  We study two notions of stability in multiwinner elections that are based on
the Condorcet criterion. The first notion was introduced by Gehrlein: A
committee is stable if each committee member is preferred to each non-member by
a (possibly weak) majority of voters. The second notion is called local
stability (introduced in this paper): A size-$k$ committee is locally stable in
an election with $n$ voters if there is no candidate $c$ and no group of more
than $\frac{n}{k+1}$ voters such that each voter in this group prefers $c$ to
each committee member. We argue that Gehrlein-stable committees are appropriate
for shortlisting tasks, and that locally stable committees are better suited
for applications that require proportional representation. The goal of this
paper is to analyze these notions in detail, explore their compatibility with
notions of proportionality, and investigate the computational complexity of
related algorithmic tasks.
