A Linear Programming Approach to Estimate the Core in Cooperative Games
J Camacho, JC Gonçalves-Dosantos, J Sánchez-Soriano
TL;DR
This work tackles the computational challenge of the core in TU games by introducing an iterative linear-programming framework that approximates the core through extreme-point sampling. By solving $k$ LPs with randomly or deterministically chosen objective vectors, it collects extreme points and forms their convex hull, guaranteeing the estimated set is a subset of the true core and converges to the core as $k$ grows. The authors formalize the approach, analyze its polynomial-time complexity for fixed dimensionality, and evaluate it on diverse game models, showing high core-volume recovery (often >95%) with modest runtime (seconds) for up to 13 players. The method provides a practical, deterministic alternative to sampling-based relaxations, facilitates checking core membership for candidate allocations, and supports bounded-rationality considerations in cooperative decision-making.
Abstract
This paper proposes a novel algorithm to approximate the core of transferable utility (TU) cooperative games via linear programming. Given the computational hardness of determining the full core, our approach provides a tractable approximation by sampling extreme points through randomized linear problems (LPs). We analyze its convergence and computational complexity, and validate its effectiveness through extensive simulations on various game models. Our results show that the method is scalable and achieves high accuracy in terms of core reconstruction.