  In the system of approval voting, individuals vote for all candidates they
find acceptable. Many approval voting situations can be modeled geometrically,
and thus geometric concepts such as the piercing number have a natural
interpretation. In this paper, we explore piercing numbers in the setting where
voter preferences can be modeled by congruent arcs on a circle -- i.e., in
fixed-length circular societies. Given a number of voters and the length of the
voter preference arcs, we give bounds on the possible piercing number of the
society. Further, we explore which piercing numbers are more likely.
Specifically, under the assumption of uniformly distributed voter preference
arcs, we determine the probability distribution of the piercing number of
societies in which the length of the arcs is sufficiently small. We end with
simulations that give estimated probabilities of piercing number for societies
with larger voter preference arcs.
