Combinatorial Walrasian Equilibrium

TL;DR

This paper introduces Combinatorial Walrasian Equilibrium (CWE), a relaxed market concept allowing item bundling, which guarantees at least half of the optimal social welfare and can be computed efficiently, addressing limitations of classic Walrasian Equilibrium.

Contribution

It proposes CWE as a new market equilibrium concept, providing approximation guarantees for social welfare and revenue, with a polynomial-time algorithm for finding high-welfare equilibria.

Findings

Every valuation profile admits a CWE with at least half of the optimal welfare.

A polynomial-time algorithm computes a CWE achieving at least half the welfare of any given allocation.

CWE can extract a logarithmic fraction of the optimal welfare as revenue.

Abstract

We study a combinatorial market design problem, where a collection of indivisible objects is to be priced and sold to potential buyers subject to equilibrium constraints.The classic solution concept for such problems is Walrasian Equilibrium (WE), which provides a simple and transparent pricing structure that achieves optimal social welfare. The main weakness of the WE notion is that it exists only in very restrictive cases. To overcome this limitation, we introduce the notion of a Combinatorial Walrasian equilibium (CWE), a natural relaxation of WE. The difference between a CWE and a (non-combinatorial) WE is that the seller can package the items into indivisible bundles prior to sale, and the market does not necessarily clear. We show that every valuation profile admits a CWE that obtains at least half of the optimal (unconstrained) social welfare. Moreover, we devise a poly-time algorithm that, given an arbitrary allocation X, computes a CWE that achieves at least half of the welfare of X. Thus, the economic problem of finding a CWE with high social welfare reduces to the algorithmic problem of social-welfare approximation. In addition, we show that every valuation profile admits a CWE that extracts a logarithmic fraction of the optimal welfare as revenue. Finally, these results are complemented by strong lower bounds when the seller is restricted to using item prices only, which motivates the use of bundles. The strength of our results derives partly from their generality - our results hold for arbitrary valuations that may exhibit complex combinations of substitutes and complements.