Consistency, unanimity, and the Borda rule in social ranking

Abstract: The social ranking is a recently proposed framework for evaluating the power of individuals according to the performance ranking of their coalitions. Although its origin can be traced to the classical power indices in simple games, social ranking approaches carry out this evaluation within the ordinal framework of social choice theory. This article introduces the Borda rule into social ranking. Specifically, we focus on two essential properties of the Borda rule--consistency and closeness to unanimity--and investigate the social ranking solutions (SRSs) satisfying these properties. Among several possible definitions of the Borda rule as an SRS, we characterize one of such solutions by (a weak version of) consistency, closeness to unanimity (under the linear and symmetric domain), neutrality (i.e., names of the individuals do not matter), and independence of perfunctory participation (i.e., adding a perfunctory coalition into the worst class of the coalitional ranking does not affect the social ranking). We therefore propose a new Borda-type SRS for evaluating the competence of individuals in coalitional contexts.