To understand and summarize approval preferences and other binary evaluation data, it is useful to order the items on an axis which explains the data. In a political election using approval voting, this could be an ideological left-right axis such that each voter approves adjacent candidates, an analogue of single-peakedness. In a perfect axis, every approval set would be an interval, which is usually not possible, and so we need to choose an axis that gets closest to this ideal. The literature has developed algorithms for optimizing several objective functions (e.g., minimize the number of added approvals needed to get a perfect axis), but provides little help with choosing among different objectives. In this paper, we take a social choice approach and compare 5 different axis selection rules axiomatically, by studying the properties they satisfy. We establish some impossibility theorems, and characterize (within the class of scoring rules) the rule that chooses the axes that maximize the number of votes that form intervals, using the axioms of ballot monotonicity and resistance to cloning. Finally, we study the behavior of the rules on data from French election surveys, on the votes of justices of the US Supreme Court, and on synthetic data.

翻译：为理解并归纳批准偏好及其他二元评价数据，将项目沿某一解释数据的轴线排序是有益的。在采用批准投票的政治选举中，这可能是意识形态上的左右轴线，使得每位选民批准相邻的候选人，即单峰性的类似情形。在完美轴线中，每个批准集应是一个区间，但这通常难以实现，因此我们需要选择最接近这一理想的轴线。现有文献已开发出优化多个目标函数的算法（例如，最小化获得完美轴线所需增加的批准数量），但关于如何选择不同目标提供的指导有限。本文采用社会选择方法，通过研究满足的特性，公理比较了5种不同的轴线选择规则。我们建立了若干不可能性定理，并（在评分规则类中）利用选票单调性和抗克隆性公理刻画了选择最大化构成区间选票数的轴线规则。最后，我们基于法国选举调查数据、美国最高法院大法官投票数据及合成数据研究了这些规则的行为表现。