Walrasian Dynamics in Multi-unit Markets

TL;DR

This paper analyzes the dynamics of Walrasian envy-free pricing mechanisms in multi-unit markets, showing convergence to equilibria with minimal revenue and welfare loss, and introduces a mechanism with promising convergence properties.

Contribution

It demonstrates convergence of best response dynamics in multi-unit markets for Walrasian envy-free prices and proposes a mechanism with strong convergence conjectures.

Findings

Best response dynamics converge to pure Nash equilibria with small revenue and welfare loss.

The minimum Walrasian envy-free price mechanism converges for two buyers and conjectured for more.

General bounds are established for all Nash equilibria in monotone pricing mechanisms.

Abstract

In a multi-unit market, a seller brings multiple units of a good and tries to sell them to a set of buyers that have monetary endowments. While a Walrasian equilibrium does not always exist in this model, natural relaxations of the concept that retain its desirable fairness properties do exist. We study the dynamics of (Walrasian) envy-free pricing mechanisms in this environment, showing that for any such pricing mechanism, the best response dynamic starting from truth-telling converges to a pure Nash equilibrium with small loss in revenue and welfare. Moreover, we generalize these bounds to capture all the Nash equilibria for a large class of (monotone) pricing mechanisms. We also identify a natural mechanism, which selects the minimum Walrasian envy-free price, in which for $n=2$ buyers the best response dynamic converges from any starting profile, and for which we conjecture convergence for any number of buyers.