Theory and Decision

, Volume 78, Issue 2, pp 273–287 | Cite as

A proportional value for cooperative games with a coalition structure

Article

Abstract

We introduce a solution concept for cooperative games with transferable utility and a coalition structure that is proportional for two-player games. Our value is obtained from generalizing a proportional value for cooperative games with transferable utility (Ortmann 2000) in a way that parallels the extension of the Shapley value to the Owen value. We provide two characterizations of our solution concept, one that employs a property that can be seen as the proportional analog to Myerson’s balanced contribution property; and a second one that relies on a consistency property.

Keywords

Shapley value Owen value Proportional value Consistency 

Notes

Acknowledgments

I would like to thank Sylvain Béal, André Casajus, Philippe Solal, Winfried Hochstättler, Harald Wiese, and an anonymous referee for helpful comments on previous versions of this paper. Financial support from the Université Charles-de-Gaulle—Lille 3 is gratefully acknowledged.

References

  1. Alonso-Meijide, J. M., & Carreras, F. (2011). The proportional coalitional Shapley value. Expert Systems with Applications, 6, 6967–6979.CrossRefGoogle Scholar
  2. Calvo, E., & Gutiérrez, E. (2010). Solidarity in games with a coalition structure. Mathematical Social Sciences, 60, 196–203.CrossRefGoogle Scholar
  3. Calvo, E., Lasaga, J., & Winter, E. (1996). The principle of balanced contributions and hierarchies of cooperation. Mathematical Social Sciences, 31, 171–182.CrossRefGoogle Scholar
  4. Hart, S., & Mas-Colell, A. (1989). Potential, value, and consistency. Econometrica, 57(3), 589–614.CrossRefGoogle Scholar
  5. Khmelnitskaya, A. B., & Driessen, T. S. H. (2003). Semiproportional values for TU games. Mathematical Method, 57, 495–511.Google Scholar
  6. Moulin, H. (1987). Equal or proportional division of a surplus, and other methods. International Journal of Game Theory, 16(3), 161–186.CrossRefGoogle Scholar
  7. Myerson, R. B. (1980). Conference structures and fair allocation rules. International Journal of Game Theory, 9, 169–182.CrossRefGoogle Scholar
  8. Ortmann, K. (2000). The proportional value for positive cooperative games. Mathematical Methods of Operations Research, 51(2), 235–248.CrossRefGoogle Scholar
  9. Owen, G. (1977). Values of games with a priori unions. In R. Henn & O. Moeschlin (Eds.), Essays in mathematical economics & game theory (pp. 76–88). Berlin: Springer.CrossRefGoogle Scholar
  10. Rawls, J. (1955). Two concepts of rules. The Philosophical Review, 64(1), 3–32.CrossRefGoogle Scholar
  11. Segerson, K. (1988). Uncertainty and incentives for nonpoint pollution control. Journal of Environmental Economics and Management, 15, 87–98.CrossRefGoogle Scholar
  12. Shapley, L. S. (1953). A value for \(n\)-person games. In H. Kuhn & A. Tucker (Eds.), Contributions to the theory of games (Vol. II, pp. 307–317). Princeton: Princeton University Press.Google Scholar
  13. Straffin, P., & Heaney, J. P. (1981). Game theory and the Tennessee valley authority. International Journal of Game Theory, 10(1), 35–43.CrossRefGoogle Scholar
  14. van den Brink, R., Levínsky, R., & Zeleny, M. (2007). The balanced solution for cooperative transferable utility games. Tinbergen Institute Discussion Paper TI 2007–073/1.Google Scholar
  15. Vorob’ev, N. N., & Liapounov, A. N. (1998). The proper Shapley value. In L. A. Petrosjan & V. V. Mazalov (Eds.), Game theory and applications IV (pp. 155–159). New York: Nova Science Publishers.Google Scholar
  16. Winter, E. (1992). The consistency and potential for values of games with coalition structure. Games and Economic Behavior, 4, 132–144.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Université de Franche-Comté, CRESE (EA 3190)BesanconFrance
  2. 2.LSI Leipziger Spieltheoretisches InstitutLeipzigGermany
  3. 3.HHL Leipzig Graduate School of ManagementLeipzigGermany

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