumption that the underlying profiles are drawn from the Condorcet domain, i.e. a set of preference profiles where a Condorcet winner exists. The motivation for this restriction is the often stated assumption that profiles with a Condorcet winner are more likely than those without it. We further restrict the profiles by assuming that the starting point of our analysis is that the Condorcet winner coincides with the choice of the voting rule under scrutiny. The reason for making this additional restriction is that—intuitively—the outcomes that coincide with the Condorcet winner make those outcomes stable and, thus, presumably less vulnerable to various voting paradoxes. It will be seen that this is, indeed, the case for some voting rules and some voting paradoxes, but not for all of them. Keywords Voting rule · Voting paradox · Condorcet domain · Profile restrictions
1 Introduction Many, if not not all, voting rules can be seen as attempts to overcome specific problems, anomalies or puzzles observed in conducting elections or analyzing their results. Sometimes the problems faced with are so grave that they acquire the status of paradoxes. Voting paradoxes have, indeed, played an important role in the development of voting theory. The best known of these are known as Borda’s and Condorcet’s paradoxes, illustrated in Tables 1 and 2, respectively.
* Hannu Nurmi [email protected] 1
Department of Contemporary History, Philosophy and Political Science, University of Turku, Turku, Finland
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H. Nurmi Table 1 Borda’s paradox Grazia (1953)
Table 2 Condorcet’s paradox
4 Voters
3 Voters
2 Voters
A
B
C
B
C
B
C
A
A
1 Voter
1 Voter
1 Voter
A
B
C
C
A
B
B
C
A
In these tables the letters A, B and C stand for candidates (or policy alternatives, for that matter) and the columns represent the preference rankings of voters from top to bottom. Borda’s paradox pertains to the plurality (or first-past-the-post) voting where the candidate receiving more votes than any of his/her competitors is the winner. Assuming that Table 1 depicts the preferences of the nine voters, A can be expected to win under plurality voting, but would be defeated by both B and C in pairwise majority voting. In fact, the plurality winner A is an absolute loser in the sense of being the last-ranked candidate of an absolute majority voters. Thus, the paradox boils down to an incompatibility of two intuitive views on what winning means, viz. one which determines the winner by looking at the distribution of first ranks among the alternatives and one based on pairwise majority contests.1 The latter intuition may lead
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