  Perpetual voting was recently introduced as a framework for long-term
collective decision making. In this framework, we consider a sequence of
subsequent approval-based elections and try to achieve a fair overall outcome.
To achieve fairness over time, perpetual voting rules take the history of
previous decisions into account and identify voters that were dissatisfied with
previous decisions. In this paper, we look at perpetual voting rules from an
axiomatic perspective and study two main questions. First, we ask how simple
such rules can be while still meeting basic desiderata. For two simple but
natural classes, we fully characterize the axiomatic possibilities. Second, we
ask how proportionality can be formalized in perpetual voting. We study
proportionality on simple profiles that are equivalent to the apportionment
setting and show that lower and upper quota axioms can be used to distinguish
(and sometimes characterize) perpetual voting rules. Furthermore, we show a
surprising connection between a perpetual rule called Perpetual Consensus and
Frege's apportionment method.

Computer Science and Game Theory