  Approval-based committee (ABC) voting rules elect a fixed size subset of the
candidates, a so-called committee, based on the voters' approval ballots over
the candidates. While these rules have recently attracted significant
attention, axiomatic characterizations are largely missing so far. We address
this problem by characterizing ABC voting rules within the broad and intuitive
class of sequential valuation rules. These rules compute the winning committees
by sequentially adding candidates that increase the score of the chosen
committee the most. In more detail, we first characterize almost the full class
of sequential valuation rules based on mild standard conditions and a new axiom
called consistent committee monotonicity. This axiom postulates that the
winning committees of size k can be derived from those of size k-1 by only
adding candidates and that these new candidates are chosen consistently. By
requiring additional conditions, we derive from this result also a
characterization of the prominent class of sequential Thiele rules. Finally, we
refine our results to characterize three well-known ABC voting rules, namely
sequential approval voting, sequential proportional approval voting, and
sequential Chamberlin-Courant approval voting.
