William Hamilton developed the biological theory of kin selection before game theory became familiar to biologists. Thus he implicitly confined his analysis to a rather special subclass of games, with linear structure. This paper shows that while Hamilton's rule does not apply to a more general class of games, there is a useful generalization that does apply. This paper also generalizes results in my 1995 AER paper on sibling interaction from symmetric two-player games to multiplayer games that may be asymmetric.