# On the Likelihood of the Borda Effect: The Overall Probabilities for General Weighted Scoring Rules and Scoring Runoff Rules

- 25 Downloads

## Abstract

The *Borda Effect*, first introduced by Colman and Poutney (Behav Sci 23:15–20, 1978), occurs in a preference aggregation process using the Plurality rule if given the (unique) winner there is at least one loser that is preferred to the winner by a majority of the electorate. Colman and Poutney (1978) distinguished two forms of the Borda Effect: the *Weak Borda Effect*, describing a situation under which the unique winner of the Plurality rule is majority dominated by only one loser; and the *Strong Borda Effect*, under which the Plurality winner is majority dominated by each of the losers. The *Strong Borda Effect* is well documented in the literature as the Strong Borda Paradox. Colman and Poutney (1978) showed that the probability of the *Weak Borda Effect* is not negligible; but they only focused on the Plurality rule. In this note, we extend the work of Colman and Poutney (1978) by providing, for three-candidate elections, representations of the limiting probabilities of the *(Weak) Borda Effect* for the whole family of scoring rules and scoring runoff rules. Our analysis leads us to highlight that there is a relation between the *(Weak) Borda Effect* and Condorcet efficiency. We perform our analysis under the assumptions of Impartial Culture and Impartial Anonymous Culture, which are two well-known assumptions often used for such a study.

## Keywords

Borda effect Rankings Scoring rules Probability Impartial culture Impartial and anonymous culture## Notes

### Acknowledgements

The author would like to thank Bill Gehrlein for his help and for his suggestions for literature on the representations of quadrivariate orthant probabilities. This work benefited from the support of “Investissements d’Avenir” of the French National Agency for Research (CEBA, ref. ANR-10-LABX-25-01).

