The Shapley value and the strength of weak players in Big Boss games

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.