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Ranking Multidimensional Alternatives and Uncertain Prospects

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  • Mongin, Philippe
  • Pivato, Marcus

Abstract

We introduce a two-stage ranking of multidimensional alternatives, including uncertain prospects as particular case, when these objects can be given a suitable matrix form. The first stage defines a ranking of rows and a ranking of columns, and the second stage ranks matrices by applying natural monotonicity conditions to these auxiliary rankings. Owing to the Debreu-Gorman theory of additive separability, this framework is sufficient to generate very precise numerical representations. We apply them to three main types of multidimensional objects: streams of commodity baskets through time, monetary input-output matrices, and most extensively, uncertain prospects either in a social or an individual context of decision. Among other applications, the new approach delivers the strongest existing form of Harsanyi's (1955) Aggregation Theorem and casts light on the classic comparison between the ex ante and ex post Pareto principle. It also provides a novel derivation of subjective probability from preferences, in the style of Anscombe and Aumann (1963).

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 42515.

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Date of creation: 07 Nov 2012
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Handle: RePEc:pra:mprapa:42515

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Keywords: additively separable; multiattribute decisions; utilitarian; social welfare; social aggregation; ex ante Pareto; input-output matrix; subjective expected utility;

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References

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  1. Mongin Philippe, 1995. "Consistent Bayesian Aggregation," Journal of Economic Theory, Elsevier, vol. 66(2), pages 313-351, August.
  2. Hammond, Peter J, 1981. "Ex-ante and Ex-post Welfare Optimality under Uncertainty," Economica, London School of Economics and Political Science, vol. 48(191), pages 235-50, August.
  3. Blackorby, Charles & Donaldson, David & Weymark, John A., 1999. "Harsanyi's social aggregation theorem for state-contingent alternatives1," Journal of Mathematical Economics, Elsevier, vol. 32(3), pages 365-387, November.
  4. Itzhak Gilboa & Dov Samet & David Schmeidler, 2004. "Utilitarian Aggregation of Beliefs and Tastes," Journal of Political Economy, University of Chicago Press, vol. 112(4), pages 932-938, August.
  5. MONGIN, Philippe, 1996. "The Paradox of the Bayesian Experts and State-Dependent Utility Theory," CORE Discussion Papers 1996026, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  6. MONGIN, Philippe & d ASPREMONT, Claude, 1996. "Utility theory and ethics," CORE Discussion Papers 1996063, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  7. Browning, Martin, 1991. "A Simple Nonadditive Preference Structure for Models of Household Behavior over Time," Journal of Political Economy, University of Chicago Press, vol. 99(3), pages 607-37, June.
  8. F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
  9. Crès, Hervé & Gilboa, Itzhak & Vieille, Nicolas, 2011. "Aggregation of multiple prior opinions," Journal of Economic Theory, Elsevier, vol. 146(6), pages 2563-2582.
  10. Chambers, Christopher & Takashi Hayashi, 2003. "Preference Aggregation under Uncertainty: Savage vs. Pareto," Working Papers 1184, California Institute of Technology, Division of the Humanities and Social Sciences.
  11. Gajdos, T. & Tallon, J.-M. & Vergnaud, J.-C., 2008. "Representation and aggregation of preferences under uncertainty," Journal of Economic Theory, Elsevier, vol. 141(1), pages 68-99, July.
  12. Segal, Uzi, 1992. "Additively separable representations on non-convex sets," Journal of Economic Theory, Elsevier, vol. 56(1), pages 89-99, February.
  13. Ralph Keeney & Robert Nau, 2011. "A theorem for Bayesian group decisions," Journal of Risk and Uncertainty, Springer, vol. 43(1), pages 1-17, August.
  14. Sarin, Rakesh & Wakker, Peter P, 1998. "Dynamic Choice and NonExpected Utility," Journal of Risk and Uncertainty, Springer, vol. 17(2), pages 87-119, November.
  15. John C. Harsanyi, 1955. "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility," Journal of Political Economy, University of Chicago Press, vol. 63, pages 309.
  16. Charles Blackorby & David Donaldson & Philippe Mongin, 2004. "Social Aggregation Without the Expected Utility Hypothesis," Working Papers hal-00242932, HAL.
  17. Chateauneuf, Alain & Wakker, Peter, 1993. "From local to global additive representation," Journal of Mathematical Economics, Elsevier, vol. 22(6), pages 523-545.
  18. Chew, Soo Hong, 1983. "A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox," Econometrica, Econometric Society, vol. 51(4), pages 1065-92, July.
  19. Dekel, Eddie, 1986. "An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom," Journal of Economic Theory, Elsevier, vol. 40(2), pages 304-318, December.
  20. Wakker, Peter, 1993. "Additive representations on rank-ordered sets : II. The topological approach," Journal of Mathematical Economics, Elsevier, vol. 22(1), pages 1-26.
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