G T ] 2 9 Ju l 2 01 6 Envy-Free Pricing in Multi-unit Markets ∗

We study the envy-free pricing problem in multi-unit market s with budgets, where there is a seller who brings multiple units of a good, while several buyers bri ng monetary endowments. Our goal is to compute an envy-free (item) price and allocation, i.e. an ou tcome where all the demands of the buyers are met given their budget constraints, which additionally chieves a desirable objective. We analyze markets with linear valuations, where the buyers are price tak rsand pricemakers, respectively. In theprice takingscenario, for the problem of computing a welfare maximizing e vy-free pricing we provide a polynomial time algorithm, while for the proble m of computing a revenue optimal envyfree pricing we provide an FPTAS and exact algorithm (which i s polynomial for a constant number of types of buyers). In the price makingscenario, 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 approxim ation ratio is a function of the market share,s, which can roughly be understood as the maximum buying power f any individual buyer in the market. When the market is even mildly competitive—i.e. wit h 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 t he strategyproof mechanisms for both objectives on competitive markets. Finally, for price taking buyers with general valuations, w e provide fully polynomial time approximation schemes as well as hardness results for both revenue a nd welfare. keywords: envy-free pricing, market share, revenue, social welfare, strategyproofness, efficiency The authors aknowledge support from the Danish National Res earch Foundation and The National Science Foundation of China (under the grant 61361136003) for the Sino-Danish Cen ter for the Theory of Interactive Computation and from the Ce nter for Research in Foundations of Electronic Markets (CFEM), s upported by the Danish Strategic Research Council. Simina B rânzei was also supported by (I) ISF grant 1435/14 administered by t he Israeli Academy of Sciences and Israel-USA Bi-national S cience Foundation (BSF) grant 2014389, (II) the I-CORE Program of t he Planning and Budgeting Committee and The Israel Science Foundation, and (III) the Simons Institute for the Theory of Computing. Aris Filos-Ratsikas was also partially support ed by the ERC Advanced Grant 321171 (ALGAME). Finally, Simina Brânz ei and Aris Filos-Ratsikas received partial support from th e COST Action IC1205 on “Computational Social Choice”. Hebrew University of Jerusalem, Israel. E-mail: s mina.branzei@gmail.com University of Oxford, United Kingdom. E-mail: aris.filos-ratsikas@cs.ox.ac.uk Aarhus University, Denmark. E-mail: bromille@cs.au.dk Tsinghua University, China. E-mail: 297410725@qq.com