Aggregating Opinions on Sets of Alternatives: Characterization and Applications

Evaluating player contributions requires aggregating the results from a variety of sets of players. Football players are sometimes evaluated in terms of individual abilities (e.g., running speed), but other times, they are evaluated in terms of their collaborative performance with players of relevant positions (i.e., the collaboration of the forward players) or evaluated as a whole team. In this paper, we study methods for aggregating opinions on various sets of alternatives by extending the existing opinion aggregation model. In particular, we propose two rules. First, we propose the lexicographic support rule, which counts the number of opinions in favor of each set and then compares the numbers of sets, including particular players, lexicographically. Our second rule is the support rule, which ranks alternatives by the number of opinions in favor of sets including the alternative. Our contribution is twofold. The first contribution is the axiomatic characterization of these rules. The lexicographic support rule is characterized by neutrality, positive responsiveness, completeness, anonymity, cancellation of opinions in favor of sets with equal supports and dominance. The support rule is characterized by substituting the last two axioms with cancellation. The connections between these two rules and the social ranking solutions are also revealed.