Review of Economic Design

, Volume 16, Issue 4, pp 269–282 | Cite as

Nash bargaining in ordinal environments

Original Paper
First Online:
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Accepted:

Abstract

We analyze the implications of Nash’s (Econometrica 18:155–162, 1950) axioms in ordinal bargaining environments; there, the scale invariance axiom needs to be strenghtened to take into account all order-preserving transformations of the agents’ utilities. This axiom, called ordinal invariance, is a very demanding one. For two-agents, it is violated by every strongly individually rational bargaining rule. In general, no ordinally invariant bargaining rule satisfies the other three axioms of Nash. Parallel to Roth (J Econ Theory 16:247–251, 1977), we introduce a weaker independence of irrelevant alternatives (IIA) axiom that we argue is better suited for ordinally invariant bargaining rules. We show that the three-agent Shapley–Shubik bargaining rule uniquely satisfies ordinal invariance, Pareto optimality, symmetry, and this weaker IIA axiom. We also analyze the implications of other independence axioms.

Keywords

Bargaining Shapley–Shubik rule Ordinal invariance Independence of irrelevant alternatives Brace 

JEL Classification 

C78 
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References

  1. Bennett E (1997) Multilateral bargaining problems. Games Econ Behav 19:151–179Google Scholar
  2. Kalai E, Smorodinsky M (1975) Other solutions to Nash’s bargaining problem. Econometrica 43:513–518Google Scholar
  3. Kalai E (1977) Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45:1623–1630CrossRefGoogle Scholar
  4. Kıbrıs Ö (2004a) Ordinal invariance in multicoalitional bargaining. Games Econ Behav 46(1):76–87CrossRefGoogle Scholar
  5. Kıbrıs Ö (2004b) Egalitarianism in Ordinal Bargaining: the Shapley-Shubik Rule. Games Econ Behav 49(1):157–170CrossRefGoogle Scholar
  6. Luce RD, Raiffa H (1957) Games and decisions. Wiley, New YorkGoogle Scholar
  7. Nash JF (1950) The bargaining problem. Econometrica 18:155–162CrossRefGoogle Scholar
  8. Roth AE (1977) Independence of irrelevant alternatives and solutions to Nash’s bargaining problem. J Econ Theory 16:247–251CrossRefGoogle Scholar
  9. Roth AE (1979) Axiomatic models of bargaining. Springer, BerlinCrossRefGoogle Scholar
  10. Safra Z, Samet D (2004) An ordinal solution to bargaining problems with many players. Games Econ Behav 46:129–142CrossRefGoogle Scholar
  11. Safra Z, Samet D (2005) A family of ordinal solutions to bargaining problems with many players. Games Econ Behav 50:89–106CrossRefGoogle Scholar
  12. Shapley L (1969) Utility comparison and the theory of games. La décision: agrégation et dynamique des ordres de préférence, PARIS: Editions du CNRS pp 251–263Google Scholar
  13. Shapley L, Shubik M (1974) Game theory in economics—chapter 4. preferences and utility. Rand Corporation Report R-904/4Google Scholar
  14. Shubik M (1982) Game theory in the social sciences. MIT Press, CambridgeGoogle Scholar
  15. Sprumont Y (2000) A note on ordinally equivalent Pareto surfaces. J Math Econ 34:27–38CrossRefGoogle Scholar
  16. Thomson W (forthcoming) Bargaining theory: the axiomatic approach. Academic PressGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Faculty of Arts and Social SciencesSabancı UniversityIstanbulTurkey

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