# Monotonicity Violations by Borda’s Elimination and Nanson’s Rules: A Comparison

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## Abstract

This paper compares the vulnerability of Borda Elimination Rule (BER) and of Nanson Elimination Rule (NER) to monotonicity paradoxes under both fixed and variable electorates. It is shown that while NER is totally immune and BER is vulnerable to monotonicity failure in 3-candidate elections, neither of these two rules dominates the other in *n*-candidate elections (*n *> 3) when no Condorcet Winner exists. When the number of competing alternatives is larger than three and no Condorcet Winner exists, we find profiles where NER violates monotonicity while BER does not, profiles where BER violates monotonicity while NER does not, as well as profiles where both NER and BER violate monotonicity. These findings extend to both fixed and variable electorates, as well as to situations where the initial winners under both rules are the same, as well as to situations where the initial winners under both rules are different. So, which of the two rules should be preferred in terms of monotonicity in *n*-candidate elections (*n *> 3) where no Condorcet Winner exists, depends on the kind of profiles one can expect to encounter in practice most often. Nevertheless, in view of the results of 3-candidate elections under other scoring elimination rules, we conjecture that inasmuch as BER and NER exhibit monotonicity failures, it is more likely to occur in closely contested elections.

## Keywords

Elections Borda Elimination Rule Nanson Elimination Rule Non-monotonicity Voting paradoxes## Notes

### Acknowledgements

The authors are very grateful to Nicholas R. Miller for his thoughtful and constructive suggestions on earlier versions of this article, as well as to two anonymous referees of this article for their helpful comments.

## References

- Black D (1958) The theory of committees and elections. Cambridge University Press, CambridgeGoogle Scholar
- Borda J-C de (1784 [1995]). Mémoire sur les élections au scrutin. In: Histoire de l’academie royale des sciences année 1781, pp. 651–665. Reprinted in McLean I, Urken AB (1995) Classics of social choice. University of Michigan Press, Ann Arbor, pp 83–89Google Scholar
- Brandt F, Geist C, Strobel M (2016) Analyzing the practical relevance of voting paradoxes via Ehrhart theory, computer simulations, and empirical data. In: Thangarajah J, Tuyls K, Jonker C, Marsella S (eds) Proceedings of the 15th international conference on autonomous agents and multiagent systems (AAMAS 2016), 9–13 May 2016, Singapore, pp 385–393Google Scholar
- Condorcet M (1785) Essai sur l’application de l’analyse à la probabilité des decisions rendues à la pluralité des voix. L’Imprimerie Royale, ParisGoogle Scholar
- Coombs CH (1964) A theory of data. Wiley, New YorkGoogle Scholar
- Diss M, Doghmi A (2016) Multi-winner scoring election methods: Condorcet consistency and paradoxes. Public Choice 169:97–116CrossRefGoogle Scholar
- Felsenthal DS (2012) Review of paradoxes afflicting procedures for electing a single candidate. In: Felsenthal DS, Machover M (eds) Electoral systems: paradoxes, assumptions, and procedures. Springer, Heidelberg, pp 19–92CrossRefGoogle Scholar
- Felsenthal DS, Nurmi H (2016) Two types of participation failure under nine voting methods in variable electorates. Public Choice 168:115–135CrossRefGoogle Scholar
- Felsenthal DS, Nurmi H (2017) Monotonicity failures afflicting procedures for electing a single candidate. Springer, ChamCrossRefGoogle Scholar
- Felsenthal DS, Tideman N (2013) Varieties of failure of monotonicity and participation under five voting methods. Theor Decis 75:59–77CrossRefGoogle Scholar
- Felsenthal DS, Tideman N (2014) Interacting double monotonicity failure with direction of impact under five voting methods. Math Soc Sci 67:57–66CrossRefGoogle Scholar
- Fishburn PC (1977) Condorcet social choice functions. SIAM J Appl Math 33:469–489CrossRefGoogle Scholar
- Fishburn PC, Brams SJ (1983) Paradoxes of preferential voting. Math Mag 56:207–214CrossRefGoogle Scholar
- Gehrlein WV (2001) Condorcet winners on four candidates with anonymous voters. Econ Lett 71:335–340CrossRefGoogle Scholar
- Gehrlein WV, Lepelley D (2011) Voting paradoxes and group coherence: the Condorcet efficiency of voting rules. Springer, BerlinCrossRefGoogle Scholar
- Geller C (2005) Single transferable vote with Borda elimination: proportional representation, moderation, quasi-chaos and stability. Elect Stud 24:265–280CrossRefGoogle Scholar
- Green-Armytage J, Tideman TN, Cosman R (2016) Statistical evaluation of voting rules. Soc Choice Welf 46:183–212CrossRefGoogle Scholar
- Lepelley D, Chantreuil F, Berg S (1996) The likelihood of monotonicity paradoxes in run-off elections. Math Soc Sci 3:133–146CrossRefGoogle Scholar
- Lepelley D, Moyouwou I, Smaoui H (2018) Monotonicity paradoxes in three-candidate elections using scoring elimination rules. Soc Choice Welf 50:1–33CrossRefGoogle Scholar
- McLean I, Urken AB (eds) (1995) Classics of social choice. University of Michigan Press, Ann ArborGoogle Scholar
- Miller NR (2017) Closeness matters: monotonicity failure in IRV elections with three candidates. Public Choice 173:91–108CrossRefGoogle Scholar
- Moulin H (1988) Condorcet’s principle implies the no-show paradox. J Econ Theory 45:53–64CrossRefGoogle Scholar
- Nanson EJ (1883) Methods of election. Trans Proc R Soc Victoria 19:197–240Google Scholar
- Niou EMS (1987) A note on Nanson’s rule. Public Choice 54:191–193CrossRefGoogle Scholar
- Nurmi H (1989) On Nanson’s method. In: Borg O, Apunen O, Hakovirta H, Paastela J (eds) Democracy in the modern world. Essays for Tatu Vanhanen. Acta Universitatis Tamperensis, Series A, vol 260. University of Tampere, Tampere, pp 199–210Google Scholar
- Plassmann F, Tideman TN (2014) How frequently do different voting rules encounter voting paradoxes? Soc Choice Welf 42:31–75CrossRefGoogle Scholar
- Schürmann A (2013) Exploiting polyhedral symmetries in social choice. Soc Choice Welf 40:1097–1110CrossRefGoogle Scholar
- Smaoui H, Lepelley D, Moyouwou I (2016) Borda elimination rule and monotonicity paradoxes in three-candidate elections. Econ Bull 36:1722–1728Google Scholar
- Smith JH (1973) Aggregation of preferences with variable electorate. Econometrica 41:1027–1041CrossRefGoogle Scholar