D ec 2 01 7 Approximate Revenue Maximization with Multiple Items ∗

Maximizing the revenue from selling more than one good (or item) to a single buyer is a notoriously difficult problem, in stark contrast to the one-good case. For two goods, we show that simple “onedimensional” mechanisms, such as selling the goods separately, guarantee at least 73% of the optimal revenue when the valuations of the two goods are independent and identically distributed, and at least 50% when they are independent. For the case of k > 2 independent goods, we show that selling them separately guarantees at least a c/ log k fraction of the optimal revenue; and, for independent and identically distributed goods, we show that selling them as one bundle guarantees at least a c/ log k fraction of the optimal revenue. This version: September 2017. Previous versions: February 2012; April 2012 (arXiv 1204.1846 and Center for Rationality DP-606); May 2014; March 2017. Research partially supported by a European Research Council Advanced Investigator grant (Hart) and by an Israel Science Foundation grant (Nisan). We thank Motty Perry and Phil Reny for introducing us to the subject and for many helpful discussions, and the referees for their very careful reading and useful comments. A presentation that covers some of this work is available at http://www.ma.huji.ac.il/hart/abs/2good-p.html The Hebrew University of Jerusalem (Federmann Center for the Study of Rationality, Department of Economics, and Institute of Mathematics). E-mail : hart@huji.ac.il Web site: http://www.ma.huji.ac.il/hart The Hebrew University of Jerusalem (Federmann Center for the Study of Rationality, and School of Computer Science and Engineering), and Microsoft Research. E-mail : noam@cs.huji.ac.il Web site: http://www.cs.huji.ac.il/~noam

[1]  Elias Koutsoupias,et al.  Duality and optimality of auctions for uniform distributions , 2014, SIAM J. Comput..

[2]  Moshe Babaioff,et al.  The menu-size complexity of revenue approximation , 2016, STOC.

[3]  Ron Kupfer.  A Note on the Ratio of Revenues Between Selling in a Bundle and Separately , 2016, ArXiv.

[4]  Tim Roughgarden,et al.  Learning Simple Auctions , 2016, COLT.

[5]  D. Menicucci,et al.  On the Optimality of Pure Bundling for a Monopolist , 2015 .

[6]  S. Hart,et al.  Maximal revenue with multiple goods: Nonmonotonicity and other observations , 2015 .

[7]  S. Matthew Weinberg,et al.  Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity , 2018, ACM Trans. Economics and Comput..

[8]  Christos Tzamos,et al.  Strong Duality for a Multiple-Good Monopolist , 2014, EC.

[9]  Andrew Chi-Chih Yao,et al.  An n-to-1 Bidder Reduction for Multi-item Auctions and its Applications , 2014, SODA.

[10]  S. Matthew Weinberg,et al.  A Simple and Approximately Optimal Mechanism for an Additive Buyer , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[11]  Yiannis Giannakopoulos.  Bounding Optimal Revenue in Multiple-Items Auctions , 2014, ArXiv.

[12]  Noam Nisan,et al.  Sampling and Representation Complexity of Revenue Maximization , 2014, WINE.

[13]  Pingzhong Tang,et al.  Optimal mechanisms with simple menus , 2014, EC.

[14]  Christos Tzamos,et al.  The Complexity of Optimal Mechanism Design , 2012, SODA.

[15]  Andrew Chi-Chih Yao,et al.  On revenue maximization for selling multiple independently distributed items , 2013, Proceedings of the National Academy of Sciences.

[16]  Christos Tzamos,et al.  Mechanism design via optimal transport , 2013, EC '13.

[17]  Noam Nisan,et al.  The menu-size complexity of auctions , 2013, EC '13.

[18]  Alejandro M. Manelli,et al.  Multidimensional mechanism design: Revenue maximization and the multiple-good monopoly. A corrigendum , 2012, J. Econ. Theory.

[19]  Yang Cai,et al.  Optimal Multi-dimensional Mechanism Design: Reducing Revenue to Welfare Maximization , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[20]  Nima Haghpanah,et al.  Bayesian optimal auctions via multi- to single-agent reduction , 2012, EC '12.

[21]  Yang Cai,et al.  An algorithmic characterization of multi-dimensional mechanisms , 2011, STOC '12.

[22]  Nir Shabbat.  Approximate Revenue Maximization with Multiple Items , 2012 .

[23]  S. Hart,et al.  Implementation of Reduced Form Mechanisms: A Simple Approach and a New Characterization , 2011 .

[24]  Omer Lev,et al.  A two-dimensional problem of revenue maximization , 2011 .

[25]  G. Pavlov,et al.  Optimal Mechanism for Selling Two Goods , 2011 .

[26]  Shuchi Chawla,et al.  Multi-parameter mechanism design and sequential posted pricing , 2010, BQGT.

[27]  Shuchi Chawla,et al.  The power of randomness in bayesian optimal mechanism design , 2010, EC '10.

[28]  S. Matthew Weinberg,et al.  Pricing randomized allocations , 2009, SODA '10.

[29]  Shuchi Chawla,et al.  Algorithmic pricing via virtual valuations , 2007, EC '07.

[30]  Philippe Jehiel,et al.  Mixed Bundling Auctions , 2006, J. Econ. Theory.

[31]  Alejandro M. Manelli,et al.  Multidimensional Mechanism Design: Revenue Maximization and the Multiple-Good Monopoly , 2004, J. Econ. Theory.

[32]  Alejandro M. Manelli,et al.  Bundling as an optimal selling mechanism for a multiple-good monopolist , 2006, J. Econ. Theory.

[33]  M. Pycia.  Stochastic vs Deterministic Mechanisms in Multidimensional Screening , 2006 .

[34]  H. Fang,et al.  To Bundle or Not to Bundle , 2005 .

[35]  John Thanassoulis,et al.  Haggling over substitutes , 2004, J. Econ. Theory.

[36]  F. Schoenberg,et al.  Approximating the Distribution of Pareto Sums , 2003 .

[37]  M. Armstrong.  Price Discrimination by a Many-Product Firm , 1999 .

[38]  Yannis Bakos,et al.  Bundling Information Goods: Pricing, Profits and Efficiency , 1998 .

[39]  Kim C. Border.  IMPLEMENTATION OF REDUCED FORM AUCTIONS: A GEOMETRIC APPROACH , 1991 .

[40]  R. McAfee,et al.  Multidimensional incentive compatibility and mechanism design , 1988 .

[41]  Richard P. McLean,et al.  FULL EXTRACTION OF THE SURPLUS IN BAYESIAN AND DOMINANT STRATEGY AUCTIONS , 1988 .

[42]  J. Rochet.  The taxation principle and multi-time Hamilton-Jacobi equations☆ , 1985 .

[43]  R. Zeckhauser,et al.  Optimal Selling Strategies: When to Haggle, When to Hold Firm , 1983 .

[44]  Roger B. Myerson,et al.  Optimal Auction Design , 1981, Math. Oper. Res..