Abstract
The distinction – in fact, rivalry – between two intuitive notions about what constitutes the winning candidates or policy alternatives has been present in the social choice literature from its Golden Age, i.e. in the late 18’th century [13]. According to one of them, the winners can be distinguished by looking a the performance of candidates in one-on-one, that is, pairwise contests. According to the other, the winners are in general best situated in the evaluators’ rankings over all candidates. The best known class of rules among those conforming to the first intuitive notion are those that always elect the Condorcet winner whenever one exists. These rules are called Condorcet extensions for the obvious reason that they extend Condorcet’s well-known winner criterion beyond the domain where it can be directly applied. A candidate is a Condorcet winner whenever it defeats all other candidates in pairwise contests with a majority of votes. Condorcet extensions specify winners in all settings including those where a Condorcet winner is not to be found. Of course, in those settings where there is a Condorcet winner they all end up with electing it.
Useful remarks of the referees on an earlier version are gratefully acknowledged.
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Notes
- 1.
Several variations of the criterion can be envisioned. To wit, one could call a candidate the Condorcet winner whenever he/she wins all contestants with more than 50% of the votes, or by some other qualified majority of votes. One can also determine the winner is pairwise contests by some other criterion like, for instance, the number of goals in football or time spent in skating a fixed distance, etc.
- 2.
The descriptive accuracy of assuming complete and transitive preference relations can be and has been questioned, but in the present context we shall not dwell on the issue (see, e.g. [20]).
- 3.
Fishburn makes the valid point that it is not entirely fair to Dodgson to name the rule after him since he suggested several other rules and suggested counting preference switches only as a component of a more complex procedure [10]. Also Tideman questions the plausibility of associating Dodgson with this rule [24]. See [3]. Keeping these caveats in mind we shall, however, conform to the standard usage of the concept of Dodgson’s rule.
- 4.
Note that the NER winner C is defeated by B in a pairwise majority comparison. Hence C is not in the core.
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Nurmi, H. (2018). Reflections on Two Old Condorcet Extensions. In: Nguyen, N., Kowalczyk, R., Mercik, J., Motylska-Kuźma, A. (eds) Transactions on Computational Collective Intelligence XXXI. Lecture Notes in Computer Science(), vol 11290. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-58464-4_2
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