## References

- Bezembinder T (1996) The plurality majority converse under single peakedness. Soc Choice Welf 13:365–380CrossRefGoogle Scholar
- Cervone D, Gehrlein WV, Zwicker W (2005) Which scoring rule maximizes Condorcet efficiency under IAC? Theory Decis 58:145–185CrossRefGoogle Scholar
- Colman AM (1980) The likelihood of the Borda effect in small decision-making committees. Br J Math Stat Psychol 33:50–56CrossRefGoogle Scholar
- Colman AM (1984) The weak Borda effect and plurality-majority disagreement. Br J Math Stat Psychol 37:288–292CrossRefGoogle Scholar
- Colman AM (1986) Rejoinder to Gillet. Br J Math Stat Psychol 39:87–89CrossRefGoogle Scholar
- Colman AM, Poutney I (1978) Borda’s voting paradox: theoretical likelihood and electoral occurrences. Behav Sci 23:15–20CrossRefGoogle Scholar
- David FN, Mallows CL (1961) The variance of Spearman’s rho in normal samples. Biometrika 48:19–28CrossRefGoogle Scholar
- de Borda JC (1784) A paper on elections by ballot. In: Sommerlad F, McLean I (1989, eds) The political theory of Condorcet, University of Oxford Working Paper, Oxford, pp 122–129Google Scholar
- de Borda JC (1781) Mémoire sur les élections au scrutin. Histoire de l’Académie Royale des Sciences, ParisGoogle Scholar
- de Condorcet M (1785) Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. National Government Publication, ParisGoogle Scholar
- Diss M, Gehrlein WV (2012) Borda’s paradox with weighted scoring rules. Soc Choice Welf 38:121–136CrossRefGoogle Scholar
- Diss M, Gehrlein WV (2015) The true impact of voting rule selection on Condorcet efficiency. Econ Bull 35(4):2418–2426Google Scholar
- Diss M, Tlidi A (2018) Another perspective on Borda’s paradox. Theory Decis 84(1):99–121CrossRefGoogle Scholar
- Diss M, Merlin V, Valognes F (2010) On the Condorcet efficiency of approval voting and extended scoring rules for three alternatives. In: Laslier JF, Sanver M (eds) Handbook on approval voting. Springer, Berlin, pp 255–283CrossRefGoogle Scholar
- Diss M, Louichi A, Merlin V, Smaoui H (2012) An example of probability computations under the IAC assumption: the stability of scoring rules. Math Soc Sci 64:57–66CrossRefGoogle Scholar
- Diss M, Kamwa E, Tlidi A (2018) A note on the likelihood of the absolute majority paradoxes. Econ Bull 38(4):1727–1734Google Scholar
- Favardin P, Lepelley D (2006) Some further results on the manipulability of social choice rules. Soc Choice Welf 26:485–509CrossRefGoogle Scholar
- Fishburn PC, Gehrlein WV (1976) Borda’s rule, positional voting and Condorcet’s simple majority principle. Public Choice 28:79–88CrossRefGoogle Scholar
- Gehrlein WV (1979) A representation for quadrivariate normal positive orthant probabilities. Commun Stat 8:349–358CrossRefGoogle Scholar
- Gehrlein WV (2002) Obtaining representations for probabilities of voting outcomes with effectively unlimited precision integer arithmetic. Soc Choice Welf 19:503–512CrossRefGoogle Scholar
- Gehrlein WV (2004) Consistency in measures of social homogeneity: a connection with proximity to single peaked preferences. Qual Quant 38:147–171CrossRefGoogle Scholar
- Gehrlein WV (2017) Computing multivariate normal positive orthant probabilities with 4 and 5 variables. technical report. https://www.researchgate.net/publication/320467212_Computing_Multivariate_Normal_Positive_Orthant_Probabilities_with_4_and_5_Variables. Accessed 5 Feb 2018
- Gehrlein WV, Fishburn PC (1976) The probability of the paradox of voting: a computable solution. J Econ Theory 13:14–25CrossRefGoogle Scholar
- Gehrlein WV, Fishburn PC (1978a) Coincidence probabilities for simple majority and positional voting rules. Soc Sci Res 7:272–283CrossRefGoogle Scholar
- Gehrlein WV, Fishburn PC (1978b) Probabilities of election outcomes for large electorates. J Econ Theory 19(1):38–49CrossRefGoogle Scholar
- Gehrlein WV, Lepelley D (1998) The Condorcet efficiency of approval voting and the probability of electing the Condorcet loser. J Math Econ 29:271–283CrossRefGoogle Scholar
- Gehrlein WV, Lepelley D (2010b) On the probability of observing Borda’s paradox. Soc Choice Welf 35:1–23CrossRefGoogle Scholar
- Gehrlein WV, Lepelley D (2011) Voting paradoxes and group coherence. Springer, BerlinCrossRefGoogle Scholar
- Gehrlein WV, Lepelley D (2017) Elections, voting rules and paradoxical outcomes. Springer, BerlinCrossRefGoogle Scholar
- Gehrlein WV, Lepelley D, Moyouwou I (2015) Voters preference diversity, concepts of agreement and Condorcet’s paradox. Qual Quant 49(6):2345–2368CrossRefGoogle Scholar
- Gillett R (1984) The weak Borda effect is an unsatisfactory index of plurality/majority disagreement. Br J Math Stat Psychol 37:128–130CrossRefGoogle Scholar
- Gillett R (1986) The weak Borda effect is of little relevance to social choice theory. Br J Math Stat Psychol 39:79–86CrossRefGoogle Scholar
- Kamwa E, Valognes F (2017) Scoring rules and preference restrictions: the strong Borda paradox revisited. Revue d’Economie Politique 127(3):375–395CrossRefGoogle Scholar
- Kim KH, Roush FW (1996) Statistical manipulability of social choice functions. Group Decis Negotia 5:262–282CrossRefGoogle Scholar
- Lepelley D (1993) On the probability of electing the Condorcet loser. Math Soc Sci 25:105–116CrossRefGoogle Scholar
- Lepelley D (1996) Constant scoring rules, Condorcet criteria and single-peaked preferences. Econ Theory 7(3):491–500CrossRefGoogle Scholar
- Lepelley D, Valognes F (2003) Voting rules, manipulability and social homogeneity. Public Choice 116:165–184CrossRefGoogle Scholar
- Lepelley D, Louichi A, Valognes F (2000a) Computer simulations of voting systems. In: Ballot G, Weisbuch G (eds) Applications of simulations to social sciences. Hermes, Oxford, pp 181–194Google Scholar
- Lepelley D, Pierron P, Valognes F (2000b) Scoring rules, Condorcet efficiency and social homogeneity. Theory Decis 49:175–196CrossRefGoogle Scholar
- Lepelley D, Louichi A, Smaoui H (2008) On Ehrhart polynomials and probability calculations in voting theory. Soc Choice Welf 30:363–383CrossRefGoogle Scholar
- Moyouwou I, Tchantcho H (2015) Asymptotic vulnerability of positional voting rules to coalitional manipulation. Math Soc Sci 89:70–82CrossRefGoogle Scholar
- Nanson EJ (1883) Methods of election. Trans Proc R Soc Vic 18:197–240Google Scholar
- Nurmi H, Suojanen M (2004) Assessing contestability of electoral outcomes: an illustration of Saari’s geometry of elections in the light of the 2001 British parliamentary elections. Qual Quant 38:719–733CrossRefGoogle Scholar
- Riker WH (1982) Liberalism against populism: a confrontation between the theory of democracy and the theory of social choice. Freeman Press, New YorkGoogle Scholar
- Saari DG (1994) Geometry of voting. Springer, New YorkCrossRefGoogle Scholar
- Saari D, Valognes F (1999) The geometry of Black’s single peakedness and related conditions. J Math Econ 32:429–456CrossRefGoogle Scholar
- Tataru M, Merlin V (1997) On the relationship of the Condorcet winner and positional voting rules. Math Soc Sci 34:81–90CrossRefGoogle Scholar
- Taylor AD (1997) A glimpse of impossibility: Kenneth Arrow’s impossibility theory and voting. Perspect Polit Sci 26:23–26CrossRefGoogle Scholar
- Van Newenhizen J (1992) The Borda method is most likely to respect the Condorcet principle. Econ Theory 2:69–83CrossRefGoogle Scholar
- Weber RJ (1978) Comparison of voting systems. Yale University, unpublished manuscriptGoogle Scholar
- Wilson MC, Pritchard G (2007) Probability calculations under the IAC hypothesis. Math Soc Sci 54:244–256CrossRefGoogle Scholar
- Young HP (1988) Condorcet’s theory of voting. Am Polit Sci Rev 82–4:1231–1244CrossRefGoogle Scholar