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Annals of Operations Research

, Volume 264, Issue 1–2, pp 41–56 | Cite as

Efficient extensions of communication values

  • Sylvain Béal
  • André Casajus
  • Frank Huettner
Original Paper
  • 134 Downloads

Abstract

We study values for transferable utility games enriched by a communication graph. The most well-known such values are component-efficient and characterized by some deletion link property. We study efficient extensions of such values: for a given component-efficient value, we look for a value that (i) satisfies efficiency, (ii) satisfies the link-deletion property underlying the original component-efficient value, and (iii) coincides with the original component-efficient value whenever the underlying graph is connected. Béal et al. (Soc Choice Welf 45:819–827, 2015) prove that the Myerson value (Myerson in Math Oper Res 2:225–229, 1977) admits a unique efficient extension, which has been introduced by van den Brink et al. (Econ Lett 117:786–789, 2012). We pursue this line of research by showing that the average tree solution (Herings et al. in Games Econ Behav 62:77–92, 2008) and the compensation solution (Béal et al. in Int J Game Theory 41:157–178, 2012b) admit similar unique efficient extensions, and that there exists no efficient extension of the position value (Meessen in Communication games, 1988; Borm et al. in SIAM J Discrete Math 5:305–320, 1992). As byproducts, we obtain new characterizations of the average tree solution and the compensation solution, and of their efficient extensions.

Keywords

Efficient extension Average tree solution Compensation solution Position value Component fairness Relative fairness Balanced link contributions Myerson value Component-wise egalitarian solution 

Mathematics Subject Classification

91A12 

JEL Classification

C71 

References

  1. Aumann, R. J., & Dreze, J. H. (1974). Cooperative games with coalition structures. International Journal of Game Theory, 3, 217–237.CrossRefGoogle Scholar
  2. Béal, S., Casajus, A., & Huettner, F. (2015). Efficient extensions of the Myerson value. Social Choice and Welfare, 45, 819–827.CrossRefGoogle Scholar
  3. Béal, S., Casajus, A., & Huettner, F. (2016). On the existence of efficient and fair extensions of communication values for connected graphs. Economics Letters, 146, 103–106.CrossRefGoogle Scholar
  4. Béal, S., Lardon, A., Rémila, E., & Solal, P. (2012a). The average tree solution for multichoice forest games. Annals of Operations Research, 196, 27–51.CrossRefGoogle Scholar
  5. Béal, S., Rémila, E., & Solal, P. (2012b). Compensations in the Shapley value and the compensation solutions for graph games. International Journal of Game Theory, 41, 157–178.CrossRefGoogle Scholar
  6. Béal, S., Rémila, E., & Solal, P. (2012c). Fairness and fairness for neighbors: The difference between the Myerson value and component-wise egalitarian solutions. Economics Letters, 117, 263–267.CrossRefGoogle Scholar
  7. Borm, P., Owen, G., & Tijs, S. (1992). On the position value for communication situations. SIAM Journal on Discrete Mathematics, 5, 305–320.CrossRefGoogle Scholar
  8. Casajus, A. (2007). An efficient value for TU games with a cooperation structure. Working Paper, Universität Leipzig, Germany.Google Scholar
  9. Demange, G. (2004). On group stability in hierarchies and networks. Journal of Political Economy, 112, 754–778.CrossRefGoogle Scholar
  10. Ghintran, A., González-Arangüena, E., & Manuel, C. (2012). A probabilistic position value. Annals of Operations Research, 201, 183–196.CrossRefGoogle Scholar
  11. Hamiache, G. (2012). A matrix approach to TU games with coalition and communication structures. Social Choice and Welfare, 38, 85–100.CrossRefGoogle Scholar
  12. Hart, S., & Kurz, M. (1983). Endogenous formation of coalitions. Econometrica, 51, 1047–1064.CrossRefGoogle Scholar
  13. Herings, P. J.-J., van der Laan, G., & Talman, A. J. J. (2008). The average tree solution for cycle-free graph games. Games and Economic Behavior, 62, 77–92.CrossRefGoogle Scholar
  14. Khmelnitskaya, A. B. (2014). Values for games with two-level communication structures. Discrete Applied Mathematics, 166, 34–50.CrossRefGoogle Scholar
  15. Meessen, R. (1988). Communication games. Master’s thesis, Department of Mathematics, University of Nijmegen, The Netherlands (in Dutch).Google Scholar
  16. Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of Operations Research, 2, 225–229.CrossRefGoogle Scholar
  17. Owen, G. (1977). Values of games with a priori unions. In R. Henn & O. Moeschlin (Eds.), Essays in mathematical economics and game theory (pp. 76–88). Berlin: Springer.CrossRefGoogle Scholar
  18. Shapley, L. S. (1953). A value for \(n\)-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Contribution to the theory of games, vol II. Annals of mathematics studies 28. Princeton: Princeton University Press.Google Scholar
  19. Slikker, M. (2005). A characterization of the position value. International Journal of Game Theory, 33, 505–514.CrossRefGoogle Scholar
  20. Slikker, M. (2007). Bidding for surplus in network allocation problems. Journal of Economic Theory, 137, 493–511.CrossRefGoogle Scholar
  21. van den Brink, R., Khmelnitskaya, A. B., & van der Laan, G. (2012). An efficient and fair solution for communication graph games. Economics Letters, 117, 786–789.CrossRefGoogle Scholar
  22. Zhang, G., Shan, E., Kang, L., & Dong, Y. (2017). Two efficient values of cooperative games with graph structure based on \(\tau \)-values. Journal of Combinatorial Optimization, 34, 462–482.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.CRESE EA3190Univ. Bourgogne Franche-ComtéBesançonFrance
  2. 2.HHL Leipzig Graduate School of ManagementLeipzigGermany
  3. 3.ESMT European School of Management and TechnologyBerlinGermany

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