  Consider a society of voters, each of whom specify an approval set over a
linear political spectrum. We examine double-interval societies, in which each
person's approval set is represented by two disjoint closed intervals, and
study this situation where the approval sets are pairwise-intersecting: every
pair of voters has a point in the intersection of their approval sets. The
approval ratio for a society is, loosely speaking, the popularity of the most
popular position on the spectrum. We study the question: what is the minimal
guaranteed approval ratio for such a society? We provide a lower bound for the
approval ratio, and examine a family of societies that have rather low approval
ratios. These societies arise from double-n strings: arrangements of n symbols
in which each symbol appears exactly twice.
