  We consider a voting scenario in which the resource to be voted upon may
consist of both indivisible and divisible goods. This setting generalizes both
the well-studied model of multiwinner voting and the recently introduced model
of cake sharing. Under approval votes, we propose two variants of the extended
justified representation (EJR) notion from multiwinner voting, a stronger one
called EJR for mixed goods (EJR-M) and a weaker one called EJR up to 1 (EJR-1).
We extend three multiwinner voting rules to our setting -- GreedyEJR, the
method of equal shares (MES), and proportional approval voting (PAV) -- and
show that while all three generalizations satisfy EJR-1, only the first one
provides EJR-M. In addition, we derive tight bounds on the proportionality
degree implied by EJR-M and EJR-1, and investigate the proportionality degree
of our proposed rules.

Computer Science and Game Theory