  Behavioural equivalences can be characterized via bisimulations, modal logics
and spoiler-defender games. In this paper we review these three perspectives in
a coalgebraic setting, which allows us to generalize from the particular
branching type of a transition system. We are interested in qualitative notions
(classical bisimulation) as well as quantitative notions (bisimulation
metrics).
  Our first contribution is to introduce a spoiler-defender bisimulation game
for coalgebras in the classical case. Second, we introduce such games for the
metric case and furthermore define a real-valued modal coalgebraic logic, from
which we can derive the strategy of the spoiler. For this logic we show a
quantitative version of the Hennessy-Milner theorem.
