Stackelberg Network Pricing Games

TL;DR

This paper analyzes a Stackelberg network pricing game where a leader sets prices on certain edges, followers choose minimum cost solutions, and the goal is to maximize revenue, with new approximation algorithms and hardness results.

Contribution

It provides tight approximation bounds for single-price algorithms, extends results to multiple followers, and introduces a polynomial-time algorithm for Stackelberg bipartite vertex cover.

Findings

Single-price algorithm yields a (1+ε) log m approximation.

Problem is hard to approximate within certain logarithmic factors.

Polynomial-time algorithm for Stackelberg bipartite vertex cover.

Abstract

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of $m$ priceable edges in a graph. The other edges have a fixed cost. Based on the leader's decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leader's prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions, and the problem is to find revenue maximizing prices. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a $(1+\epsilon) \log m$-approximation for any $\epsilon >0$. This can be extended to provide a $(1+\epsilon)(\log k + \log m)$-approximation for the general problem and $k$ followers. The latter result is essentially best possible, as the problem is shown to be hard to approximate within $\mathcal{O(\log^\epsilon k + \log^\epsilon m)$. If followers have demands, the single-price algorithm provides a $(1+\epsilon)m^2$-approximation, and the problem is hard to approximate within $\mathcal{O(m^\epsilon)$ for some $\epsilon >0$. Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex cover, which is based on non-trivial max-flow and LP-duality techniques. Our results can be extended to provide constant-factor approximations for any constant number of followers.