Big Boss Games represent a specific class of cooperative games where a single veto player, known as the Big Boss, plays a central role in determining resource allocation and maintaining coalition stability. In this paper, we introduce a novel allocation scheme for Big Boss games, based on two classical solution concepts: the Shapley value and the $\tau$-value. This scheme generates a coalitionally stable allocation that effectively accounts for the contributions of weaker players. Specifically, we consider a diagonal of the core that includes the Big Boss's maximum aspirations, the $\tau$-value, and those of the weaker players. From these allocations, we select the one that is closest to the Shapley value, referred to as the Projected Shapley Value allocation (PSV allocation). Through our analysis, we identify a new property of Big Boss games, particularly the relationship between the allocation discrepancies assigned by the $\tau$-value and the Shapley value, with a particular focus on the Big Boss and the other players. Additionally, we provide a new characterization of convexity within this context. Finally, we conduct a statistical analysis to assess the position of the PSV allocation within the core, especially in cases where computing the Shapley value is computationally challenging.

翻译：Big Boss博弈是一类特殊的合作博弈，其中存在一个具有否决权的参与者（称为Big Boss），在资源分配与联盟稳定性维持中扮演核心角色。本文基于沙普利值与$\tau$值这两个经典解概念，为Big Boss博弈提出了一种新颖的分配方案。该方案生成的联盟稳定分配能有效考量弱势参与者的贡献。具体而言，我们考察包含Big Boss最高期望、$\tau$值以及弱势参与者分配的核心对角线，从中选取最接近沙普利值的分配方案，称为投影沙普利值分配（PSV分配）。通过分析，我们揭示了Big Boss博弈的新性质，特别是$\tau$值与沙普利值分配差异之间的关系，并重点关注Big Boss与其他参与者的互动。此外，我们在此框架下给出了凸性的新特征描述。最后，通过统计分析评估了PSV分配在核心中的位置，特别针对沙普利值计算复杂度较高的情况进行了探讨。