Reducing Revenue to Welfare Maximization: Approximation Algorithms and other Generalizations

TL;DR

This paper extends a known reduction from revenue to welfare maximization in auction design to approximate settings, enabling efficient near-optimal mechanisms using welfare approximations and advanced optimization techniques.

Contribution

It introduces an approximation-preserving reduction from revenue to welfare maximization, accommodating approximation algorithms and generalizing previous structural results.

Findings

Provides a polynomial-time reduction for approximate revenue maximization

Enables the design of approximately optimal mechanisms using welfare approximations

Employs novel optimization techniques over non-convex regions

Abstract

It was recently shown in [http://arxiv.org/abs/1207.5518] that revenue optimization can be computationally efficiently reduced to welfare optimization in all multi-dimensional Bayesian auction problems with arbitrary (possibly combinatorial) feasibility constraints and independent additive bidders with arbitrary (possibly combinatorial) demand constraints. This reduction provides a poly-time solution to the optimal mechanism design problem in all auction settings where welfare optimization can be solved efficiently, but it is fragile to approximation and cannot provide solutions to settings where welfare maximization can only be tractably approximated. In this paper, we extend the reduction to accommodate approximation algorithms, providing an approximation preserving reduction from (truthful) revenue maximization to (not necessarily truthful) welfare maximization. The mechanisms output by our reduction choose allocations via black-box calls to welfare approximation on randomly selected inputs, thereby generalizing also our earlier structural results on optimal multi-dimensional mechanisms to approximately optimal mechanisms. Unlike [http://arxiv.org/abs/1207.5518], our results here are obtained through novel uses of the Ellipsoid algorithm and other optimization techniques over {\em non-convex regions}.