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Minimal Exact Balancedness

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  • Lohmann, E.R.M.A.

    (Tilburg University, School of Economics and Management)

  • Borm, P.E.M.

    (Tilburg University, School of Economics and Management)

  • Herings, P.J.J.

    (Tilburg University, School of Economics and Management)

Abstract

To verify whether a transferable utility game is exact, one has to check a linear inequality for each exact balanced collection of coalitions. This paper studies the structure and properties of the class of exact balanced collections. Comparing the definition of exact balanced collections with the definition of balanced collections, the weight vector of a balanced collection must be positive whereas the weight vector for an exact balanced collection may contain one negative weight. We investigate minimal exact balanced collections, and show that only these collections are needed to obtain exactness. The relation between minimality of an exact balanced collection and uniqueness of the corresponding weight vector is analyzed. We show how the class of minimal exact balanced collections can be partitioned into three basic types each of which can be systematically generated.
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Suggested Citation

  • Lohmann, E.R.M.A. & Borm, P.E.M. & Herings, P.J.J., 2011. "Minimal Exact Balancedness," Other publications TiSEM 9255deed-69d2-4d64-adbe-5, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:9255deed-69d2-4d64-adbe-55a3ede1cbdf
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    References listed on IDEAS

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    1. 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.
    2. 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).
    3. Csóka, Péter & Herings, P. Jean-Jacques & Kóczy, László Á., 2009. "Stable allocations of risk," Games and Economic Behavior, Elsevier, vol. 67(1), pages 266-276, September.
    4. Calleja, Pedro & Borm, Peter & Hendrickx, Ruud, 2005. "Multi-issue allocation situations," European Journal of Operational Research, Elsevier, vol. 164(3), pages 730-747, August.
    5. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    7. Lorenzo-Freire, S. & Alonso-Meijide, J.M. & Casas-Mendez, B. & Hendrickx, R.L.P., 2005. "Balanced Contributions for Multi-Issue Allocation Situations," Discussion Paper 2005-93, Tilburg University, Center for Economic Research.
    8. Bezalel Peleg, 1965. "An inductive method for constructing mimmal balanced collections of finite sets," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 12(2), pages 155-162, June.
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    10. Branzei, R. & Tijs, S. & Zarzuelo, J., 2009. "Convex multi-choice games: Characterizations and monotonic allocation schemes," European Journal of Operational Research, Elsevier, vol. 198(2), pages 571-575, October.
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    Cited by:

    1. Tomáš Kroupa & Milan Studený, 2019. "Facets of the cone of totally balanced games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(2), pages 271-300, October.
    2. Milan Studený & Václav Kratochvíl, 2022. "Facets of the cone of exact games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 35-80, February.

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    JEL classification:

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

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