Approximation of the Shapley value for the Euclidean travelling salesman game

The travelling salesman problem (TSP) consists of finding a minimal route to serve a set customers using one vehicle. This naturally leads to the problem of finding a fair way to subdivide the overall cost of a trip between all participating customers. The Shapley value associated with the Euclidean travelling salesman game (TSG) has been proven to provide a solution to the fair cost allocation problem that satisfies several very important axiomatic properties. Unfortunately the calculation of the Shapley value involves high computational complexity, which makes it impractical for many real applications. This has led to substantial research effort dedicated to finding approximations with lower computational complexity. We develop a novel methodology of approximating the Shapley value of the Euclidean TSG, which is inspired by an extension of the 1D case. From this we derive two approximation methods having polynomial computational complexity, and also indicate how they could, in principle, be further refined. We provide experimental results which show that our proposed approximations compare favorably with the state-of-the-art approximations of the Euclidean TSG found in the literature.