Nash equilibria in Fisher market

TL;DR

This paper models the strategic behavior of buyers in Fisher markets as a game, characterizing Nash equilibria and showing that all equilibria are symmetric with payoffs forming a piecewise linear concave curve.

Contribution

It introduces the Fisher market game, providing a complete polyhedral characterization of Nash equilibria for the two-buyer case and analyzing the role of correlated equilibria.

Findings

All Nash equilibria are symmetric.

Equilibria payoffs form a piecewise linear concave curve.

Third-party mediation does not improve payoffs.

Abstract

Much work has been done on the computation of market equilibria. However due to strategic play by buyers, it is not clear whether these are actually observed in the market. Motivated by the observation that a buyer may derive a better payoff by feigning a different utility function and thereby manipulating the Fisher market equilibrium, we formulate the {\em Fisher market game} in which buyers strategize by posing different utility functions. We show that existence of a {\em conflict-free allocation} is a necessary condition for the Nash equilibria (NE) and also sufficient for the symmetric NE in this game. There are many NE with very different payoffs, and the Fisher equilibrium payoff is captured at a symmetric NE. We provide a complete polyhedral characterization of all the NE for the two-buyer market game. Surprisingly, all the NE of this game turn out to be symmetric and the corresponding payoffs constitute a piecewise linear concave curve. We also study the correlated equilibria of this game and show that third-party mediation does not help to achieve a better payoff than NE payoffs.