Data valuation using Shapley value has emerged as a prevalent research domain
in machine learning applications. However, it is a challenge to address the
role of order in data cooperation as most research lacks such discussion. To
tackle this problem, this paper studies the definition of the partial ordinal
Shapley value by group theory in abstract algebra. Besides, since the
calculation of the partial ordinal Shapley value requires exponential time,
this paper also gives three algorithms for approximating the results. The
Truncated Monte Carlo algorithm is derived from the classic Shapley value
approximation algorithm. The Classification Monte Carlo algorithm and the
Classification Truncated Monte Carlo algorithm are based on the fact that the
data points in the same class provide similar information, then we can
accelerate the calculation by leaving out some data points in each class.

本文研究了抽象代数中通过群论定义偏序 Shapley 价值的概念及其在数据合作中的角色。提出了三种算法来近似计算该价值，其中包括基于经典 Shapley 价值算法的截断蒙特卡罗算法，以及基于同一类中数据点提供类似信息这一事实的蒙特卡罗和截断蒙特卡罗算法，以加快计算速度。