Aggregating incomplete rankings

Abstract

This study considers a method for deriving a ranking of alternatives by aggregating the rankings submitted by multiple individuals, each of whom need not evaluate all of the alternatives. We call the collection of subsets of alternatives that individuals can evaluate an evaluability profile. For a given evaluability profile, we define an aggregating ranking function whose inputs are the rankings provided by individuals on the alternatives that they evaluate. We investigate the properties of such functions, focusing on modified versions of the properties originally introduced by Arrow and his followers. Whether there exists an aggregating ranking function that satisfies a given combination of the properties depends on the evaluability profile. Accordingly, we identify the necessary and sufficient conditions on evaluability profiles to ensure the existence of functions that satisfy four different combinations of the properties. Finally, we discuss whether these properties are satisfied in a real-world scenario.