Networks of Complements

TL;DR

This paper analyzes how the structure of networks of complementary products influences seller revenues and social welfare in pricing games, providing theoretical bounds and dynamics for equilibrium outcomes.

Contribution

It introduces a comprehensive analysis of equilibria in networks of complements, including bounds on efficiency and convergence dynamics for various graph structures.

Findings

Positive and negative bounds on Price of Anarchy and Price of Stability for different graph families

Existence of approximately-efficient equilibria through best-reply dynamics

Convergence of dynamics to non-trivial equilibria in several network structures

Abstract

We consider a network of sellers, each selling a single product, where the graph structure represents pair-wise complementarities between products. We study how the network structure affects revenue and social welfare of equilibria of the pricing game between the sellers. We prove positive and negative results, both of "Price of Anarchy" and of "Price of Stability" type, for special families of graphs (paths, cycles) as well as more general ones (trees, graphs). We describe best-reply dynamics that converge to non-trivial equilibrium in several families of graphs, and we use these dynamics to prove the existence of approximately-efficient equilibria.