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Minimal exact balancedness

Author

Listed:
  • Lohmann, E.
  • Borm, P.J.A.
  • Herings, P.J.J.

    (Microeconomics & Public Economics)

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. & Borm, P.J.A. & Herings, P.J.J., 2011. "Minimal exact balancedness," Research Memorandum 009, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    • Handle: RePEc:unm:umamet:2011009
    DOI: 10.26481/umamet.2011009
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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. 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.
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    5. 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).
    6. Calleja, Pedro & Borm, Peter & Hendrickx, Ruud, 2005. "Multi-issue allocation situations," European Journal of Operational Research, Elsevier, vol. 164(3), pages 730-747, August.
    7. 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.
    8. Hans Haller & Jean Derks, 1999. "Weighted nucleoli," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(2), pages 173-187.
    9. 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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    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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