Homogeneity and monotonicity of distance-rationalizable voting rules

Distance rationalizability is a framework for classifying voting rules by interpreting them in terms of distances and consensus classes. It can also be used to design new voting rules with desired properties. A particularly natural and versatile class of distances that can be used for this purpose is that of votewise distances [12], which "lift" distances over individual votes to distances over entire elections using a suitable norm. In this paper, we continue the investigation of the properties of votewise distance-rationalizable rules initiated in [12]. We describe a number of general conditions on distances and consensus classes that ensure that the resulting voting rule is homogeneous or monotone. This complements the results of [12], where the authors focus on anonymity, neutrality and consistency. We also introduce a new class of voting rules, that can be viewed as "majority variants" of classic scoring rules, and have a natural interpretation in the context of distance rationalizability.