Lipschitz Continuous Allocations for Optimization Games
Soh Kumabe, Yuichi Yoshida
TL;DR
The paper addresses robustness of allocations in cooperative optimization games under perturbations by introducing Lipschitz-continuous allocation schemes and leveraging the $\alpha$-core. It develops polynomial-time algorithms that produce $\left(\tfrac{1}{2}-\epsilon\right)$-approximate core allocations for the Matching Game with Lipschitz constant $O(\epsilon^{-1})$ and a $4$-approximate core allocation for the Minimum Spanning Tree Game with constant Lipschitz, thereby balancing core-approximation with stability. It further analyzes the Shapley value, showing a spectrum of Lipschitz behavior across games: constant for MSTG but $\Omega(\log n)$ for MG, even though Shapley computation can be #P-hard. Together, these results offer a practical framework for perturbation-robust, core-like allocations in graph-based optimization games with clear implications for fairness and resilience in real-world networks.
Abstract
In cooperative game theory, the primary focus is the equitable allocation of payoffs or costs among agents. However, in the practical applications of cooperative games, accurately representing games is challenging. In such cases, using an allocation method sensitive to small perturbations in the game can lead to various problems, including dissatisfaction among agents and the potential for manipulation by agents seeking to maximize their own benefits. Therefore, the allocation method must be robust against game perturbations. In this study, we explore optimization games, in which the value of the characteristic function is provided as the optimal value of an optimization problem. To assess the robustness of the allocation methods, we use the Lipschitz constant, which quantifies the extent of change in the allocation vector in response to a unit perturbation in the weight vector of the underlying problem. Thereafter, we provide an algorithm for the matching game that returns an allocation belonging to the $\left(\frac{1}{2}-ε\right)$-approximate core with Lipschitz constant $O(ε^{-1})$. Additionally, we provide an algorithm for a minimum spanning tree game that returns an allocation belonging to the $4$-approximate core with a constant Lipschitz constant. The Shapley value is a popular allocation that satisfies several desirable properties. Therefore, we investigate the robustness of the Shapley value. We demonstrate that the Lipschitz constant of the Shapley value for the minimum spanning tree is constant, whereas that for the matching game is $Ω(\log n)$, where $n$ denotes the number of vertices.