  We study the problem of fair sequential decision making given voter
preferences. In each round, a decision rule must choose a decision from a set
of alternatives where each voter reports which of these alternatives they
approve. Instead of going with the most popular choice in each round, we aim
for proportional representation. We formalize this aim using axioms based on
Proportional Justified Representation (PJR), which were proposed in the
literature on multi-winner voting and were recently adapted to multi-issue
decision making. The axioms require that every group of $\alpha\%$ of the
voters, if it agrees in every round (i.e., approves a common alternative), then
those voters must approve at least $\alpha\%$ of the decisions. A stronger
version of the axioms requires that every group of $\alpha\%$ of the voters
that agrees in a $\beta$ fraction of rounds must approve $\beta\cdot\alpha\%$
of the decisions. We show that three attractive voting rules satisfy axioms of
this style. One of them (Sequential Phragm\'en) makes its decisions online, and
the other two satisfy strengthened versions of the axioms but make decisions
semi-online (Method of Equal Shares) or fully offline (Proportional Approval
Voting). The first two are polynomial-time computable, and the latter is based
on an NP-hard optimization, but it admits a polynomial-time local search
algorithm that satisfies the same axiomatic properties. We present empirical
results about the performance of these rules based on synthetic data and U.S.
political elections. We also run experiments where votes are cast by preference
models trained on user responses from the moral machine dataset about ethical
dilemmas.
