International Journal of Game Theory

, Volume 43, Issue 2, pp 351–368 | Cite as

Constrained core solutions for totally positive games with ordered players

  • René van den BrinkEmail author
  • Gerard van der Laan
  • Valeri A. Vasil’ev


In many applications of cooperative game theory to economic allocation problems, such as river-, polluted river- and sequencing games, the game is totally positive (i.e., all dividends are nonnegative), and there is some ordering on the set of the players. A totally positive game has a nonempty core. In this paper we introduce constrained core solutions for totally positive games with ordered players which assign to every such a game a subset of the core. These solutions are based on the distribution of dividends taking into account the hierarchical ordering of the players. The Harsanyi constrained core of a totally positive game with ordered players is a subset of the core of the game and contains the Shapley value. For special orderings it coincides with the core or the Shapley value. The selectope constrained core is defined for acyclic orderings and yields a subset of the Harsanyi constrained core. We provide a characterization for both solutions.


Totally positive TU-game Digraph Harsanyi dividends Core Shapley value Harsanyi set Selectope Polluted river games 

JEL Classification

C71 (Cooperative games) 



This research has been done while the third author was visiting the Tinbergen Institute, VU University Amsterdam, on NWO-Grant 047.017.017 within the framework of Dutch-Russian Cooperation. This author also appreciates financial support from the Russian Leading Scientific Schools Fund (Grant 4113.2008.6) and Russian Fund for Basic Research (Grant 13-06-00311). We thank two anonymous referees and the associate editor for their valuable comments.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • René van den Brink
    • 1
    Email author
  • Gerard van der Laan
    • 1
  • Valeri A. Vasil’ev
    • 2
  1. 1.Department of Econometrics and Tinbergen InstituteVU UniversityAmsterdamThe Netherlands
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia

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