The Complexity of Optimal Mechanism Design

TL;DR

This paper proves that computing revenue-optimal auctions for multiple items is computationally intractable under standard complexity assumptions, even in very simple settings, highlighting fundamental limits in auction design.

Contribution

It establishes the first complexity-theoretic hardness result for multi-item auction revenue optimization, showing no efficient algorithm exists unless major complexity classes collapse.

Findings

Optimal multi-item auctions are computationally hard to find.

Hardness holds even for simple, independent item values.

Approximation is the best feasible approach under current complexity assumptions.

Abstract

Myerson's seminal work provides a computationally efficient revenue-optimal auction for selling one item to multiple bidders. Generalizing this work to selling multiple items at once has been a central question in economics and algorithmic game theory, but its complexity has remained poorly understood. We answer this question by showing that a revenue-optimal auction in multi-item settings cannot be found and implemented computationally efficiently, unless ZPP contains P^#P. This is true even for a single additive bidder whose values for the items are independently distributed on two rational numbers with rational probabilities. Our result is very general: we show that it is hard to compute any encoding of an optimal auction of any format (direct or indirect, truthful or non-truthful) that can be implemented in expected polynomial time. In particular, under well-believed complexity-theoretic assumptions, revenue-optimization in very simple multi-item settings can only be tractably approximated. We note that our hardness result applies to randomized mechanisms in a very simple setting, and is not an artifact of introducing combinatorial structure to the problem by allowing correlation among item values, introducing combinatorial valuations, or requiring the mechanism to be deterministic (whose structure is readily combinatorial). Our proof is enabled by a flow-interpretation of the solutions of an exponential-size linear program for revenue maximization with an additional supermodularity constraint.