Understanding Incentives: Mechanism Design Becomes Algorithm Design

We provide a computationally efficient black-box reduction from mechanism design to algorithm design in very general settings. Specifically, we give an approximation-preserving reduction from truthfully maximizing any objective under arbitrary feasibility constraints with arbitrary bidder types to (not necessarily truthfully) maximizing the same objective plus virtual welfare (under the same feasibility constraints). Our reduction is based on a fundamentally new approach: we describe a mechanism's behavior indirectly only in terms of the expected value it awards bidders for certain behavior, and never directly access the allocation rule at all. Applying our new approach to revenue, we exhibit settings where our reduction holds both ways. That is, we also provide an approximation-sensitive reduction from (non-truthfully) maximizing virtual welfare to (truthfully) maximizing revenue, and therefore the two problems are computationally equivalent. With this equivalence in hand, we show that both problems are NP-hard to approximate within any polynomial factor, even for a single monotone sub modular bidder. We further demonstrate the applicability of our reduction by providing a truthful mechanism maximizing fractional max-min fairness.

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

[2]  Elias Koutsoupias,et al.  A Lower Bound for Scheduling Mechanisms , 2007, SODA '07.

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

[4]  Piotr Krysta,et al.  Buying cheap is expensive: hardness of non-parametric multi-product pricing , 2007, SODA '07.

[5]  Christos H. Papadimitriou,et al.  On the Hardness of Being Truthful , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[6]  Elchanan Mossel,et al.  Inapproximability for VCG-based combinatorial auctions , 2010, SODA '10.

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

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

[9]  Shahar Dobzinski An impossibility result for truthful combinatorial auctions with submodular valuations , 2010, STOC '11.

[10]  Yang Cai,et al.  Extreme-Value Theorems for Optimal Multidimensional Pricing , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[11]  Saeed Alaei,et al.  Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[12]  Robert D. Kleinberg,et al.  Bayesian incentive compatibility via matchings , 2011, SODA '11.

[13]  Shahar Dobzinski,et al.  Optimal auctions with correlated bidders are easy , 2010, STOC '11.

[14]  Xiaohui Bei,et al.  Bayesian incentive compatibility via fractional assignments , 2010, SODA '11.

[15]  Shahar Dobzinski,et al.  The computational complexity of truthfulness in combinatorial auctions , 2012, EC '12.

[16]  Noam Nisan,et al.  Approximate revenue maximization with multiple items , 2012, EC '12.

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

[18]  Nicole Immorlica,et al.  On the limits of black-box reductions in mechanism design , 2012, STOC '12.

[19]  Itai Ashlagi,et al.  Optimal Lower Bounds for Anonymous Scheduling Mechanisms , 2012, Math. Oper. Res..

[20]  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.

[21]  S. Matthew Weinberg,et al.  Symmetries and optimal multi-dimensional mechanism design , 2012, EC '12.

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

[23]  Kamesh Munagala,et al.  Optimal auctions via the multiplicative weight method , 2012, EC '13.

[24]  Shuchi Chawla,et al.  Prior-independent mechanisms for scheduling , 2013, STOC '13.

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

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

[27]  Shuchi Chawla,et al.  Bayesian algorithmic mechanism design , 2014, SECO.

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