# Selling Complementary Goods: Dynamics, Efficiency and Revenue

## TL;DR

This paper analyzes a model of price competition between two sellers of perfect complements, showing the existence of pure Nash equilibria, quantifying welfare and revenue impacts, and exploring the convergence properties of best response dynamics.

## Contribution

It establishes the existence of pure Nash equilibria in a classic complementary goods pricing game and provides new insights into welfare, revenue, and dynamic convergence.

## Key findings

- Pure Nash equilibria always exist in the model.
- Quantitative analysis of welfare and revenue compared to monopoly.
- Negative results on convergence of best response dynamics.

## Abstract

We consider a price competition between two sellers of perfect-complement goods. Each seller posts a price for the good it sells, but the demand is determined according to the sum of prices. This is a classic model by Cournot (1838), who showed that in this setting a monopoly that sells both goods is better for the society than two competing sellers. We show that non-trivial pure Nash equilibria always exist in this game. We also quantify Cournot's observation with respect to both the optimal welfare and the monopoly revenue. We then prove a series of mostly negative results regarding the convergence of best response dynamics to equilibria in such games.