  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.
