Lipschitz Continuous Allocations for Optimization Games

TL;DR

This paper investigates robust allocation methods in cooperative optimization games, focusing on Lipschitz continuity to ensure stability against game perturbations, and provides algorithms with provable bounds for specific game types.

Contribution

It introduces algorithms for matching and spanning tree games with Lipschitz bounds and analyzes the robustness of the Shapley value in these contexts.

Findings

Matching game allocation has Lipschitz constant O(ε^{-1})

Spanning tree game allocation has constant Lipschitz constant

Shapley value's Lipschitz constant varies, being constant for spanning trees and logarithmic for matchings.

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 $\left(\frac{1}{2}-\epsilon\right)$-approximate core with Lipschitz constant $O(\epsilon^{-1})$. 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 $\Omega(\log n)$, where $n$ denotes the number of vertices.