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Values for two-stage games: Another view of the Shapley axioms
Authors:A. Beja  I. Gilboa
Affiliation:1. Tel Aviv University, Israel
2. Massachusetts Institute of Technology, USA
3. Department of Managerial Economics and Decision Sciences, J. L. Kellogg Graduate School of Management, Northwestern University, USA
Abstract:This short study reports an application of the Shapley value axioms to a new concept of ldquotwo-stage games.rdquo 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 ldquotwo-stage majority gamesrdquo (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 ldquolambda-transfer valuerdquo 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 ldquomulti-stage games.rdquoEarlier versions of this study were presented at the International Conference on Game Theory and its Applications, organized by Ohio State University in 1987, and at the Workshop on Mathematical Economics and Game Theory at Tel Aviv Unversity. We gratefully acknowledge the valuable comments received on both occasions, especially those of Robert J. Aumann, Roy Gardner, Sergiu Hart, Ehud Kalai, Michael Maschler, Alvin E. Roth, and Lloyd S. Shapley, and also those ofIJGT's anonymous referees. Of course, all responsibility lies with us.
Keywords:Shapley Values  Shapley Axioms  Majority Games  Two-Stage Games
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