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.

评分规则是一种流行的汇总排名方法；它们经常用于许多场合，包括社会选择、信息检索和体育。评分规则由权重向量（评分向量）参数化，每个位置一个，并将在汇总与每个投票者的位置相对应的权重时获得的得分最大化的候选人宣布为获胜者。众所周知，正确设置权重是一项至关重要的任务，因为不同的候选人可以通过不同的评分向量获胜。在本文中，我们提供了几种方法来考虑所有可能的权重来确定获胜者。我们首先提出了VolumetricTop，这是一种基于超多形体对备选方案进行排名的规则，该超多面体表示给备选方案评分最高的权重集，并从社会选择理论的角度对该规则进行了详细分析。为了克服它的一些局限性，我们提出了另外两种方法：体积径流，一种迭代消除与最小区域相关的替代方案直到找到获胜者的规则，以及体积锦标赛，其中替代方案在成对比较中进行匹配; 我们提供了有关这些规则的一些见解。最后，我们提供了一些使用所提出的方法进行排名聚合的测试用例。