  This paper is devoted to the study of a discrepancy-type characteristic --
the fixed volume discrepancy -- of the Fibonacci point set in the unit square.
It was observed recently that this new characteristic allows us to obtain
optimal rate of dispersion from numerical integration results. This observation
motivates us to thoroughly study this new version of discrepancy, which seems
to be interesting by itself. The new ingredient of this paper is the use of the
average over the shifts of hat functions instead of taking the supremum over
the shifts. We show that this change in the setting results in an improvement
of the upper bound for the smooth fixed volume discrepancy, similarly to the
well-known results for the usual $L_p$-discrepancy. Interestingly, this shows
that ``bad boxes'' for the usual discrepancy cannot be ``too small''. The known
results on smooth discrepancy show that the obtained bounds cannot be improved
in a certain sense.
