Envy-Free Allocations for Budgeted Bidders

We study the problem of identifying prices to support a given allocation of items to bidders in an envy-free way. A bidder will envy another bidder if she would prefer to obtain the other bidder's item at the price paid by that bidder. Envy-free prices for allocations have been studied extensively; here: we focus oil the impact of budget: beyond their willingness to pay for items; bidders are also constrained by their ability to pay, which may be lower than their willingness. In a recent Paper; Aggarwal et at. show dial; a variant of the Ascending Auction finds a feasible and bidder-optimal assignment and supporting envy-free prices in polynomial time so long as the input satisfies certain non-degeneracy conditions. While this settles the problem of finding a. feasible allocation; an auctioneer might sometimes also be interested in a specific allocation of items to bidders. We present two polynomial-time algorithms for this problem; one which finds maximal prices supporting the given allocation (if such prices exist); and another which finds minimal prices. We also prove a structural result characterizing when different allocations are supported by the same minimal price vector.