  We consider a setting where $n$ buyers, with combinatorial preferences over
$m$ items, and a seller, running a priority-based allocation mechanism,
repeatedly interact. Our goal, from observing limited information about the
results of these interactions, is to reconstruct both the preferences of the
buyers and the mechanism of the seller. More specifically, we consider an
online setting where at each stage, a subset of the buyers arrive and are
allocated items, according to some unknown priority that the seller has among
the buyers. Our learning algorithm observes only which buyers arrive and the
allocation produced (or some function of the allocation, such as just which
buyers received positive utility and which did not), and its goal is to predict
the outcome for future subsets of buyers. For this task, the learning algorithm
needs to reconstruct both the priority among the buyers and the preferences of
each buyer. We derive mistake bound algorithms for additive, unit-demand and
single minded buyers. We also consider the case where buyers' utilities for a
fixed bundle can change between stages due to different (observed) prices. Our
algorithms are efficient both in computation time and in the maximum number of
mistakes (both polynomial in the number of buyers and items).
