Understanding Incentives: Mechanism Design Becomes Algorithm Design

Abstract

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 submodular bidder.We further demonstrate the applicability of our reduction by providing a truthful mechanism maximizing fractional max-min fairness. This is the first instance of a truthful mechanism that optimizes a non-linear objective.

Simultaneous bayesian auctions and computational complexity

“…Even though novel algorithmic techniques have occasionally helped advance the state of the art in important fronts [Cai et al 2012[Cai et al , 2013Papadimitriou and Pierrakos 2011], more often computational complexity considerations have shown that the constraint of truthfulness mixes poorly with algorithmic efficiency [Papadimitriou et al 2008;Dobzinski and Vondrak 2012;Cai et al 2013]. Roughly speaking, we now know that the auctions of Vickrey and Myerson are isolated areas of light in a sea of dark, while the new computationally efficient auctions discovered by computer scientists generally lack the compelling simplicity of those archetypes.…”

Separation in Correlation-Robust Monopolist Problem with Budget

“…Indeed, the input to this problem can be specified with n one-dimensional distributions {F i } n i=1 , each described with |V i | parameters, where V i is a support of F i . This is in contrast with the traditional computational Bayesian framework [6,5,7], where the input to a single-buyer monopoly problem (distribution D of types v = (v 1 , . .…”

“…The computational framework of Cai-DaskalakisWeinberg addresses the problem of multidimensional mechanism design [6,5,7] from a computational complexity perspective. This line of work proposes a computationally tractable Bayesian incentive compatible (BIC) solution for mechanism design problems with multiple buyers.…”

The Complexity of Contracts

“…A major contribution of theoretical computer science to game theory and economics has been the articulation of natural classes of succinctly representable settings and a thorough study of the computational complexity of optimal design problems in such settings. Examples include work on multi-dimensional mechanism design that has emphasized succinct type distributions [9,10,11,12], succinct signalling schemes with an exponential number of states of nature [20], and the efficient computation of correlated equilibria in succinctly representable multi-player games [46,36]. The goal of this paper is to initiate an analogous line of work for succinctly described agency problems in contract theory.…”