On bargaining position descriptions in non-transferable utility games

Symmetry versus asymmetry


We present a generalization to the Harsanyi solution for non-transferable utility (NTU) games based on non-symmetry among the players. Our notion of non-symmetry is presented by a configuration of weights which correspond to players' relative bargaining power in various coalitions. We show not only that our solution (i.e., the bargaining position solution) generalizes the Harsanyi solution, (and thus also the Shapley value), but also that almost all the non-symmetric generalizations of the Shapley value for transferable utility games known in the literature are in fact bargaining position solutions. We also show that the non-symmetric Nash solution for the bargaining problem is also a special case of our general solution. We use our general representation of non-symmetry to make a detailed comparison of all the recent extensions of the Shapley value using both a direct and an axiomatic approach.

This is a preview of subscription content, access via your institution.


  1. Aumann R (1985) On the Non-Transferable Utility Value, A comment on Roth-Shafer Examples. Econometrica 53, 667–678.

    Google Scholar 

  2. Aumann RJ (1985) An Axiomatization of the Non-Transferable Utility Value. Econometrica 53, 599–612.

    Google Scholar 

  3. Aumann RJ (1986) Rejoinder. Econometrica 54, 981–984.

    Google Scholar 

  4. Harsanyi JC (1963) A simplified Bargaining Model for the Cooperative n-Person Game. International Economic Review 4, 194–220.

    Google Scholar 

  5. Hart S, Kurz M (1983) Endogenous formation of Coalitions. Econometrica 51, 1047–1064.

    Google Scholar 

  6. Hart S, Kurz M Stable Coalition Structures. In: Coalition and Collective Action (1984) MT Holler (ed). Physica-Verlag, Viena, 236–258.

    Google Scholar 

  7. Imai H (1983) On Harsanyi's Solution. International Journal of Game Theory 12, 161–179.

    Google Scholar 

  8. Kalai E, Samet D (1985) Monotonic Solutions to General Cooperative Games. Econometrica 53, 307–328.

    Google Scholar 

  9. Kalai E, Samet D (1988) Weighted Shapley Values. In The Shapley Value Essays in Honaor of Lloyd S. Shapley, AE Roth (ed). Cambridge University Press. 83–99.

  10. Levy A, Mclean R, Weighted Coalition Structure Values. (1988) to apper in The International Journal of Game Theory.

  11. Myerson R (1977) Graphs and Cooperation. Mathematics of Operations Research 2, 225–229.

    Google Scholar 

  12. Osborne JM, Rubinstein A (1990) Bargaining and Markets. Academic Press, Inc.

  13. Owen G (1977) Values of Games with Priori Unions, in Essays in Mathematical Economics and Game Theory R Henn and O Moschlin (ed) Springer-Verlag, New York, 76–88.

    Google Scholar 

  14. Roth AE (1980) Values for Games Without Side Payments. Some difficulties with Current Concepts. Econometrica 48, 457–465.

    Google Scholar 

  15. Roth AE (1986) On the Non-Transferable Utility Value: A reply to Aumann, Econometrica 54, 981–984.

    Google Scholar 

  16. Roth AE (1979) Axiomatic Models of Bargaining. Lecture Notes in Economics and Mathematical Systems. Springer Verlag, Berlin Heidelberg New York.

    Google Scholar 

  17. Shafer W (1980) On the Existence and Interpretation of Value Allocations, Econometrica 48, 467–477.

    Google Scholar 

  18. Shapley LS (1953) A value for n-Person Games. Contribution to the Theory of Games II. Annals of Mathematics Studies, Princeton, 307–317.

  19. Shapley LS (1977) A Comparison of Power Indices and Nosymmetric Generalization. The Rand Paper Series.

  20. Shapley LS (1988) A value for n-Person Games (the same 1953 paper) The Shapley Value, Essays in Honor of Lloyd S Shapley, A Roth (ed) Cambridge University Press. 31–40.

  21. Weber RJ. Probabilistic Values for Games. (1988) in The Shapley Value, Essays in Honor of Lloyd S, Shapley A, Roth E (ed) Cambridge University Press. 101–119.

  22. Winter E (1989) A Value for Cooperative Games with Level Structure of Cooperation International Journal of Games Theory 18, 227–240.

    Google Scholar 

  23. Winter E. On Non Transferable Utility Games with Coalition Structure. The International Journal of Game Theory 1991, 20, 1, pp 53–64.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Winter, E. On bargaining position descriptions in non-transferable utility games. Int J Game Theory 21, 191–211 (1992). https://doi.org/10.1007/BF01245461

Download citation


  • General Solution
  • General Representation
  • Economic Theory
  • Game Theory
  • Detailed Comparison