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Theory and Decision

, Volume 60, Issue 2–3, pp 137–174 | Cite as

Representability of Ordinal Relations on a Set of Conditional Events

  • Giulianella Coletti
  • Barbara VantaggiEmail author
Article

Abstract

Any dynamic decision model should be based on conditional objects and must refer to (not necessarily structured) domains containing only the elements and the information of interest. We characterize binary relations, defined on an arbitrary set of conditional events, which are representable by a coherent generalized decomposable conditional measure and we study, in particular, the case of binary relations representable by a coherent conditional probability.

Keywords

comparative conditional degree of belief comparative coherent conditional probability representability conditional capacities coherent conditional probabilities 

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References

  1. Aumann, R.J. 1962Utility theory without the completeness axiomEconometrica30445462Google Scholar
  2. Bouchon-Meunier, B., Coletti, G., Marsala, C. 2002Independence and possibilistic conditioningAnnals of Mathematics and Artificial Intelligence35107124CrossRefGoogle Scholar
  3. Capotorti, A., Coletti, G., Vantaggi, B. 1998Non-additive ordinal relations representable by lower or upper probabilitiesKybernetika347990Google Scholar
  4. Chateauneuf, A., Jaffray, J.Y. 1984Archimedean qualitative probabilityJournal of Mathematical Psychology28191204CrossRefGoogle Scholar
  5. Chateauneuf, A. 1985On the existence of a probability measure compatible with a total preorder on a boolean algebraJournal of Mathematical Economics144352CrossRefGoogle Scholar
  6. Chateauneuf, A., Kast, R., Lapied, A. 2001Conditioning capacities and choquet integrals: the role of comonotonyTheory and Decision51367386CrossRefGoogle Scholar
  7. Coletti, G. 1990Coherent qualitative probabilityJournal of Mathematical Psychology34297310CrossRefGoogle Scholar
  8. Coletti, G., Regoli, G. 1992How can an expert system help in choosing the optimal decision?Theory and Decision33253264CrossRefGoogle Scholar
  9. Coletti, G., Gilio, A., Scozzafava, R. 1993Comparative probability for conditional events: a new look through coherenceTheory and Decision35237258CrossRefGoogle Scholar
  10. Coletti, G. 1994Coherent numerical and ordinal Probabilistic assessmentsIEEE Trasactions on Systems, Man, and Cybernetics2417471754CrossRefGoogle Scholar
  11. Coletti, G. 1996Coherence principles for handling qualitative and quantitative partial probabilistic assessmentsMathware and Soft Computing3159172Google Scholar
  12. Coletti, G., Scozzafava, R. 2001From conditional events to conditional measures: a new axio matic approachAnnals of Mathematics and Artificial Intelligence32373392CrossRefGoogle Scholar
  13. Coletti, G. and Scozzafava, R. (2001b). Locally additive comparative probabilities, Proceedings of the 2nd International Symposium on Imprecise Probabilities and their Applications, Ithaca, pp. 83–92.Google Scholar
  14. Coletti, G., Scozzafava, R. 2002Probabilistic Logic in a Coherent SettingTrends in logic no 15, KluwerDordrecht/Boston/LondonGoogle Scholar
  15. Coletti, G. and Scozzafava R. (2003). Toward a general theory of conditional beliefs. Proceedings of the 6th Workshop on Uncertainty Processing, Hejnice, pp. 65–76.Google Scholar
  16. Finetti, B. 1931Sul significato soggettivo della, probabilitàFundamenta Matematicae17293329Google Scholar
  17. de Finetti, B. (1949). Sull’Impostazione Assiomatica del Calcolo delle Probabilità, Annali Univ. Trieste 19, 3–55. (Engl. transl: Ch. 5 in , Piobability, Induction, Statistics, Wiley, London, 1972).Google Scholar
  18. de Finetti, B. (1970). Teoria della Probabilità. Torino: Einaudi (Engl. transl. (1974) Theory of probability, London: Wiley & Sons).Google Scholar
  19. Dubins, L.E. 1975Finitely additive conditional probabilities, conglomerability and disintegrationAnnals of Probability38999Google Scholar
  20. Dubra, J., Maccheroni, J., Ok, E.A. 2004Expected utility theory without the completeness axiomJournal of Economic Theory115118133CrossRefGoogle Scholar
  21. Krauss, P.H. 1968Representation of conditional probability measures on Boolean algebrasActa Math. Acad. Scient. Hungar19229241CrossRefGoogle Scholar
  22. Fenchel, W. 1951Convex Cones Sets and FunctionsLectures at Princeton UniversityPrinceton NJGoogle Scholar
  23. Fine, T.L. 1973Theories of ProbabilityAcademic PressNew YorkGoogle Scholar
  24. Halpern, J.Y. 1999A counterexample to theorems of Cox and FineJournal of Artificial Intelligence Research106785Google Scholar
  25. Holzer, S. 1985On coherence and conditional previsionBollettino UMI, SerieVI–C IV441460Google Scholar
  26. Lehmann, D. 1996Generalized qualitative probability: Savage revisitedHorvitz, Jensen,  eds. Proceeding of the Twelfth Conference on Uncertainty in Artificial Intelligence PortlandOregonMorgan, Kaufmann381388Google Scholar
  27. Lehman, R.S. 1955On confirmation and rational bettingThe Journal of Symbolic Logic20251262Google Scholar
  28. Kannai, Y. 1963Existence of a utility in infinite dimensional partially ordered spacesIsrael Journal of Mathematics1229234Google Scholar
  29. Koopman, B.O. 1940The axioms and algebra of intuitive probabilityAnnals of Mathematics41269292CrossRefGoogle Scholar
  30. Kraft, C.H., Pratt, J.W., Seidenberg, A. 1959Intuitive probability on finite setsAnnals of Mathematical Statistics30408419Google Scholar
  31. Popper, K.R. 1959The Logic of Scientific DiscoveryRoutledgeLondonGoogle Scholar
  32. Seidenfeld, T.S., Schervish, M.J., Kadane, J.B. 1995A representation of partially ordered preferencesThe Annals of Statistics2321682217CrossRefGoogle Scholar
  33. Regazzini, E. 1985Finitely additive conditional probabilitiesRendiconti Sem. Mat. Fis. Milano556989CrossRefGoogle Scholar
  34. Savage, L.J. 1954The Foundations of StatisticsWileyNew YorkGoogle Scholar
  35. Scott, D. 1964Measurement structures and linear inequalitiesJournal of Mathematical Psychology1233247CrossRefGoogle Scholar
  36. Vind, K. 2000von Neumann-Morgenstern preferencesJournal of Mathematical Economics33109122CrossRefGoogle Scholar
  37. Wakker, P. 1981Agreeing probability measures for comparative probability structuresAnnals of Statistics9658662Google Scholar
  38. Williams, P.M. (1975). Notes on conditional previsions, School of Mathematical and Physical Sciences, Working Paper, The University of Sussex.Google Scholar
  39. Wong, S.K.M., Yao, Y.Y., Bollmann, P., Burger, H.C. 1991Axiomatization of qualitative belief structureIEEE Transactions on Systems Man Cybernetics21726734CrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Dip. Metodi e Modelli MatematiciUniversità “La Sapienza” RomaRomaItaly

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