Equilibrium in the Canonical Stackelberg Triopoly via Response Functions and Fixed Point Theory

TL;DR

This paper studies the existence and uniqueness of market equilibrium in a three-firm Stackelberg triopoly using fixed point theory, highlighting convergence issues and recursive formulations as firms increase.

Contribution

It introduces a reformulation of equilibrium conditions for triopoly and analyzes convergence and limiting behavior with increasing participants.

Findings

Existence and uniqueness of equilibrium are established via coupled fixed-point theory.

Convergence of best-response dynamics is not guaranteed even with linear demand.

Recursive equilibrium formulation helps analyze the limit as the number of firms grows.

Abstract

We analyze a canonical extension of the Stackelberg duopoly to a sequential framework, where each firm strategically anticipates the reactions of all subsequent players. In a triopoly (three-firm) settings, we obtain existence and uniqueness of market equilibrium via a reformulation of the equilibrium conditions that draws on coupled fixed-point theory. Even with linear demand, convergence of myopic best-response dynamics is not guaranteed. A recursive equilibrium formulation enables the analysis of the limiting case as the number of participants grow.