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Monotonicity violations under plurality with a runoff: the case of French presidential elections


A voting rule is monotonic if a winning candidate never becomes a loser by being raised in voters’ rankings of candidates, ceteris paribus. Plurality with a runoff is known to fail monotonicity. To see how widespread this failure is, we focus on French presidential elections since 1965. We identify mathematical conditions that allow a logically conceivable scenario of vote shifts between candidates that may lead to a monotonicity violation. We show that eight among the ten elections held since 1965 (those in 1965 and 1974 being the exceptions) exhibit this theoretical vulnerability. To be sure, the conceived scenario of vote shifts that enables a monotonicity violation may not be plausible under the political context of the considered election. Thus, we analyze the political landscape of these eight elections and argue that for two of them (2002 and 2007 elections), the monotonicity violation scenario was plausible within the conjuncture of the time.

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  1. Black (1958) and Brams and Fishburn (2002) give a comprehensive account of this condition, which should be distinguished from other stronger monotonicity conditions that are more oriented to the implementability of collective choice rules. For those different monotonicities of implementation theory, once can see Maskin and Sjöström (2002).

  2. Felsenthal and Nurmi (2017) present a detailed account of monotonicity failures.

  3. See Doron and Kronick (1977) for arguments against using non-monotonic voting rules.

  4. Golder (2005) discusses the use of PwR in different elections around the world.

  5. The qualification “restricted” implicitly assumes that voters’ preferences do not change between the two rounds and that strategic voting is not a concern. See Conitzer and Sandholm (2005) for a formalization of PwR as a protocol.

  6. Under this adaptation that overlooks preferences, the question on whether voters’ preferences change between the two rounds vanish. In a similar vein, strategic voting is not a concern. See Footnote 5.

  7. Although we postpone the discussion on the runoff outcome to the next section where we use real election data, we note right away that the condition “\({C}_{1}\) not exceeding the threshold of 50 votes” ensures that \({C}_{1}\) does not become the PwR winner without needing a runoff.

  8. on that particularly prevails when the elec.

  9. Although in French politics candidates typically stand out more than the party they are associated with, to be more informative to a larger audience, we gave the party names in the tables and used their English translations.

  10. The data is from:.

  11. The data is from:.

  12. The data is from.

  13. The data is from.

  14. The data is from.

  15. Although the inequality in Proposition 1 requires \(x\) < 30.06, the actual vote of Chirac which is 19.94 is the binding upper bound on \(x\). This issue applies to the 1995, 2002 and 2017 elections as well.

  16. The data is from.

  17. The data is from.

  18. The data is from.

  19. The data is from.

  20. The data is from.

  21. Some changes may be more unlikely than others but for the purpose of our analysis a single category that reflects unlikeliness suffices.

  22. Among the reasons for such a shift was the steady move of Christian west towards left and the popularity of socialism among youngsters who vote for the first time in 1981.






  28. At the end of our analysis, we are able to qualify the 2002 election as one which is practically open to an upward 1.1 monotonicity violation.Thus, we do not analyze the plausibility of other 1.k violation scenarios for k > 1.





  33. We thank an anonymous referee for suggesting this perspective.

  34. A well-known instance of manipulative vote transfers under non-monotonic voting rules is the referendum in Italy held on 12 June 2005 under the (non-monotonic) majority with quorum rule, where the Catholic Church, being against the withdrawal of a law dealing with medically assisted procreation, asked its supporters to abstain rather than casting a negative vote (Houy (2009)).


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We thank Denis Bouyssou, Olivier Cailloux, Uğur Özdemir and Jérôme Lang for their precious comments on earlier version of this paper; Jean Lainé, Jean-François Laslier, Vincent Merlin and Ali Özkes for very useful discussions; an anonymous referee and an anonymous associate editor for their thoughtful reviews that improved our work. This paper is a part of ‘Polarization viewed from a social choice perspective’ (POSOP) research project that is carried on under the RDI program funded by İstanbul Bilgi University. We would like to thank POSOP for the support.

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Correspondence to M. Remzi Sanver.

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Keskin, U., Sanver, M.R. & Tosunlu, H.B. Monotonicity violations under plurality with a runoff: the case of French presidential elections. Soc Choice Welf 59, 305–333 (2022).

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