Tie-breaking the highest median: alternatives to the majority judgment

Abstract

The paper deals with voting rules that require voters to rate the candidates on a finite evaluation scale and then elect a candidate whose median grade is maximum. These rules differ by the way they choose among candidates with the same median grade. Call proponents (resp. opponents) of a candidate the voters who rate this candidate strictly above (resp. strictly below) her median grade. A simple rule, called the typical judgment, orders tied candidates by the difference between their share of proponents and opponents. An appealing rule, called the usual judgment, divides this difference by the share of median votes. An alternative rule, called the central judgment, compares the relative shares of proponents and opponents. The usual judgment is continuous with respect to these shares. The majority judgment of Balinski and Laraki (Proce Natl Acad Sci 104(21):8720–8725, 2007) considers the largest of these shares and loses continuity. A result in Balinski and Laraki (Oper Res 62(3):483–511, 2014) aims to characterize the majority judgment and states that only a certain class of functions share some valuable characteristics, like monotonicity. We relativize this result, by emphasizing that it only holds true for non-discrete scales of grades. Properties remaining specific to the majority judgment in the discrete case are idiosyncratic features rather than universally sought criteria, and other median-based rules exist that are both monotonic and continuous.