Skip to main content
SpringerLink
Log in

The collective value: a new solution for games with coalition structures

  • Original Paper
  • Published:
TOP Aims and scope Submit manuscript

Abstract

In this study, we provide a new solution for cooperative games with coalition structures. The collective value of a player is defined as the sum of the equal division of the pure surplus obtained by his coalition from the coalitional bargaining and of his Shapley value for the internal coalition. The weighted Shapley value applied to a game played by coalitions with coalition-size weights is assigned to each coalition, reflecting the size asymmetries among coalitions. We show that the collective value matches exogenous interpretations of coalition structures and provide an axiomatic foundation of this value. A noncooperative mechanism that implements the collective value is also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Amer R, Carreras F (1995) Cooperation indices and coalitional value. Top 3:117–135

    Article  Google Scholar 

  • Aumann RJ, Dreze JH (1974) Cooperative games with coalition structures. Int J Game Theory 3:217–237

    Article  Google Scholar 

  • Calvo E, Lasaga JJ, Winter E (1996) The principle of balanced contributions and hierarchies of cooperation. Math Soc Sci 31:171–182

    Article  Google Scholar 

  • Casajus A (2009) Outside options, component efficiency, and stability. Games Econ Behav 65:49–61

    Article  Google Scholar 

  • Gómez-Rúa M, Vidal-Puga J (2010a) Balanced per capita contributions and level structure of cooperation. Top. doi:10.1007/s11750-009-0122-3

    Google Scholar 

  • Gómez-Rúa M, Vidal-Puga J (2010b) The axiomatic approach to three values in games with coalition structure. Eur J Oper Res 207:795–806

    Article  Google Scholar 

  • Hamiache G (2006) A value for games with coalition structures. Soc Choice Welf 26:93–105

    Article  Google Scholar 

  • Hart S, Kurz M (1983) Endogenous formation of coalitions. Econometrica 51:1047–1064

    Article  Google Scholar 

  • Hart S, Mas-Colell A (1989) Potential, value and consistency. Econometrica 57:589–614

    Article  Google Scholar 

  • Kalai E (1977) Nonsymmetric Nash solutions and replications of 2-person bargaining. Int J Game Theory 6:129–133

    Article  Google Scholar 

  • Kalai E, Samet D (1987) On weighted Shapley values. Int J Game Theory 16:205–222

    Article  Google Scholar 

  • Kamijo Y (2008) Implementation of weighted values in hierarchical and horizontal cooperation structures. Math Soc Sci 56:336–349

    Article  Google Scholar 

  • Kamijo Y (2009) A two-step Shapley value for cooperative games with coalition structures. Int Game Theory Rev 11:207–214

    Article  Google Scholar 

  • Levy A, McLean RP (1989) Weighted coalition structure values. Games Econ Behav 1:234–249

    Article  Google Scholar 

  • Myerson RB (1977) Graphs and cooperation in games. Math Oper Res 2:225–229

    Article  Google Scholar 

  • Myerson RB (1980) Conference structures and fair allocation rules. Int J Game Theory 9:169–182

    Article  Google Scholar 

  • Owen G (1977) Values of games with a priori unions. In: Henn R, Moeschlin O (eds) Essays in mathematical economics and game theory. Springer, Berlin, pp 76–88

    Chapter  Google Scholar 

  • Peleg B, Sudhölter P (2003) Introduction to the theory of cooperative games. Kluwer Academic, Boston

    Book  Google Scholar 

  • Pérez-Castrillo D, Wettstein D (2001) Bidding for the surplus: a non-cooperative approach to the Shapley value. J Econ Theory 100:274–294

    Article  Google Scholar 

  • Schmeidler D (1969) The nucleolus of a characteristic function Game. SIAM Journal of Applied Mathematics 1163–1169

  • Shapley LS (1953a) Additive and non-additive set functions. PhD thesis, Princeton University

  • Shapley LS (1953b) A value for n-person game. In: Kuhn H, Tucker A (eds) Contributions to the theory of games, vol 2. Princeton University Press, Princeton, pp 307–317

    Google Scholar 

  • Thomson W (1986) Replication invariance of bargaining solutions. Int J Game Theory 15:59–63

    Article  Google Scholar 

  • van den Brink R, van der Laan G (2005) A class of consistent share functions for games in coalition structure. Games Econ Behav 51:193–212

    Article  Google Scholar 

  • Vidal-Puga JJ (2005) The Harsanyi paradox and the “right to talk” in bargaining among coalitions. Working Paper

  • Vidal-Puga JJ, Bergantiños G (2003) An implementation of the Owen value. Games Econ Behav 44:412–427

    Article  Google Scholar 

  • Winter E (1989) A value for cooperative games with levels structures of cooperation. Int J Game Theory 18:227–240

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Waseda Institute for Advanced Study, 1-6-1, Nishiwaseda, Shunjuku-ku, Tokyo, Japan

    Yoshio Kamijo

Authors
  1. Yoshio Kamijo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshio Kamijo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kamijo, Y. The collective value: a new solution for games with coalition structures. TOP 21, 572–589 (2013). https://doi.org/10.1007/s11750-011-0191-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11750-011-0191-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation