Aggregating Incomplete Rankings

Abstract

This study considers the method to derive a ranking of alternatives by aggregating the rankings submitted by several individuals who may not evaluate all of them. The collection of subsets of alternatives that individuals (can) evaluate is referred to as an evaluability profile. For a given evaluability profile, we define an aggregating ranking function whose inputs are the rankings of individuals on the alternatives they evaluate. We investigate the properties of aggregating ranking functions, which are modifications of those introduced in previous studies. Whether an aggregating ranking function satisfying a combination of properties exists depends on the evaluability profile. Thus, we identify the necessary and sufficient conditions on evaluability profiles to ensure the existence of the functions satisfying four different combinations of properties. Furthermore, to examine how frequently possible or impossible evaluability profiles occur, we derive the proportion of each type in specific cases.