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Modularity and Monotonicity of Games

Author

Listed:
  • Takao Asano

    (Faculty of Economics, Okayama University)

  • Hiroyuki Kojima

    (Department of Economics, Teikyo University)

Abstract

The purpose of this paper is twofold. First, we generalize Kajii et al. (2007), and provide a condition under which for a game v, its Mobius inversion is equal to zero within the framework of the k-modularity of v for k >= 2. This condition is more general than that in Kajii et al. (2007). Second, we provide a condition under which for a game v for k >= 2, its Mobius inversion takes non-negative values, and not just zero. This paper relates the study of totally monotone games to that of kmonotone games. Furthermore, the modularity of a game can be related to k-additive capacities proposed by Grabisch (1997). As applications of our results to economics, this paper shows that a Gini index representation of Ben-Porath and Gilboa (1994) can be characterized by using our results directly. Our results can also be applied to potential functions proposed by Hart and Mas-Colell (1989) and further analyzed by Ui et al. (2011). *>= is greater than or equal to.

Suggested Citation

  • Takao Asano & Hiroyuki Kojima, 2013. "Modularity and Monotonicity of Games," KIER Working Papers 871, Kyoto University, Institute of Economic Research.
    • Handle: RePEc:kyo:wpaper:871
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    References listed on IDEAS

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    1. Miranda, Pedro & Grabisch, Michel & Gil, Pedro, 2006. "Dominance of capacities by k-additive belief functions," European Journal of Operational Research, Elsevier, vol. 175(2), pages 912-930, December.
    2. van den Nouweland, Anne & Borm, Peter & Tijs, Stef, 1992. "Allocation Rules for Hypergraph Communication Situations," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(3), pages 255-268.
    3. Chateauneuf, Alain & Rebille, Yann, 2004. "A Yosida-Hewitt decomposition for totally monotone games," Mathematical Social Sciences, Elsevier, vol. 48(1), pages 1-9, July.
    4. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    5. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    6. Chateauneuf, Alain & Jaffray, Jean-Yves, 1989. "Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 263-283, June.
    7. Gilboa, Itzhak, 1989. "Expectation and Variation in Multi-period Decisions," Econometrica, Econometric Society, vol. 57(5), pages 1153-1169, September.
    8. Takashi Ui & Hiroyuki Kojima & Atsushi Kajii, 2011. "The Myerson value for complete coalition structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(3), pages 427-443, December.
    9. Gajdos, Thibault, 2002. "Measuring Inequalities without Linearity in Envy: Choquet Integrals for Symmetric Capacities," Journal of Economic Theory, Elsevier, vol. 106(1), pages 190-200, September.
    10. Jürgen Eichberger & David Kelsey, 1999. "E-Capacities and the Ellsberg Paradox," Theory and Decision, Springer, vol. 46(2), pages 107-138, April.
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    More about this item

    Keywords

    Belief Functions; Mobius Inversion; Totally Monotone Games; k-additive capacities; Gini Index; Potential Functions;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General

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