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On Concavification and Convex Games

Author

Listed:
  • Yaron Azrieli

    (Tel Aviv University)

  • Ehud Lehrer

    (Tel Aviv University)

Abstract
We propose a new geometric approach for the analysis of cooperative games. A cooperative game is viewed as a real valued function $u$ defined on a finite set of points in the unit simplex. We define the \emph{concavification} of $u$ on the simplex as the minimal concave function on the simplex which is greater than or equal to $u$. The concavification of $u$ induces a game which is the \emph{totally balanced cover} of the game. The concavification of $u$ is used to characterize well-known classes of games, such as balanced, totally balanced, exact and convex games. As a consequence of the analysis it turns out that a game is convex if and only if each one of its sub-games is exact.

Suggested Citation

  • Yaron Azrieli & Ehud Lehrer, 2004. "On Concavification and Convex Games," Game Theory and Information 0408002, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpga:0408002
    Note: Type of Document - pdf; pages: 13
    as

    Download full text from publisher

    File URL: https://econwpa.ub.uni-muenchen.de/econ-wp/game/papers/0408/0408002.pdf
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    Citations

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    Cited by:

    1. Bochet, O.L.A. & Klaus, B.E., 2007. "A note on Dasgupta, Hammond, and Maskin's (1979) domain richness condition," Research Memorandum 039, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    2. Ehud Lehrer, 2009. "A new integral for capacities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 39(1), pages 157-176, April.
    3. Rodica Branzei & Dinko Dimitrov & Stef Tijs, 2008. "Convex Games Versus Clan Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 10(04), pages 363-372.
    4. Péter Csóka & P. Herings & László Kóczy, 2011. "Balancedness conditions for exact games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(1), pages 41-52, August.
    5. Yaron Azrieli & Ehud Lehrer, 2005. "Cooperative investment games or population games," Game Theory and Information 0503007, University Library of Munich, Germany.
    6. Branzei, Rodica & Dimitrov, Dinko & Tijs, Stef, 2011. "Convex games, clan games, and their marginal games," Center for Mathematical Economics Working Papers 368, Center for Mathematical Economics, Bielefeld University.

    More about this item

    Keywords

    concavification; convex games; core; totally balanced; exact games;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D8 - Microeconomics - - Information, Knowledge, and Uncertainty

    NEP fields

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