Values for two-stage games: Another view of the Shapley axioms

This short study reports an application of the Shapley value axioms to a new concept of "two-stage games." In these games, the formation of a coalition in the first stage entitles its members to play a prespecified cooperative game at the second stage. The original Shapley axioms have natural equivalents in the new framework, and we show the existence of (non-unique) values and semivalues for two stage games, analogous to those defined by the corresponding axioms for the conventional (one-stage) games. However, we also prove that all semivalues (hence, perforce, all values) must give patently unacceptable solutions for some "two-stage majority games" (where the members of a majority coalition play a conventional majority game). Our reservations about these prescribed values are related to Roth's (1980) criticism of Shapley's "λ-transfer value" for non-transferable utility (NTU) games. But our analysis has wider scope than Roth's example, and the argument that it offers appears to be more conclusive. The study also indicates how the values and semivalues for two-stage games can be naturally generalized to apply for "multi-stage games."