Stackelberg Network Pricing Games

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. 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+ϵ)log m‐approximation. This can be extended to provide a (1+ϵ)(log k+log m)‐approximation for the general problem and k followers. The problem is also shown to be hard to approximate within $𝒪({\log }^{\epsilon }k+{\log }^{\epsilon }m)$ for some ϵ>0. If followers have demands, the single‐price algorithm provides an $𝒪(m^2)$ ‐approximation, and the problem is hard to approximate within $𝒪(m^{ϵ})$ for some ϵ>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. This approach can be extended to provide constant‐factor approximations for any constant number of followers.