On Envy-Free Revenue Approximation for Combinatorial Buyers with Budgets

We study the computation of revenue-maximizing envy-free outcomes, in a monopoly market with budgeted buyers. Departing from previous works, we focus on buyers with combinatorial valuation functions, that are defined over subsets of distinct goods. Our results concern several classes of valuation functions. For single-minded budgeted buyers, we show a best possible polynomial-time approximation of the optimum envy-free revenue. Subsequently, we establish a hardness result showing that, even for two identical budgeted additive buyers, the problem is not approximable in polynomial time. In an attempt to identify tractable families of the problem's instances, we introduce and elaborate on the notion ofbudget compatiblebuyers, that have at least enough budget to"cover"their value for any single good. Under this assumption, we establish approximation upper bounds for buyers with submodular valuation functions over preference subsets, as well as for buyers with identical subadditive valuation functions. Subsequently, we analyze an algorithm for budget compatible buyers with arbitrary additive valuation functions, which approximates the optimum envy-free revenue within a constant factor, for a constant number of buyers. We conclude with several interesting open questions, regarding budget compatible buyers with combinatorial valuation functions.