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

The Borda Effect, first introduced by Colman and Poutney (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; they only focused on the Plurality rule. In this note, we extend the work of Colman and Poutney (1978) by providing in three-candidate elections, the representations for the limiting probabilities of the (Weak) Borda Effect for the whole family of the scoring rules and scoring runoff rules. We highlight that there is a relation between the (Weak) Borda Effect and the Condorcet efficiency. We perform our analysis under the Impartial Culture and the Impartial Anonymous Culture which are two well-known assumptions often used for such a study.