Hardness of an Asymmetric 2-Player Stackelberg Network Pricing Game

Davide Bilò,Luciano Gualà,Guido Proietti

doi:10.3390/a14010008

Abstract

Consider a communication network represented by a directed graph G=(V,E) of n nodes and m edges. Assume that edges in E are partitioned into two sets: a set C of edges with a fixed non-negative real cost, and a set P of edges whose costs are instead priced by a leader. This is done with the final intent of maximizing a revenue that will be returned for their use by a follower, whose goal in turn is to select for his communication purposes a subnetwork of Gminimizing a given objective function of the edge costs. In this paper, we study the natural setting in which the follower computes a single-source shortest paths tree of G, and then returns to the leader a payment equal to the sum of the selected priceable edges. Thus, the problem can be modeled as a one-round two-player Stackelberg Network Pricing Game, but with the novelty that the objective functions of the two players are asymmetric, in that the revenue returned to the leader for any of her selected edges is not equal to the cost of such an edge in the follower’s solution. As is shown, for any ϵ>0 and unless P=NP, the leader’s problem of finding an optimal pricing is not approximable within n1/2−ϵ, while, if G is unweighted and the leader can only decide which of her edges enter in the solution, then the problem is not approximable within n1/3−ϵ. On the positive side, we devise a strongly polynomial-time O(n)-approximation algorithm, which favorably compares against the classic approach based on a single-price algorithm. Finally, motivated by practical applications, we consider the special cases in which edges in C are unweighted and happen to form two popular network topologies, namely stars and chains, and we provide a comprehensive characterization of their computational tractability.

Highlights

Leader–follower games were introduced by von Stackelberg in 1934 [1], with the aim of modeling heterogeneous markets, namely markets in which one or more players are in a leadership position, and can in practice manipulate the market to their own advantage, by directly influencing the choices of the remaining subjects
We focus on a natural asymmetric SNPG, namely that in which the follower aims at building a single-source shortest paths tree (SPT) of G rooted at a given node r
We turn our attention to the development of an approximation algorithm for the Asymmetric Stackelberg Shortest Paths Tree (ASSPT) game, and we devise a pricing strategy that in O(m + n log n) time returns an

Summary

Introduction

Leader–follower games were introduced by von Stackelberg in 1934 [1], with the aim of modeling heterogeneous markets, namely markets in which one or more players are in a leadership position, and can in practice manipulate the market to their own advantage, by directly influencing the choices of the remaining subjects. The strategic aspect of the game consists of the fact that the follower computes a solution by optimizing an objective (public) function, while the leader has her own objective function, which is, by the way, computed over the solution selected by the follower. The Internet is a vast, pervasive electronic market mainly composed of millions of independent end-users, whose actions are by the way influenced by the owners of physical/logical portions of the network, for instance service providers. Under this perspective, it turns out to be intriguing the problem of analyzing the antagonism emerging between leaders and followers whenever a communication subnetwork must be allocated

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Conclusion