Nucleolus, Happy Nucleolus, and Vehicle Routing

TL;DR

The paper investigates computing and relating the nucleolus $\mathcal{N}$ and the happy nucleolus $\mathcal{N}_h$ in cooperative games, with emphasis on monotone and subadditive costs. It shows the relation between the two concepts is nuanced: $\mathcal{N}_h$ need not be dominated by $\mathcal{N}$, nor equal to the nucleolus of the fractional game, and that intuitive conjectures can fail. It studies the separation problem in the Maschler–Peleg–Shapley (MPS) scheme, proving that subspace-avoidance constraints can be removed with arbitrarily small additive error for monotone games and relating this to dynamic-programming approaches. The authors then develop a constraint-generation heuristic for vehicle routing games that leverages the MPS framework and packing-LP solvers, achieving practical, scalable performance with modest average error on sizable instances. Overall, the work provides theoretical counterexamples, algorithmic techniques, and a practical toolset for applying fair cost allocations in routing contexts.

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 Köhnemann 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.

Figures (8)

• Figure 1: Set covering instance proving \ref{['thm:counterexp_domination']}. Edges illustrate sets of size 2. We have $\mathcal{N}_h \equiv \frac{1}{2}$, $\mathcal{N}(p_i)=\frac{7}{15}$ for $i \in \{4,5,6\}$, and $\mathcal{N}(p_i)=\frac{3}{5}$ otherwise. Especially, $\mathcal{N}_h(p_4) > \mathcal{N}(p_4)$.
• Figure 2: A set covering instance with a fractionally dominated set (blue), proving \ref{['thm:frac_cost_function']}. Edges illustrate sets of size 2. Sets with the same cost are drawn in the same color.
• Figure 3: MPS Scheme
• Figure 4: An illustration of the proof of Theorem \ref{['thm:red-subspace-avoiding-coal']}. Players in $\bm{P'}$ are drawn as black points, all others gray. $\bm{S^*}$ is an optimum solution to the subspace-avoiding problem. The solution(s) found by our algorithm are shown in blue.
• Figure 5: Sketched happy nucleolus heuristic for vehicle routing games

Theorems & Definitions (19)

• Definition 1: Cooperative game
• Definition 2: Nucleolus schmeidler1969nucleolus
• Definition 3: Happy nucleolus blauth2024cost
• Definition 4
• Remark 5: Value functions
• Theorem 6: Negative answers to conjectures in meir2011subsidies
• proof
• Definition 7
• Theorem 8
• proof