A Note on the Likelihood of the Absolute Majority Paradoxes

International audienceFor three-candidate elections, we compute under the Impartial Anonymous Culture assumption, the conditional probabilities of the Absolute Majority Winner Paradox (AMWP) and the Absolute Majority Loser Paradox (AMLP) under the Plurality rule, the Borda rule, and the Negative Plurality rule for a given number of voters. We also provide a representation of the conditional probability of these paradoxes for the whole family of weighted scoring rules with large electorates. The AMWP occurs when a candidate who is ranked first by more than half of the voters is not selected by a given voting rule; the AMLP appears when a candidate who is ranked last by more than half of the voters is elected. As no research papers have tried to evaluate the likelihood of these paradoxes, this note is designed to fill this void. Our results allow us to claim that ignoring these two paradoxes in the literature, particularly AMWP, is not justified