  We compute the one-loop quantum corrections to the interactions between the
two metrics of the ghost-free massive bigravity. When considering gravitons
running in the loops, we show how the structure of the interactions gets
destabilized at the quantum level, exactly in the same way as in its massive
gravity limit. A priori one might have expected a better quantum behavior,
however the broken diffeomorphism invariance out of the two initial
diffeomorphisms in bigravity has similar consequences at the quantum level as
the broken diffeomorphism in massive gravity. From lessons of the generated
quantum corrections through matter loops we propose yet other types of
effective composite metrics to which the matter fields can couple. Among these
new effective metrics there might be one or more that could provide interesting
phenomenology and important cosmological implications.
