Volumetric Aggregation Methods for Scoring Rules with Unknown Weights

Scoring rules are a popular method for aggregating rankings; they are frequently used in many settings, including social choice, information retrieval and sports. Scoring rules are parametrized by a vector of weights (the scoring vectors), one for each position, and declare as winner the candidate that maximizes the score obtained when summing up the weights corresponding to the position of each voter. It is well known that properly setting the weights is a crucial task, as different candidates can win with different scoring vectors. In this paper, we provide several methods to identify the winner considering all possible weights. We first propose VolumetricTop, a rule that ranks alternatives based on the hyper-polytope representing the set of weights that give the alternative the highest score, and provide a detailed analysis of the rule from the point-of-view of social choice theory. In order to overcome some of its limitations, we then propose two other methods: Volumetric-runoff, a rule that iteratively eliminates the alternative associated with the smallest region until a winner is found, and Volumetric-tournament, where alternatives are matched in pairwise comparisons; we provide several insights about these rules. Finally we provide some test cases of rank aggregation using the proposed methods.