Tech Reports

ULCS-07-022

Stackelberg Network Pricing Games

Patrick Briest, Martin Hoefer, and Piotr Krysta

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 pricable 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 pricable 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+eps)log m)-approximation for any eps > 0. This can be extended to provide a ((1+eps)(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 O(log^eps k + log^eps m). If followers have demands, the single-price algorithm provides a ((1+eps)m^2)-approximation, and the problem is hard to approximate within O(m^eps) for some eps >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.

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