Borda’s Paradox with weighted scoring rules

Abstract

Representations are obtained for the probabilities that a Strict Borda Paradox and a Strong Borda Paradox are observed for large electorates with three candidates under the standard assumptions of Impartial Culture and Impartial Anonymous Culture. These representations are obtained for general weighted scoring rules (WSRs), and the probabilities are found to be maximized for voting rules like plurality rule and negative plurality rule. It is found that these paradox probabilities are not reduced for every scoring rule with the introduction of some degree of dependence among voters’ preferences with IAC. It is concluded that actual observances of a Strict Borda Paradox should be extremely rare, and that while observances of a Strong Borda Paradox should also be rare, they might occasionally be witnessed.