[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Minimum norm solutions for cooperative games

  • Original Paper
  • Published:
International Journal of Game Theory Aims and scope Submit manuscript

Abstract

A solution f for cooperative games is a minimum norm solution, if the space of games has a norm such that f(v) minimizes the distance (induced by the norm) between the game v and the set of additive games. We show that each linear solution having the inessential game property is a minimum norm solution. Conversely, if the space of games has a norm, then the minimum norm solution w.r.t. this norm is linear and has the inessential game property. Both claims remain valid also if solutions are required to be efficient. A minimum norm solution, the least square solution, is given an axiomatic characterization.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Charnes A, Rousseau J, Seiford L (1978) Complements, mollifiers and the propensity to disrupt. Int J Game Theory 7:37–50

    Article  Google Scholar 

  • Dubey P, Neyman A, Weber J (1981) Value theory without efficiency. Math Oper Res 4:99–131

    Google Scholar 

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

    Article  Google Scholar 

  • Keane M (1969) Some topics in n-person game theory. Ph.D. dissertation, Northwestern University, Evanston

  • Kleinberg NL, Weiss JH (1985) A new formula for the Shapley value. Econ Lett 17:311–315

    Article  Google Scholar 

  • Kleinberg NL, Weiss JH (1986) The orthogonal decomposition of games and an averaging formula for the Shapley value. Math Oper Res 11:117–124

    Article  Google Scholar 

  • Kleinberg NL, Weiss JH (1988) The orthogonal decomposition of games and an averaging formula for the Shapley value: a correction. Math Oper Res 13:190

    Article  Google Scholar 

  • Luenberger D (1969) Optimization by vector space methods. Wiley, New York

    Google Scholar 

  • Maschler M, Peleg B (1996) A characterization, existence proof and dimension bounds for the kernel of a game. Pac J Math 18:289–328

    Google Scholar 

  • Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, Berlin

    Google Scholar 

  • Rothblum UG (1985) A simple proof for the Kleinberg-Weiss representation of the Shapley value. Econ Lett 19:137–139

    Article  Google Scholar 

  • Ruiz LM, Valenciano F, Zarzuelo FM (1996) The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector. Int J Game Theory 25:113–134

    Article  Google Scholar 

  • Ruiz LM, Valenciano F, Zarzuelo FM (1998a) The family of least square values for transferable utility games. Games Econ Behav 24:109–130

    Article  Google Scholar 

  • Ruiz LM, Valenciano F, Zarzuelo FM (1998b) Some new results on least square values for TU games. TOP 6:139–158

    Article  Google Scholar 

  • Shapley LS (1953) A value for n-person games. Ann Math Stud 28:307–317

    Google Scholar 

  • Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26

    Article  Google Scholar 

  • Winter E (2002) The Shapley value. Chap 53 In: Aumann R, Hart S (eds) Handbook of game theory with economic applications, vol 3. Elsevier, Amsterdam

Download references

Author information

Authors and Affiliations

  1. Department of Economics, University of Helsinki, 00014, Helsinki, Finland

    Klaus Kultti

  2. Department of Economics, University of Turku, 20014, Turku, Finland

    Hannu Salonen

Authors
  1. Klaus Kultti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hannu Salonen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hannu Salonen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kultti, K., Salonen, H. Minimum norm solutions for cooperative games. Int J Game Theory 35, 591–602 (2007). https://doi.org/10.1007/s00182-007-0070-9

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00182-007-0070-9

Keywords

JEL Classification