The Annals of Statistics

A representation of partially ordered preferences

Teddy Seidenfeld, Mark J. Schervish, and Joseph B. Kadane

Full-text: Open access

Abstract

This essay considers decision-theoretic foundations for robust Bayesian statistics. We modify the approach of Ramsey, de Finetti, Savage and Anscombe and Aumann in giving axioms for a theory of robust preferences. We establish that preferences which satisfy axioms for robust preferences can be represented by a set of expected utilities. In the presence of two axioms relating to state-independent utility, robust preferences are represented by a set of probability/utility pairs, where the utilities are almost state-independent (in a sense which we make precise). Our goal is to focus on preference alone and to extract whatever probability and/or utility information is contained in the preference relation when that is merely a partial order. This is in contrast with the usual approach to Bayesian robustness that begins with a class of "priors" or "likelihoods," and a single loss function, in order to derive preferences from these probability/utility assumptions.

Article information

Source
Ann. Statist., Volume 23, Number 6 (1995), 2168-2217.

Dates
First available in Project Euclid: 15 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034713653

Digital Object Identifier
doi:10.1214/aos/1034713653

Mathematical Reviews number (MathSciNet)
MR1389871

Zentralblatt MATH identifier
0871.62008

Subjects
Primary: 62C05: General considerations
Secondary: 62A15

Keywords
Robust statistics axioms of decision theory state-dependent utility partial order

Citation

Seidenfeld, Teddy; Schervish, Mark J.; Kadane, Joseph B. A representation of partially ordered preferences. Ann. Statist. 23 (1995), no. 6, 2168--2217. doi:10.1214/aos/1034713653. https://projecteuclid.org/euclid.aos/1034713653


Export citation

References

  • ANSCOMBE, F. J. and AUMANN, R. J. 1963. A definition of subjective probability. Ann. Math. Statist. 34 199 205. Z.
  • AUMANN, R. J. 1962. Utility theory without the completeness axiom. Econometrica 30 445 462. Z.
  • AUMANN, R. J. 1964. Utility theory without the completeness axiom: a correction. Econometrica 32 210 212. Z.
  • BERGER, J. 1985. Statistical Decision Theory and Bayesian Analy sis, 2nd ed. Springer, New York.Z.
  • DE FINETTI, B. 1937. La prevision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincare 7 1 68. ´ Z.
  • DEGROOT, M. 1974. Reaching a consensus. J. Amer. Statist. Assoc. 69 118 121. Z.
  • ELLSBERG, D. 1961. Risk, ambiguity, and the Savage axioms. Quart. J. Econom. 75 643 669. Z.
  • FISHBURN, P. C. 1979. Utility Theory for Decision Making. Krieger, New York. Z.
  • FISHBURN, P. C. 1982. The Foundations of Expected Utility. Reidel, Dordrecht. Z.
  • GIRON, F. J. and RIOS, S. 1980. Quasi-Bayesian behaviour: A more realistic approach to Z decision making? In Bayesian Statistics J. M. Bernardo, M. H. DeGroot, D. V. Lindley. and A. F. M. Smith, eds. 17 38. Univ. Valencia Press. Z.
  • HARTIGAN, J. A. 1983. Bay es Theory. Springer, New York. Z. Z
  • HAUSNER, M. 1954. Multidimensional utilities. In Decision Processes R. M. Thrall, C. H.. Coombs and R. L. Davis, eds. 167 180. Wiley, New York.
  • KADANE, J. B., Ed. 1984. Robustness of Bayesian Analy sis. North-Holland, Amsterdam. Z.
  • KADANE, J. B. 1986. Progress toward a more ethical method for clinical trials. Journal of Medicine and Philosophy 11 385 404. Z.
  • KADANE, J. B. and SEDRANSK, N. 1980. Toward a more ethical clinical trial. In Bayesian Z. Statistics J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds. 329 338. Univ. Valencia Press. Z.
  • KANNAI, Y. 1963. Existence of a utility in infinite dimensional partially ordered spaces. Israel J. Math. 1 229 234. Z.
  • LEVI, I. 1974. On indeterminate probabilities. J. Philos. 71 391 418. Z.
  • LEVI, I. 1980. The Enterprise of Knowledge. MIT Press. Z.
  • LEVI, I. 1990. Pareto unanimity and consensus. J. Philos. 87 481 492. Z. Z.
  • MOSKOWITZ, H., WONG, R. T. and CHU, P.-Y. 1988. Robust interactive decision-analysis RID : Concepts, methodology, and sy stem principles. Paper 948, Krannert Graduate School of Management, Purdue Univ. Z.
  • NAU, R. F. 1992. Indeterminate probabilities on finite sets. Ann. Statist. 20 1737 1767. Z.
  • NAU, R. F. 1993. The shape of incomplete preferences. Paper 9301, The Fuqua School of Business, Duke Univ. Z.
  • RAMSEY, F. P. 1931. Truth and probability. In The Foundations of Mathematics and Other Z. Logical Essay s R. B. Braithwaite, ed. 156 198. Kegan, Paul, Trench, Trubner and Co. Ltd., London. Z.
  • RIOS INSUA, D. 1990. Sensitivity Analy sis in Multi-Objective Decision Making. Springer, New York.Z.
  • RIOS INSUA, D. 1992. On the foundations of decision making under partial information. Theory and Decision 33 83 100. Z.
  • SAVAGE, L. J. 1954. The Foundations of Statistics. Wiley, New York. Z.
  • SCHERVISH, M. J., SEIDENFELD, T. and KADANE, J. B. 1990. State-dependent utilities. J. Amer. Statist. Assoc. 85 840 847. Z.
  • SCHERVISH, M. J., SEIDENFELD, T. and KADANE, J. B. 1991. Shared preferences and statedependent utilities. Management Sci. 37 1575 1589. Z.
  • SEIDENFELD, T., KADANE, J. B. and SCHERVISH, M. J. 1989. On the shared preferences of two Bayesian decision makers. J. Philos. 86 225 244. Z.
  • SEIDENFELD, T. and SCHERVISH, M. J. 1983. A conflict between finite additivity and avoiding Dutch Book. Philos. Sci. 50 398 412. Z.
  • SEIDENFELD, T., SCHERVISH, M. J. and KADANE, J. B. 1990. Decisions without ordering. In Z. Acting and Reflecting W. Sieg, ed. 143 170. Kluwer, Dordrecht. Z.
  • SMITH, C. A. B. 1961. Consistency in statistical inference and decision. J. Roy. Statist. Soc. Ser. B 23 1 25. Z.
  • SZPILRAJN, E. 1930. Sur l'extension de l'ordre partiel. Fund. Math. 16 386 389. Z.
  • VON NEUMANN, J. and MORGENSTERN, O. 1947. Theory of Games and Economic Behavior, 2nd ed. Princeton Univ. Press. Z.
  • WALLEY, P. 1991. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London. Z.
  • WHITE, C. C. 1986. A posteriori representations based on linear inequality descriptions of a priori and conditional probabilities. IEEE Trans. Sy stems Man Cy bernet. 16 570 573. Z.
  • WILLIAMS, P. 1976. Indeterminate probabilities. In Formal Methods in the Methodology of Z. Empirical Sciences M. Przelecki, K. Szaniawski and R. Wojcicki, eds. 229 246. Reidel, Dordrecht.
  • PITTSBURGH, PENNSy LVANIA 15213