Bertrand Networks

TL;DR

This paper analyzes price competition among multiple sellers with limited buyer access, modeling the scenario with graphs, proving equilibrium existence, and exploring structural and computational properties, while leaving some questions open.

Contribution

It introduces a graph-based model for seller competition with buyer restrictions and provides existence proofs and structural analysis of equilibria.

Findings

Equilibria always exist in the modeled scenarios.

Structural and computational properties of equilibria are characterized.

Complete analysis provided for specific cases.

Abstract

We study scenarios where multiple sellers of a homogeneous good compete on prices, where each seller can only sell to some subset of the buyers. Crucially, sellers cannot price-discriminate between buyers. We model the structure of the competition by a graph (or hyper-graph), with nodes representing the sellers and edges representing populations of buyers. We study equilibria in the game between the sellers, prove that they always exist, and present various structural, quantitative, and computational results about them. We also analyze the equilibria completely for a few cases. Many questions are left open.