[Paper Review] Strategic Cooperation in Cost Sharing Games

Martin Hoefer|arXiv (Cornell University)|Mar 16, 2010

Game Theory and Voting Systems36 references4 citations

TL;DR

This paper establishes a tight link between strong equilibria in strategic cost sharing games and the core in coalitional cost sharing games, showing that the existence of strong equilibria is characterized by the integrality gap of underlying LP formulations. It provides a unified framework to derive exact and approximate strong equilibria for set cover, facility location, and network design games via primal-dual algorithms, proving the strong price of anarchy is always 1.

ABSTRACT

In this paper we consider strategic cost sharing games with so-called arbitrary sharing based on various combinatorial optimization problems, such as vertex and set cover, facility location, and network design problems. We concentrate on the existence and computational complexity of strong equilibria, in which no coalition can improve the cost of each of its members. Our main result reveals a connection between strong equilibrium in strategic games and the core in traditional coalitional cost sharing games studied in economics. For set cover and facility location games this results in a tight characterization of the existence of strong equilibrium using the integrality gap of suitable linear programming formulations. Furthermore, it allows to derive all existing results for strong equilibria in network design cost sharing games with arbitrary sharing via a unified approach. In addition, we are able to show that in general the strong price of anarchy is always 1. This should be contrasted with the price of anarchy of Θ(n) for Nash equilibria. Finally, we indicate that the LP-approach can also be used to compute near-optimal and near-stable approximate strong equilibria.

Motivation & Objective

• To characterize the existence of strong equilibria in strategic cost sharing games based on combinatorial optimization problems.
• To establish a formal connection between strong equilibria in strategic games and the core in traditional coalitional cost sharing games.
• To develop a unified LP-based approach for computing approximate strong equilibria in games like set cover, facility location, and network design.
• To analyze the strong price of anarchy and show it is always 1, in contrast to the Θ(n) price of anarchy for Nash equilibria.
• To extend the applicability of primal-dual algorithms to derive constant-factor approximate strong equilibria in polynomial time.

Proposed method

• Leverages linear programming duality and integrality gap analysis to characterize the existence of strong equilibria in strategic cost sharing games.
• Uses the core of coalitional cost sharing games as a benchmark to determine whether strong equilibria exist in strategic settings.
• Applies primal-dual algorithms to compute approximate strong equilibria, with performance guarantees based on LP approximation ratios.
• Reduces the problem of finding strong equilibria to solving underlying combinatorial optimization problems via LP formulations.
• Demonstrates that deviations by coalitions can be analyzed through equivalent unilateral deviations in single-player subgames, preserving stability guarantees.
• Uses the structure of the generalized linear production model to explore broader applicability beyond the current classes of games.

Experimental results

Research questions

• RQ1 Under what conditions does a strong equilibrium exist in strategic cost sharing games based on combinatorial optimization problems?
• RQ2 How is the existence of strong equilibria related to the integrality gap of the underlying LP formulation?
• RQ3 Can the core of a coalitional cost sharing game be used to characterize or compute strong equilibria in a strategic setting?
• RQ4 What is the strong price of anarchy in strategic cost sharing games, and how does it compare to the price of anarchy in Nash equilibria?
• RQ5 Can primal-dual algorithms be adapted to compute approximate strong equilibria with bounded approximation ratios?

Key findings

• The existence of strong equilibria in set cover and facility location games is tightly characterized by the integrality gap of their respective LP formulations.
• The strong price of anarchy is always 1 in strategic cost sharing games, a significant improvement over the Θ(n) price of anarchy in Nash equilibria.
• For vertex cover games, even with an integrality gap of 1 and a non-empty core, strong equilibria may still fail to exist due to strategic incentives.
• Primal-dual algorithms can compute (2,2)-strong equilibria for vertex cover, (f,f)-strong equilibria for set cover (where f is the maximum frequency), and (3,3)-strong equilibria for metric uncapacitated facility location games in polynomial time.
• The equivalence between strong equilibria and core solutions allows a unified approach to derive results across multiple game classes, including network design games.
• The LP-based framework can be extended to approximate strong equilibria, ensuring stability and efficiency in settings where exact strong equilibria do not exist.