Envy-Free Revenue Approximation for Asymmetric 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 asymmetric combinatorial valuation functions over subsets of items. We first establish a hardness result showing that, even with two identical additive buyers, the problem is inapproximable. In an attempt to identify tractable families of the problem's instances, we introduce the notion of budget compatible buyers, placing a restriction on the budget of each buyer in terms of his valuation function. Under this assumption, we establish approximation upper bounds for buyers with submodular valuations over preference subsets as well as for buyers with identical subadditive valuation functions. Finally, we also analyze an algorithm for arbitrary additive valuation functions, which yields a constant factor approximation for a constant number of buyers. We conclude with several intriguing open questions regarding budgeted buyers with asymmetric valuation functions.