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Cominimum Additive Operators

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  • 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).

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Bibliographic Info

Paper provided by Kyoto University, Institute of Economic Research in its series KIER Working Papers with number 601.

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Length: 17 pages
Date of creation: Feb 2005
Date of revision:
Handle: RePEc:kyo:wpaper:601

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

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  1. Eichberger, J. & Kelsey, D., 1996. "E-Capacities and the Ellsberg Paradox," Discussion Papers 96-13, Department of Economics, University of Birmingham.
  2. E. Algaba & J. M. Bilbao & P. Borm & J. J. López, 2000. "The position value for union stable systems," Computational Statistics, Springer, vol. 52(2), pages 221-236, November.
  3. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
  4. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-87, May.
  5. Gilboa, Itzhak, 1989. "Expectation and Variation in Multi-period Decisions," Econometrica, Econometric Society, vol. 57(5), pages 1153-69, September.
  6. Nouweland, C.G.A.M. van den & Borm, P.E.M. & Tijs, S.H., 1992. "Allocation rules for hypergraph communication situations," Open Access publications from Tilburg University urn:nbn:nl:ui:12-146635, Tilburg University.
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Cited by:
  1. Takashi Ui & Hiroyuki Kojima & Atsushi Kajii, 2011. "The Myerson value for complete coalition structures," Computational Statistics, Springer, vol. 74(3), pages 427-443, December.
  2. 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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