Stackelberg Network Pricing Games

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