The Complexity of Matching Games: A Survey

TL;DR

This survey reviews the computational complexity of solution concepts in matching games and their variants, highlighting recent advances and applications like kidney exchange.

Contribution

It provides a comprehensive overview of the complexity results for core, nucleolus, and Shapley value in matching games and their extensions.

Findings

Complexity results vary across different matching game variants.

Recent applications include international kidney exchange.

The survey summarizes key open problems in the field.

Abstract

Matching games naturally generalize assignment games, a well-known class of cooperative games. Interest in matching games has grown recently due to some breakthrough results and new applications. This state-of-the-art survey provides an overview of matching games and extensions, such as $b$-matching games and partitioned matching games; the latter originating from the emerging area of international kidney exchange. In this survey we focus on computational complexity aspects of various game-theoretical solution concepts, such as the core, nucleolus and Shapley value, when the input is restricted to a matching game or one if its variants.