We study the social aggregation problem in the preference-approval model of Brams and Sanver (The mathematics of preference, choice and order: essays in honor of Peter C. Fishburn. Springer, Berlin, 2009). Each voter reports a linear ordering of the alternatives and an acceptability threshold. A rule transforms every profile of such “opinions” into a social ordering. The approval rule ranks the alternatives according to the number of voters who find them acceptable. The broken Borda rule ranks them according to the total score they receive; the scores assigned by a voter follow the standard Borda scale except that a large break is introduced between the score of her worst acceptable alternative and the score of her best unacceptable alternative. We offer an axiomatization of this rule and other lexicographic combinations of the approval rule and a fixed social welfare function.

我们研究了 Brams 和 Sanver 的偏好批准模型中的社会聚集问题（偏好、选择和顺序的数学：纪念 Peter C. Fishburn 的论文。Springer，柏林，2009）。每个投票者报告备选方案的线性排序和可接受性阈值。规则将此类“意见”的每个配置文件转换为社会秩序。该审批规则按照谁发现他们可以接受的选民人数居的替代品。被打破的博尔达 规则根据他们得到的总分对他们进行排名；投票者分配的分数遵循标准的 Borda 量表，除了一个大的突破在她最差的可接受选项的分数和她最好的不可接受的选项的分数之间引入。我们提供了这条规则的公理化以及批准规则和固定社会福利函数的其他词典组合。