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 Freges apportionment method.
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