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Fair Stable Sets of Simple Games

Simple games are abstract representations of voting systems and other group-decision procedures. A stable set—or von Neumann–Morgenstern solution—of a simple game represents a “standard of behavior” that satisfies certain internal and external stability properties. Compound simple games are built out of component games, which are, in turn, “players” of a quotient game. I describe a method to construct fair—or symmetry-preserving—stable sets of compound simple games from fair stable sets of their quotient and components. This method is closely related to the composition theorem of Shapley (1963c), and contributes to the answer of a question that he formulated: What is the set G of simple games that admit a fair stable set? In particular, this method shows that the set G includes all simple games whose factors—or quotients in their “unique factorization” of Shapley (1967)—are in G, and suggests a path to characterize G.