This paper by R. J. Aumann and J. H. Dreze explores the relationship between various game-theoretic solution concepts and coalition structures in games with multiple players. The authors establish theorems that connect a given solution notion, defined for a coalition structure $\mathcal{B}$, with the same solution notion applied to games on each coalition within $\mathcal{B}$. These solution concepts include the kernel, nucleolus, bargaining set, value, core, and von Neumann-Morgenstern solution. A notable finding is that a single function, $v_x^*$, plays a central role in five out of these six solution notions, despite their distinct definitions. This function enters theorems in a natural and consistent manner across these different notions, highlighting an unusual and robust phenomenon in game theory. The paper also discusses the historical context of these concepts and provides definitions and analyses for each solution concept, along with examples and a general discussion.This paper by R. J. Aumann and J. H. Dreze explores the relationship between various game-theoretic solution concepts and coalition structures in games with multiple players. The authors establish theorems that connect a given solution notion, defined for a coalition structure $\mathcal{B}$, with the same solution notion applied to games on each coalition within $\mathcal{B}$. These solution concepts include the kernel, nucleolus, bargaining set, value, core, and von Neumann-Morgenstern solution. A notable finding is that a single function, $v_x^*$, plays a central role in five out of these six solution notions, despite their distinct definitions. This function enters theorems in a natural and consistent manner across these different notions, highlighting an unusual and robust phenomenon in game theory. The paper also discusses the historical context of these concepts and provides definitions and analyses for each solution concept, along with examples and a general discussion.