Automation and Remote Control

, Volume 80, Issue 9, pp 1734–1744 | Cite as

On a Cooperative Game in the Knapsack Problem

  • S. I. DotsenkoEmail author
Mathematical Game Theory and Applications


The knapsack problem with indivisible items as agents is considered. Each agent has certain weight and utility and wants to be in a knapsack. Such situation is treated as a cooperative game with transferable utility. A characteristic function of this game generalizes the characteristic function associated with the bankruptcy problem but, in contrast to the latter case, it is not convex. Nevertheless, it turns out that the core of this game is non-empty. At the end of the paper some special cases of the knapsack problem are studied. For these cases, the Shapley value, the τ-value and also the nucleolus are found in the explicit form.


knapsack problem cooperative game bankruptcy problem core Shapley value nucleolus τ-value 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mazalov, V.V., Matematicheskaya teoriya igr i prilozheniya (Mathematical Game Theory and Applications), St. Petersburg: Lan', 2010. Translated under the title Mathematical Game Theory and Applications, New York: Wiley, 2014.Google Scholar
  2. 2.
    Aumann, R. and Maschler, M., Game Theoretic Analysis of a Bankruptcy Problem from the Talmud, J. Economic Theory, 1985, no. 36, pp. 195–213.MathSciNetCrossRefGoogle Scholar
  3. 3.
    O'Neill, B., A Problem of Rights Arbitration from the Talmud, Math. Social Set., 1982, vol. 2, pp. 345–371.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Curiel, I., Maschler, B., and Tijs, S., Bankruptcy Games, Zeitshrift Operat. Res., 1987, vol. 31, no. 5, pp. 143–159.MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Taras Shevchenko National University of KyivKyivUkraine

Personalised recommendations