  Given an election, a preferred candidate p, and a budget, the SHIFT BRIBERY
problem asks whether p can win the election after shifting p higher in some
voters' preference orders. Of course, shifting comes at a price (depending on
the voter and on the extent of the shift) and one must not exceed the given
budget. We study the (parameterized) computational complexity of S HIFT BRIBERY
for multiwinner voting rules where winning the election means to be part of
some winning committee. We focus on the well-established SNTV, Bloc, k-Borda,
and Chamberlin-Courant rules, as well as on approximate variants of the
Chamberlin-Courant rule, since the original rule is NP-hard to compute. We show
that SHIFT BRIBERY tends to be harder in the multiwinner setting than in the
single-winner one by showing settings where SHIFT BRIBERY is easy in the
single-winner cases, but is hard (and hard to approximate) in the multiwinner
ones. Moreover, we show that the non-monotonicity of those rules which are
based on approximation algorithms for the Chamberlin-Courant rule sometimes
affects the complexity of SHIFT BRIBERY.
