Markets Beyond Nash Welfare for Leontief Utilities

TL;DR

This paper explores how flexible price curves can support a wider range of allocations beyond Nash welfare in markets with Leontief utilities, providing characterizations and polynomial algorithms.

Contribution

It characterizes when an allocation can be supported by increasing price curves and provides polynomial-time algorithms to find such curves or determine their non-existence.

Findings

Price curve equilibria extend beyond Nash welfare allocations.

Polynomial algorithms support or refute the existence of supporting price curves.

Any CES welfare maximizing allocation can be supported by price curves in bandwidth settings.

Abstract

We study the allocation of divisible goods to competing agents via a market mechanism, focusing on agents with Leontief utilities. The majority of the economics and mechanism design literature has focused on \emph{linear} prices, meaning that the cost of a good is proportional to the quantity purchased. Equilibria for linear prices are known to be exactly the maximum Nash welfare allocations. \emph{Price curves} allow the cost of a good to be any (increasing) function of the quantity purchased. We show that price curve equilibria are not limited to maximum Nash welfare allocations with two main results. First, we show that an allocation can be supported by strictly increasing price curves if and only if it is \emph{group-domination-free}. A similarly characterization holds for weakly increasing price curves. We use this to show that given any allocation, we can compute strictly (or weakly) increasing price curves that support it (or show that none exist) in polynomial time. These results involve a connection to the \emph{agent-order matrix} of an allocation, which may have other applications. Second, we use duality to show that in the bandwidth allocation setting, any allocation maximizing a CES welfare function can be supported by price curves.