Consistencies in Social Ranking

Ranking individuals based on their performance in different coalitions is a problem emerging in various domains (teams sports, scientific evaluation, argumentation, etc.). Often, for practical reasons, the number of comparable coalitions is limited. Therefore, the foundational principles of ranking solutions must support realistic interpretations in contexts where only certain coalitions can be compared. To address this issue, in this paper we present an axiomatic analysis of solutions for the social ranking problem centered on the notion of consistency. More precisely, we show that an appropriate notion of consistency, which specifies how to combine rankings on individuals across different rankings on coalitions, plays a key role in any axiomatic characterization, representing the true distinguishing feature of each solution. This role is further highlighted by the taxonomy of the complementary axioms used in our characterizations, which boil down to well-studied properties of invariance with respect to the label of players or coalitions, and also with respect to minor changes in a coalitional ranking. By showing the logical independence of the axioms used in each characterization, as well as a rigorous analysis of alternative notions of consistency with respect to the majority of solutions from the literature, this work attempts to provide a first systematic study of the social ranking problem over a variable domain of coalitions.