Hardness of an Asymmetric 2-Player Stackelberg Network Pricing Game

• Main Authors: Davide Bilò, Luciano Gualà, Guido Proietti
• Format: Article
• Language: English
• Published: MDPI AG 2021-12-01
• Series: Algorithms
• Subjects: communication networks; shortest paths tree; stackelberg games; network pricing games
• Online Access: https://www.mdpi.com/1999-4893/14/1/8

Consider a communication network represented by a directed graph <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></semantics></math></inline-formula> of <i>n</i> nodes and <i>m</i> edges. Assume that edges in <i>E</i> are partitioned into two sets: a set <i>C</i> of edges with a fixed non-negative real cost, and a set <i>P</i> of edges whose costs are instead priced by a <i>leader</i>. This is done with the final intent of <i>maximizing</i> a revenue that will be returned for their use by a <i>follower</i>, whose goal in turn is to select for his communication purposes a subnetwork of <i>Gminimizing</i> a given objective function of the edge costs. In this paper, we study the natural setting in which the follower computes a <i>single-source shortest paths tree</i> of <i>G</i>, and then returns to the leader a payment equal to the <i>sum</i> of the selected priceable edges. Thus, the problem can be modeled as a one-round two-player <i>Stackelberg Network Pricing Game</i>, but with the novelty that the objective functions of the two players are <i>asymmetric</i>, 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 <inline-formula><math display="inline"><semantics><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and unless <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="sans-serif">P</mi><mo>=</mo><mi mathvariant="sans-serif">NP</mi></mrow></semantics></math></inline-formula>, the leader’s problem of finding an optimal pricing is not approximable within <inline-formula><math display="inline"><semantics><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>−</mo><mi>ϵ</mi></mrow></msup></semantics></math></inline-formula>, while, if <i>G</i> is unweighted and the leader can only decide which of her edges enter in the solution, then the problem is not approximable within <inline-formula><math display="inline"><semantics><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn><mo>−</mo><mi>ϵ</mi></mrow></msup></semantics></math></inline-formula>. On the positive side, we devise a <i>strongly</i> polynomial-time <inline-formula><math display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-approximation algorithm, which favorably compares against the classic approach based on a <i>single-price</i> algorithm. Finally, motivated by practical applications, we consider the special cases in which edges in <i>C</i> are unweighted and happen to form two popular network topologies, namely <i>stars</i> and <i>chains</i>, and we provide a comprehensive characterization of their computational tractability.