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Cominimum Additive Operators

Author

Listed:
  • Atsushi Kajii

    (Institute of Economic Research, Kyoto University)

  • Hiroyuki Kojima

    (Department of Economics, Teikyo University)

  • Takashi Ui

    (Faculty of Economics, Yokohama National University)

Abstract

This paper proposes a class of weak additivity concepts for an operator on the set of real valued functions on a finite state space \omega, which include additivity and comonotonic additivity as extreme cases. Let \epsilon be a collection of subsets of \omega. Two functions x and y on \omega are \epsilon-cominimum if, for each E \subseteq \epsilon, the set of minimizers of x restricted on E and that of y have a common element. An operator I on the set of functions on is E- cominimum additive if I(x+y) = I(x)+I(y) whenever x and y are \epsilon-cominimum. The main result characterizes homogeneous E-cominimum additive operators in terms of the Choquet integrals and the corresponding non-additive signed measures. As applications, this paper gives an alternative proof for the characterization of the E-capacity expected utility model of Eichberger and Kelsey (1999) and that of the multi-period decision model of Gilboa (1989).

Suggested Citation

  • Atsushi Kajii & Hiroyuki Kojima & Takashi Ui, 2005. "Cominimum Additive Operators," KIER Working Papers 601, Kyoto University, Institute of Economic Research.
    • Handle: RePEc:kyo:wpaper:601
    as

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    References listed on IDEAS

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    1. Gilboa, Itzhak, 1989. "Expectation and Variation in Multi-period Decisions," Econometrica, Econometric Society, vol. 57(5), pages 1153-1169, September.
    2. E. Algaba & J. M. Bilbao & P. Borm & J. J. López, 2000. "The position value for union stable systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(2), pages 221-236, November.
    3. 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.
    4. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    5. Jürgen Eichberger & David Kelsey, 1999. "E-Capacities and the Ellsberg Paradox," Theory and Decision, Springer, vol. 46(2), pages 107-138, April.
    6. Jürgen Eichberger & David Kelsey, 1999. "E-Capacities and the Ellsberg Paradox," Theory and Decision, Springer, vol. 46(2), pages 107-138, April.
    7. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
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    Cited by:

    1. Takao Asano & Hiroyuki Kojima, 2019. "Consequentialism and dynamic consistency in updating ambiguous beliefs," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(1), pages 223-250, July.
    2. Daniel Li Li & Erfang Shan, 2020. "Marginal contributions and derivatives for set functions in cooperative games," Journal of Combinatorial Optimization, Springer, vol. 39(3), pages 849-858, April.
    3. Takao Asano & Hiroyuki Kojima, 2014. "Modularity and monotonicity of games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(1), pages 29-46, August.
    4. 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.
    5. Takao Asano & Hiroyuki Kojima, 2015. "An axiomatization of Choquet expected utility with cominimum independence," Theory and Decision, Springer, vol. 78(1), pages 117-139, January.
    6. Lo, Kin Chung, 2006. "Agreement and stochastic independence of belief functions," Mathematical Social Sciences, Elsevier, vol. 51(1), pages 1-22, January.

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    More about this item

    Keywords

    Choquet integral; comonotonicity; non-additive probabilities; capacities; cooperative games;
    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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