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Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games


  • Rene van den Brink

    (VU University Amsterdam)


This discussion paper resulted in a publication in 'Games and Economic Behavior', 2012, 76, 344-348. Three well-known solutions for cooperative TU-games are the Shapley value, the Banzhaf value and the equal division solution. In the literature various axiomatizations of these solutions can be found. Axiomatizations of the Shapley value often use efficiency which is not satisfied by the Banzhaf value. On the other hand, the Banzhaf value satisfies collusion neutrality which is not satisfied by the Shapley value. Both properties seem desirable. However, neither the Shapley value nor the Banzhaf value satisfy both. The equal division solution does satisfy both axioms and, moreover, together with symmetry these axioms characterize the equal division solution. Further, we show that there is no solution that satisfies efficiency, collusion neutrality and the null player property. Finally, we show that a solution satisfies efficiency, collusion neutrality and linearity if and only if there exist exogenous weights for the players such that in any game the worth of the 'grand coalition' is distributed proportional to these weights.

Suggested Citation

  • Rene van den Brink, 2009. "Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games," Tinbergen Institute Discussion Papers 09-065/1, Tinbergen Institute.
  • Handle: RePEc:tin:wpaper:20090065

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    References listed on IDEAS

    1. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    2. (*), Gerard van der Laan & RenÊ van den Brink, 1998. "Axiomatizations of the normalized Banzhaf value and the Shapley value," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 15(4), pages 567-582.
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    Cited by:

    1. Sylvain Béal & André Casajus & Frank Huettner & Eric Rémila & Philippe Solal, 2016. "Characterizations of weighted and equal division values," Theory and Decision, Springer, vol. 80(4), pages 649-667, April.

    More about this item


    Efficiency; Collusion neutrality; Shapley value; Banzhaf value; Equal division solution; Impossibility;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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