Approximately optimal mechanism design via differential privacy

We study the implementation challenge in an abstract interdependent values model and an arbitrary objective function. We design a generic mechanism that allows for approximate optimal implementation of insensitive objective functions in ex-post Nash equilibrium. If, furthermore, values are private then the same mechanism is strategy proof. We cast our results onto two specific models: pricing and facility location. The mechanism we design is optimal up to an additive factor of the order of magnitude of one over the square root of the number of agents and involves no utility transfers. Underlying our mechanism is a lottery between two auxiliary mechanisms --- with high probability we actuate a mechanism that reduces players influence on the choice of the social alternative, while choosing the optimal outcome with high probability. This is where differential privacy is employed. With the complementary probability we actuate a mechanism that may be typically far from optimal but is incentive compatible. The joint mechanism inherits the desired properties from both.

[1]  William Vickrey,et al.  Counterspeculation, Auctions, And Competitive Sealed Tenders , 1961 .

[2]  E. H. Clarke.  Multipart pricing of public goods , 1971 .

[3]  Theodore Groves,et al.  Incentives in Teams , 1973 .

[4]  A. Gibbard.  Manipulation of Voting Schemes: A General Result , 1973 .

[5]  M. Satterthwaite.  Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions , 1975 .

[6]  D. J. Roberts,et al.  THE INCENTIVES FOR PRICE-TAKING BEHAVIOR IN LARGE EXCHANGE ECONOMIES , 1976 .

[7]  M. Satterthwaite,et al.  Strategy-proofness and single-peakedness , 1976 .

[8]  H. Moulin.  On strategy-proofness and single peakedness , 1980 .

[9]  Hitoshi Matsushima.  A new approach to the implementation problem , 1988 .

[10]  Steven R. Williams,et al.  The Rate of Convergence to Efficiency in the Buyer's Bid Double Auction as the Market Becomes Large , 1989 .

[11]  R. Rob.  Pollution claim settlements under private information , 1989 .

[12]  Andrew Postlewaite,et al.  Asymmetric Information Bargaining Problems with Many Agents , 1990 .

[13]  Dilip Abreu,et al.  Subgame perfect implementation: A necessary and almost sufficient condition , 1990 .

[14]  Hitoshi Matsushima,et al.  Virtual implementation in iteratively undominated strategies: complete information , 1992 .

[15]  Steven R. Williams,et al.  Convergence to Efficiency in a Simple Market with Incomplete Information , 1994 .

[16]  D. Levine,et al.  When Are Agents Negligible , 1995 .

[17]  A. Mas-Colell,et al.  Microeconomic Theory , 1995 .

[18]  D. Fudenberg,et al.  When Are Nonanonymous Players Negligible , 1996 .

[19]  John Duggan.  Virtual Bayesian Implementation , 1997 .

[20]  Noam Nisan,et al.  Algorithmic mechanism design (extended abstract) , 1999, STOC '99.

[21]  J. Schummer.  Almost-dominant Strategy Implementation , 1999 .

[22]  Rann Smorodinsky,et al.  Pivotal Players and the Characterization of Influence , 2000, J. Econ. Theory.

[23]  Jeroen M. Swinkels.  Efficiency of Large Private Value Auctions , 2001 .

[24]  R. Serrano,et al.  Some limitations of virtual Bayesian implementation , 2001 .

[25]  Richard P. McLean,et al.  Informational Size and Incentive Compatibility , 2001 .

[26]  Rakesh V. Vohra,et al.  Strategy-proof Location on a Network , 2002, J. Econ. Theory.

[27]  Richard P. McLean,et al.  Informational Size and Incentive Compatibility with Aggregate Uncertainty , 2003 .

[28]  Roberto Serrano,et al.  A Characterization of Virtual Bayesian Implementation , 2002, Games Econ. Behav..

[29]  Maria-Florina Balcan,et al.  Mechanism design via machine learning , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[30]  Cynthia Dwork,et al.  Differential Privacy , 2006, ICALP.

[31]  Rudolf Müller,et al.  Mini special issue: Electronic Market Design , 2006, Games Econ. Behav..

[32]  Michele Bugliesi,et al.  Automata, Languages and Programming: 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part II (Lecture Notes in Computer Science) , 2006 .

[33]  Anna R. Karlin,et al.  Competitive auctions , 2006, Games Econ. Behav..

[34]  Cynthia Dwork,et al.  Calibrating Noise to Sensitivity in Private Data Analysis , 2006, TCC.

[35]  Sofya Raskhodnikova,et al.  Smooth sensitivity and sampling in private data analysis , 2007, STOC '07.

[36]  R. Vohra,et al.  Algorithmic Game Theory: Mechanism Design without Money , 2007 .

[37]  Rann Smorodinsky,et al.  The efficiency of competitive mechanisms under private information , 2007, J. Econ. Theory.

[38]  Kunal Talwar,et al.  Mechanism Design via Differential Privacy , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[39]  Moshe Tennenholtz,et al.  Approximate mechanism design without money , 2009, EC '09.

[40]  Noga Alon,et al.  Strategyproof Approximation Mechanisms for Location on Networks , 2009, ArXiv.

[41]  Cynthia Dwork,et al.  The Differential Privacy Frontier (Extended Abstract) , 2009, TCC.

[42]  Moshe Babaioff,et al.  Single-value combinatorial auctions and algorithmic implementation in undominated strategies , 2009, JACM.

[43]  Christos Tzamos,et al.  Winner-imposing strategyproof mechanisms for multiple Facility Location games , 2010, Theor. Comput. Sci..

[44]  Cynthia Dwork,et al.  Differential privacy in new settings , 2010, SODA '10.

[45]  Noga Alon,et al.  Strategyproof Approximation of the Minimax on Networks , 2010, Math. Oper. Res..

[46]  Zeyuan Allen Zhu,et al.  Asymptotically optimal strategy-proof mechanisms for two-facility games , 2010, EC '10.

[47]  S. Boicheva.  Mechanism Design without Money , 2012 .

[48]  Ariel D. Procaccia,et al.  Approximate Mechanism Design without Money , 2013, TEAC.

[49]  David Xiao,et al.  Is privacy compatible with truthfulness? , 2013, ITCS '13.