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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 theresults of these interactions, is to reconstruct both the preferences of thebuyers and the mechanism of the seller. More specifically, we consider anonline setting where at each stage, a subset of the buyers arrive and areallocated items, according to some unknown priority that the seller has amongthe buyers. Our learning algorithm observes only which buyers arrive and theallocation produced (or some function of the allocation, such as just whichbuyers received positive utility and which did not), and its goal is to predictthe outcome for future subsets of buyers. For this task, the learning algorithmneeds to reconstruct both the priority among the buyers and the preferences ofeach buyer. We derive mistake bound algorithms for additive, unit-demand andsingle minded buyers. We also consider the case where buyers' utilities for afixed bundle can change between stages due to different (observed) prices. Ouralgorithms are efficient both in computation time and in the maximum number ofmistakes (both polynomial in the number of buyers and items).

Learning What's going on: reconstructing preferences and priorities from opaque transactions