Nucleolus, Happy Nucleolus, and Vehicle Routing

TL;DR

This paper explores the computation and properties of the happy nucleolus in cost allocation, compares it with the nucleolus, and develops efficient algorithms and heuristics, especially for vehicle routing games.

Contribution

It provides new insights into the relationship between the happy nucleolus and nucleolus, and introduces efficient methods for computing the happy nucleolus in monotone games.

Findings

The relation between happy nucleolus and nucleolus is complex and counterintuitive.

Efficient algorithms are developed for the separation problem in computing the happy nucleolus.

A practical heuristic for vehicle routing games demonstrates strong performance.

Abstract

We study the recently introduced fair division concept of the happy nucleolus for cost allocation among players in a cooperative game, with special focus on its computation. The happy nucleolus applies the same fairness criterion as the well-established nucleolus but with reduced total value. Still, we show that the relation between the two concepts is quite involved, and intuitive properties do not hold - e.g., the entry of a player in the happy nucleolus can be larger than the entry of the same player in the nucleolus, even for monotone and subadditive games. This refutes conjectures of Meir, Rosenschein and Malizia (2011). Further, we study the separation problem of the linear programs appearing in the MPS scheme for computing the (happy) nucleolus. It includes linear subspace avoidance constraints, which can be handled efficiently for problems with a certain dynamic programming formulation due to K\"ohnemann and Toth (2020). We show how to get rid of these constraints for all monotone games if we allow for an arbitrarily small error of epsilon, thus conserving known approximation guarantees for the same problem without subspace avoidance. Finally, we focus on practical results at the example of vehicle routing games by designing an efficient heuristic based on our previous insights and past work, and demonstrate its power.