Non-Price Equilibria in Markets of Discrete Goods

TL;DR

This paper investigates market equilibria in discrete goods markets, showing that while pure Nash equilibria align with traditional price equilibria and are efficient, mixed equilibria always exist but can be inefficient, with bounds provided.

Contribution

It introduces the analysis of mixed Nash equilibria in markets of indivisible items, highlighting their existence, structure, and welfare properties, contrasting with pure equilibria.

Findings

Pure Nash equilibria correspond exactly to price-based equilibria.

Mixed Nash equilibria always exist in these markets.

Pure equilibria are always efficient, but mixed equilibria can be inefficient.

Abstract

We study markets of indivisible items in which price-based (Walrasian) equilibria often do not exist due to the discrete non-convex setting. Instead we consider Nash equilibria of the market viewed as a game, where players bid for items, and where the highest bidder on an item wins it and pays his bid. We first observe that pure Nash-equilibria of this game excatly correspond to price-based equilibiria (and thus need not exist), but that mixed-Nash equilibria always do exist, and we analyze their structure in several simple cases where no price-based equilibrium exists. We also undertake an analysis of the welfare properties of these equilibria showing that while pure equilibria are always perfectly efficient ("first welfare theorem"), mixed equilibria need not be, and we provide upper and lower bounds on their amount of inefficiency.