Abstract
We discuss the findings of the preceding chapters aiming at an overall judgement of the relative merits of the 18 procedures in the light of their (in)vulnerability to various voting paradoxes. No procedure is invulnerable to all the analyzed voting paradoxes, but there are differences in the variety of paradoxes that various procedures are vulnerable to. It turns out that for those emphasizing that a Condorcet Winner ought to be elected when s/he exists, the most plausible voting procedures are associated with the names of Copeland and Kemeny, while for those stressing the need to preserve the basic rationale of voting, viz., the participation condition, the most appealing system is the Borda count.
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Notes
- 1.
Campbell and Kelly (2002) devised a non–monotonic voting rule that does not exhibit the No–Show Paradox. However, as this method violates the anonymity and neutrality conditions and hence has not been considered seriously for actual use, we ignore it. The suggested method is bizarre in other respects as well, e.g. if all voters rank x last, it will be elected.
- 2.
Although all Condorcet–consistent procedures are also susceptible to the Reinforcement Paradox, there is no logical connection between this Paradox and the No–Show Paradox. As mentioned by Moulin (1988, pp. 54–55), when there are no more than three candidates there exist Condorcet–consistent procedures which are immune to both the No–Show and Twin Paradoxes, e.g., the Minimax procedure which elects the candidate to whom the smallest majority objects.
- 3.
However, in order to be able to state conclusively which of several voting procedures that are susceptible to the same paradox is more likely to display this paradox, one must know what are the necessary and/or sufficient conditions for this paradox to occur under the various compared procedures. Such knowledge is still lacking with respect to most voting procedures and paradoxes.
- 4.
When there are n candidates each of whom can be ranked by each of the v voters as either 1, 2, …, n, then the maximal average rank a candidate can obtain under Borda’s procedure is 1 (if all voters rank the same candidate at the top of their preference ordering) and the lowest average rank a candidate can obtain is n (if all voters rank the same candidate at the bottom of their preference ordering). It is of course possible that the Condorcet Winner and the Borda winner will have the same average rank.
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Appendices
Exercises
Problem 7.1
Are trivial or dictatorial voting procedures manipulable?
Problem 7.2
Show that the Plurality with Runoff procedure is manipulable.
Problem 7.3
Show that there are profiles where the agenda–controller can bring about any outcome when the Successive Elimination procedure is used under the assumption that the voters are sincere.
Answers to Exercises
Problem 7.1
Trivial procedures result in a fixed outcome regardless of preferences. Misrepresenting one’s preference cannot, therefore, improve the outcome. Dictatorial procedures always result in the outcome most preferred by the dictator. Hence, no non–dictatorial voter can improve upon the outcome that the dictator determines. The dictator, on the other hand, has no incentive to misrepresent his/her preference.
Problem 7.2
Consider the following profile:
No. of voters | Preference orderings |
2 | a ≻ b ≻ c |
1 | b ≻ a ≻ c |
2 | c ≻ b ≻ a |
With sincere voting a wins. With one voter of the last group voting as if his/her preference is b ≻ a ≻ c, ceteris paribus, the outcome is b, i.e. better than a for this voter.
Problem 7.3
The Condorcet Paradox profile is perhaps the simplest example:
No. of voters | Preference ordering |
1 | a ≻ b ≻ c |
1 | b ≻ c ≻ a |
1 | c ≻ a ≻ b |
If the agenda–controller wants a to win, he/she sets the agenda: (i) b versus c, (ii) the winner of (i) versus a. If he/she wants b to win, he/she sets the agenda: (i) a versus c, (ii) the winner of (i) versus b. If he/she wants c to win, his/her agenda is: (i) a versus b, (ii) the winner of (i) versus c.
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Felsenthal, D.S., Nurmi, H. (2018). Summary. In: Voting Procedures for Electing a Single Candidate. SpringerBriefs in Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-74033-1_7
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