Cooperative Games with Bounded Dependency Degree

TL;DR

This paper explores how bounding the dependency and supermodular degrees in cooperative games can make complex computational problems more tractable, providing efficient algorithms for certain classes of games.

Contribution

It introduces the use of dependency and supermodular degrees to analyze the complexity of cooperative games, showing tractability results for bounded degrees.

Findings

Efficient algorithms for simple games with small supermodular degree

Shapley value computation is fixed-parameter tractable with respect to dependency degree

Special classes of games allow efficient dependency determination

Abstract

Cooperative games provide a framework to study cooperation among self-interested agents. They offer a number of solution concepts describing how the outcome of the cooperation should be shared among the players. Unfortunately, computational problems associated with many of these solution concepts tend to be intractable---NP-hard or worse. In this paper, we incorporate complexity measures recently proposed by Feige and Izsak (2013), called dependency degree and supermodular degree, into the complexity analysis of cooperative games. We show that many computational problems for cooperative games become tractable for games whose dependency degree or supermodular degree are bounded. In particular, we prove that simple games admit efficient algorithms for various solution concepts when the supermodular degree is small; further, we show that computing the Shapley value is always in FPT with respect to the dependency degree. Finally, we note that, while determining the dependency among players is computationally hard, there are efficient algorithms for special classes of games.