Skip to main content
SpringerLink
Log in

A simple expression for the nucleolus in a special case

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

Abstract

We present a simple algorithm for calculating the nucleolus of a game whenever (a) the characteristic function is non-positive, ie. a “cost” function, and (b) the players can be ordered by “size” in such a way that the cost of any coalition is equal to the cost of the largest player in that coalition. The cumulative nucleolus is approximately equal to the convex envelope of this cost function. A numerical and geometric illustration is given for a game based upon Birmingham airport runway costs, where there are over 13,000 players (aircraft movements) of 11 distinct (aircraft) types.

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

  • Charnes, A., and K. O.Kortanek: On A Class of Convex and Non-Archimedean Solution Concepts forn-Person Games. System Research Memorandum, No 172, Technological Institute, Northwestern University, March 1967.

  • —: On Classes of Convex and Preemptive Nuclei forn-Person Games. In:H. W. Kuhn (ed.) Proceedings of the Princeton Symposium on Mathematical Programming, Princeton University Press, 377–390, Princeton, 1970.

    Google Scholar 

  • Grotte, J. H.: Computation of and Observations on the Nucleolus, the Normalized Nucleolus and the Central Game. MSc Thesis, Department of Applied Mathematics, Cornell University, September 1970.

  • —: Observations on the Nucleolus and the Central Game. International Journal of Game Theory,1, 3, 173–177, 1972.

    Google Scholar 

  • Keane, M. A.: Some Topics inn-Person Game Theory. PhD Dissertation, Mathematics Department, Northwestern University, June 1969.

  • Kopelowitz, A.: Computation of the Kernels of Simple Games and the Nucleolus of N-Person Games, RM No 31, Research Program in Game Theory and Mathematical Economics, Department of Mathematics, Hebrew University of Jerusalem, September 1967.

  • Littlechild, S. C., andG. Owen: A Simple Expression for the Shapley Value in a Special Case. Management Science,20, 3, 370–372, 1973.

    Google Scholar 

  • Littlechild, S. C., andG. F. Thompson: Aircraft Landing Fees: A Game Theory Approach. Working Paper Series 16, Management Centre, University of Aston, Birmingham, December 1973.

    Google Scholar 

  • Megiddo, N.: Nucleoluses of Compound Simple Games I: The Nucleolus of the Sum, RM No 76, Research Program in Game Theory and Mathematical Economics, Department of Mathematics, Hebrew University of Jerusalem, May 1972. (forthcoming, SIAM Journal of Applied Mathematics).

  • Schmeidler, D.: The Nucleolus of a Characteristic Function Game. SIAM Journal of Applied Mathematics,17, 6, 1163–1170, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Management Center, The University of Aston, 36 Wake Green Road, B 13 9 PD, Birmingham, England

    S. C. Littlechild

Authors
  1. S. C. Littlechild
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

I am indebted to R. A.Abrams and M. A.Keane for preliminary discussions on this problem.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Littlechild, S.C. A simple expression for the nucleolus in a special case. Int J Game Theory 3, 21–29 (1974). https://doi.org/10.1007/BF01766216

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01766216

Keywords

Search

Navigation