Walrasian Pricing in Multi-unit Auctions

TL;DR

This paper introduces an optimal envy-free pricing mechanism for multi-unit auctions with budgets, demonstrating strong approximation guarantees in competitive markets and highlighting the trade-offs between truthfulness and efficiency.

Contribution

It designs a novel envy-free mechanism for multi-unit auctions with budgets and analyzes its performance and limitations in terms of revenue, welfare, and truthfulness.

Findings

Mechanism achieves small constant approximation ratios in competitive markets.

Welfare approximation converges to optimal as market competitiveness increases.

Truthfulness is incompatible with Pareto efficiency without resource discarding.

Abstract

Multi-unit auctions are a paradigmatic model, where a seller brings multiple units of a good, while several buyers bring monetary endowments. It is well known that Walrasian equilibria do not always exist in this model, however compelling relaxations such as (Walrasian) envy-free pricing do. In this paper we design an optimal envy-free mechanism for multi-unit auctions with budgets. When the market is even mildly competitive, the approximation ratios of this mechanism are small constants for both the revenue and welfare objectives, and in fact for welfare the approximation converges to 1 as the market becomes fully competitive. We also give an impossibility theorem, showing that truthfulness requires discarding resources and, in particular, is incompatible with (Pareto) efficiency.