Pure Bargaining Problems and the Shapley Rule



Pure bargaining problems with transferable utility are considered. By associating a quasi-additive cooperative game with each one of them, a Shapley rule for this class of problems is derived from the Shapley value for games. The analysis of this new rule includes axiomatic characterizations and a comparison with the proportional rule.



Research partially supported by Grants SGR 2009–01029 of the Catalonia Government (Generalitat de Catalunya), and MTM 2012–34426 of the Economy and Competitiveness Spanish Ministry.


  1. Aadland, D., & Kolpin, V. (1998). Shared irrigation costs: An empirical and axiomatic analysis. Mathematical Social Sciences, 849, 203–218.CrossRefGoogle Scholar
  2. Bergantiños, G., & Vidal-Puga, J. J. (2007). The optimistic TU game in minimum cost spanning tree problems. International Journal of Game Theory, 36, 223–239.CrossRefGoogle Scholar
  3. van den Brink, R., & Funaki, Y. (2009). Axiomatizations of a class of equal surplus sharing solutions for TU-games. Theory and Decision, 67, 303–340.CrossRefGoogle Scholar
  4. Carreras, F., & Freixas, J. (2000). A note on regular semivalues. International Game Theory Review, 2, 345–352.CrossRefGoogle Scholar
  5. Driessen, T. S. H., & Funaki, Y. (1991). Coincidence of and collinearity between game theoretic solutions. OR Spektrum, 13, 15–30.CrossRefGoogle Scholar
  6. Gillies, D. B. (1953). Some theorems on n-person games. Ph.D. Thesis. Princeton University.Google Scholar
  7. Hart, S., & Mas-Colell, A. (1988). The potential of the shapley value. In A. E. Roth (Ed.), The shapley value: essays in honor of Lloyd S. Shapley (pp. 127–137). Cambridge: Cambridge University Press.Google Scholar
  8. Hart, S., & Mas-Colell, A. (1989). Potential, value and consistency. Econometrica, 57, 589–614.CrossRefGoogle Scholar
  9. Isbell, J. R. (1956). A class of majority games. Quarterly Journal of Mathematics, 7, 183–187.CrossRefGoogle Scholar
  10. Kalai, E., & Smorodinsky, M. (1975). Alternative solutions to Nash’s bargaining problem. Econometrica, 43, 513–518.CrossRefGoogle Scholar
  11. Kalai, E. (1977). Proportional solutions to bargaining situations: Interpersonal utility comparisons. Econometrica, 45, 1623–1630.CrossRefGoogle Scholar
  12. Megiddo, N. (1974). On the nonmonotonicity of the bargaining set, the kernel, and the nucleolus of a game. SIAM Journal of Applied Mathematics, 27, 355–358.CrossRefGoogle Scholar
  13. Nash, J. F. (1950). The bargaining problem. Econometrica, 18, 155–162.CrossRefGoogle Scholar
  14. von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University Press.Google Scholar
  15. Ortmann, K. M. (2000). The proportional value for positive cooperative games. Mathematical Methods of Operations Research, 51, 235–248.CrossRefGoogle Scholar
  16. Owen, G. (1995). Game theory. New York : Academic Press.Google Scholar
  17. Roth, A. E. (Ed.). (1988). The shapley vlue: Essays in honor of Lloyd S. Shapley: Cambridge University Press.Google Scholar
  18. Shapley, L. S. (1953). A value for \(n\)-person games. Annals of Mathematical Studies, 28, 307–317.Google Scholar
  19. Shapley, L. S. (1971). Cores of convex games. International Journal of Game Theory, 1, 11–26.CrossRefGoogle Scholar
  20. Shubik, M. (1962). Incentives, decentralized control, the assignment of joint costs, and internal pricing. Management Science, 8, 325–343.CrossRefGoogle Scholar
  21. Young, H. P. (1985). Monotonic solutions of cooperative games. International Journal of Game Theory, 14, 65–72.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Applied Mathematics II and Industrial and Aeronautical Engineering School of TerrassaTechnical University of CataloniaCataloniaSpain
  2. 2.ETSEIATTerrassaSpain
  3. 3.Department of MathematicsNaval Postgraduate SchoolMontereyUSA

Personalised recommendations