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On the Extreme Points of Two Polytopes associated with a Digraph and Applications to Cooperative Games

Author

Listed:
  • René van den Brink

    (Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam)

  • Gerard van der Laan

    (Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam)

  • Valeri Vasil'ev

    (Sobolev Institute of Mathematics, Novosibirsk, Russia)

Abstract

See also the publication in 'Journal of Mathematical Economics', 2008, 44, 1114-1125. In this paper we describe the extreme points of two closely related polytopes that are assigned to a digraph. The first polytope is the set of all sharing vectors (elements from the unit simplex) such that each node gets at least as much as each of its successors. The second one is the set of all fuzzy vectors (elements of the unit cube) with participation rates of players subordinated to the relationships prescribed by the digraph. We also discuss some applications in cooperative game theory.

Suggested Citation

  • René van den Brink & Gerard van der Laan & Valeri Vasil'ev, 2004. "On the Extreme Points of Two Polytopes associated with a Digraph and Applications to Cooperative Games," Tinbergen Institute Discussion Papers 04-069/1, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20040069
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    Cited by:

    1. René van den Brink & Gerard van der Laan & Valeri Vasil'ev, 2007. "Distributing Dividends in Games with Ordered Players," Tinbergen Institute Discussion Papers 06-114/1, Tinbergen Institute.

    More about this item

    Keywords

    polytope; directed graph; unit simplex; unit cube; cooperate game;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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