Suppose that $n$ computer devices are to be connected to a network via inhomogeneous Bernoulli trials. The Shapley value of a device quantifies how much the network's value increases due to the participation of that device. Characteristic functions of such games are naturally taken as the belief function (containment function) and Choquet capacity (hitting probability) of a random set (random network of devices). Traditionally, the Shapley value is either calculated as the expected marginal contribution over all possible coalitions (subnetworks), which results in exponential computational complexity, or approximated by the Monte Carlo sampling technique, where the performance is highly dependent on the stochastic sampling process. The purpose of this study is to design deterministic algorithms for games formulated via inhomogeneous Bernoulli trials that approximate the Shapley value in linear or quadratic time, with rigorous error analysis (Sections 3 and 4). Additionally, we provide a review of relevant literature on existing calculation methods in Remark 3.1 and Appendix I. A further goal is to supplement Shapley's original proof by deriving the Shapley value formula using a rigorous approach based on definite integrals and combinatorial analysis. This method explicitly highlights the roles of the Binomial Theorem and the Beta function in the proof, addressing a gap in Shapley's work (Appendix II).

翻译：假设有 $n$ 台计算机设备需要通过非齐次伯努利试验连接到网络。设备的夏普利值量化了该设备参与网络时网络价值的增加程度。此类博弈的特征函数自然地取为随机集（设备的随机网络）的信任函数（包含函数）和 Choquet 容量（命中概率）。传统上，夏普利值要么通过计算所有可能联盟（子网络）上的期望边际贡献来获得，这导致指数级计算复杂度；要么通过蒙特卡洛采样技术近似，其性能高度依赖于随机采样过程。本研究旨在为非齐次伯努利试验构建的博弈设计确定性算法，以线性或二次时间近似夏普利值，并提供严格的误差分析（第 3 节和第 4 节）。此外，我们在备注 3.1 和附录 I 中对现有计算方法的文献进行了综述。进一步的目标是通过基于定积分和组合分析的严格方法推导夏普利值公式，以补充夏普利的原始证明。该方法明确突出了二项式定理和 Beta 函数在证明中的作用，弥补了夏普利工作中的不足（附录 II）。