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

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## 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).

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