Envy-Free Pricing in Multi-unit Markets

We study the envy-free pricing problem in linear multi-unit markets with budgets, where there is a seller who brings multiple units of a good, while several buyers bring monetary endowments. Our goal is to compute an envy-free (item) price and allocation, i.e. an outcome where all the demands of the buyers are met given their budget constraints, which additionally achieves a desirable objective. In the price taking scenario, where the buyers purchase the optimal bundle given the prices they are facing, we provide a polynomial time algorithm for the problem of computing a welfare maximizing envy-free pricing, and an FPTAS and exact algorithm (which is polynomial for a constant number of types of buyers) for the problem of computing a revenue optimal envy-free pricing . In the price making scenario, where the buyers can strategize, we show a general impossibility of designing strategyproof and efficient mechanisms even with public budgets. On the positive side, we provide an optimal strategyproof mechanism whose approximation ratio is a function of the market share, s, which can roughly be understood as the maximum buying power of any individual buyer in the market. When the market is even mildly competitive—i.e. with no buyer having a market share higher than 50%—the approximation ratio of our mechanism is at most 2 for revenue and at most 1/(1 − s) for welfare. Moreover, this mechanism is optimal among all the strategyproof mechanisms for both objectives on competitive markets. Finally, for price taking buyers with general valuations, we provide fully polynomial time approximation schemes as well as hardness results for both revenue and welfare.