  For a Landau-Zener transition in a two-level system, the probability for a
particle, initially in the first level, to survive the transition and to remain
in the first level, does not depend on whether or not the second level is
broadened [V. M. Akulin and W. P. Schleicht, Phys. Rev. A {\bf 46}, 4110
(1992)]. In other words, the seminal Landau-Zener result applies regardless of
the broadening of the second level. The same question for the multistate
Landau-Zener transition is addressed in the present paper. While for integrable
multistate models, where the transition does not involve interference of the
virtual paths, it can be argued that the independence of the broadening
persists, we focus on non-integrable models involving interference. For a
simple four-state model, which allows an analytical treatment, we demonstrate
that the decay of the excited states affects the survival probability provided
that {\em the widths of the final states are different}.
