Mechanism design: from optimal transport theory to revenue maximization

A central problem in Economics and Algorithmic Game Theory is the design of auctions that maximize the auctioneer’s expected revenue. While optimal selling of a single item has been well-understood since the pioneering work of Myerson in 1981, extending his work to multi-item settings has remained a challenge. In this work, we obtain such extensions providing a mathematical framework for finding optimal mechanisms. In the first part of the work, we study revenue maximization in single-bidder multi-item settings, connecting this problem to a well-studied problem in measure theory, namely the design of optimal transport maps. By establishing strong duality between these two problems, we obtain a characterization of the structure of optimal mechanisms. As an important application, we prove that a grand bundling mechanism is optimal if and only if two measure-theoretic inequalities are satisfied. Likewise, we obtain necessary and sufficient conditions for the optimality of any mechanism in terms of a collection of measure-theoretic inequalities. Using our machinery we derive closed-form solutions in several example scenarios, illustrating the richness of mechanisms in multi-item settings, and we prove that the mechanism design problem in general is computationally intractable even for a single bidder. In the second part of the work, we study multi-bidder settings where bidders have uncertainty about the items for sale. In such settings, the auctioneer may wish to reveal some information about the item for sale in addition to running an auction. While prior work has focused only on the information design part keeping the mechanism fixed, we study the combined problem of designing the information revelation policy together with the auction format. We find that prior approaches to this problem are suboptimal and identify the optimal mechanism by connecting this setting to the multi-item mechanism design problem studied in the first part of the work. Thesis Supervisor: Constantinos Daskalakis Title: Associate Professor

[1]  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.

[2]  Yang Cai,et al.  Reducing Revenue to Welfare Maximization: Approximation Algorithms and other Generalizations , 2013, SODA.

[3]  Achim Wambach,et al.  Sequential Information Disclosure in Auctions , 2013 .

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

[5]  Ivar Ekeland,et al.  Notes on optimal transportation , 2010 .

[6]  Michael Sipser,et al.  Introduction to the Theory of Computation , 1996, SIGA.

[7]  C. Villani.  Optimal Transport: Old and New , 2008 .

[8]  Christos Tzamos,et al.  Mechanism design via optimal transport , 2013, EC.

[9]  Dirk Bergemann,et al.  Information Structures in Optimal Auctions , 2001, J. Econ. Theory.

[10]  Nicole Immorlica,et al.  Constrained Signaling in Auction Design , 2013, SODA.

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

[12]  Xi Chen,et al.  On the Complexity of Optimal Lottery Pricing and Randomized Mechanisms , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[13]  Tim Roughgarden,et al.  Algorithmic Game Theory , 2007 .

[14]  Moshe Babaioff,et al.  Peaches, lemons, and cookies: designing auction markets with dispersed information , 2013, EC.

[15]  J. Rochet,et al.  Ironing, Sweeping, and Multidimensional Screening , 1998 .

[16]  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.

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

[18]  Péter Eso,et al.  Optimal Information Disclosure in Auctions and the Handicap Auction , 2007 .

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

[20]  Xi Chen,et al.  The Complexity of Optimal Multidimensional Pricing , 2013, SODA.

[21]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[22]  Sergei Vassilvitskii,et al.  Value of Targeting , 2014, SAGT.

[23]  Peter Bro Miltersen,et al.  Send mixed signals: earn more, work less , 2012, EC '12.

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

[25]  Patrick Hummel,et al.  When Does Improved Targeting Increase Revenue? , 2016, ACM Trans. Economics and Comput..

[26]  Paul R. Milgrom,et al.  A theory of auctions and competitive bidding , 1982 .

[27]  Gabriel D. Carroll.  Robustness and Separation in Multidimensional Screening , 2017 .

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

[29]  R. Myerson.  Incentive Compatibility and the Bargaining Problem , 1979 .

[30]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

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

[32]  M. Armstrong.  Multiproduct Nonlinear Pricing , 1996 .

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

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

[35]  M. Whinston,et al.  Multiproduct Monopoly, Commodity Bundling, and Correlation of Values , 1989 .

[36]  Shahar Dobzinski,et al.  Optimal auctions with correlated bidders are easy , 2010, STOC.

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

[38]  Sergei Vassilvitskii,et al.  Ad Auctions with Data , 2012, SAGT.

[39]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.

[40]  Christos Tzamos,et al.  Optimal Pricing Is Hard , 2012, WINE.

[41]  J. Rochet.  A necessary and sufficient condition for rationalizability in a quasi-linear context , 1987 .

[42]  J. Rochet,et al.  The Economics of Multidimensional Screening , 2003 .

[43]  E. Maskin,et al.  Monopoly with Incomplete Information , 1984 .

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

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

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

[47]  George A. Akerlof.  The Market for “Lemons”: Quality Uncertainty and the Market Mechanism , 1970 .

[48]  Moshe Tennenholtz,et al.  Signaling Schemes for Revenue Maximization , 2012, TEAC.

[49]  Yang Cai,et al.  Simple and Nearly Optimal Multi-Item Auctions , 2012, SODA.

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

[51]  Alessio Figalli,et al.  When is multidimensional screening a convex program? , 2009, J. Econ. Theory.

[52]  S. Board,et al.  Revealing information in auctions: the allocation effect , 2009 .

[53]  Christos H. Papadimitriou,et al.  On optimal single-item auctions , 2010, STOC '11.

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

[55]  Li Han,et al.  Mixture Selection, Mechanism Design, and Signaling , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[56]  Nima Haghpanah,et al.  Reverse Mechanism Design , 2014, EC.

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

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

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

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

[61]  Christos Tzamos,et al.  Does Information Revelation Improve Revenue? , 2016, EC.

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

[63]  Patrick Briest,et al.  Uniform Budgets and the Envy-Free Pricing Problem , 2008, ICALP.