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Decomposable Choice Under Uncertainty

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Abstract

Savage motivated his Sure Thing Principle by arguing that, whenever an act would be preferred if an event obtains and preferred if that event did not obtain, then it should be preferred overall. The idea that it should be possible to decompose and recompose decision problems in this way has normative appeal. We show, however, that it does not require the full separability across events implicit in Savage's axiom. We formulate a weaker axiom that suffices for decomposability, and show that this implies an implicit additive representation. Our decomposability property makes local necessary conditions for optimality, globally sufficient. Thus, it is useful in computing optimal acts. It also enables Nash behavior in games of incomplete information to be decentralized to the agent-normal form. None of these results rely on probabilistic sophistication; indeed, our axiom is consistent with the Ellsberg paradox. If we assume probabilistic sophistication, however, then the axiom holds if and only if the agent's induced preferences over lotteries satisfy betweenness.

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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1207.

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Length: 33 pages
Date of creation: Jan 1999
Date of revision:
Publication status: Published in Journal of Economic Theory (2000), 92(2): 169-197
Handle: RePEc:cwl:cwldpp:1207

Note: CFP 998.
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Keywords: Sure-thing principle; decomposability; uncertainty; computation; dynamic programming solvability; agent-normal form games; non-expected utility; betweenness;

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  1. Skiadas, Costis, 1997. "Subjective Probability under Additive Aggregation of Conditional Preferences," Journal of Economic Theory, Elsevier, vol. 76(2), pages 242-271, October.
  2. Johnsen, Thore H & Donaldson, John B, 1985. "The Structure of Intertemporal Preferences under Uncertainty and Time Consistent Plans," Econometrica, Econometric Society, vol. 53(6), pages 1451-58, November.
  3. Grant, Simon & Kajii, Atsushi & Polak, Ben, 1992. "Many good choice Axioms: When can many-good lotteries be treated as money lotteries?," Journal of Economic Theory, Elsevier, vol. 56(2), pages 313-337, April.
  4. Hong Chew Soo & Epstein Larry G. & Wakker Peter, 1993. "A Unifying Approach to Axiomatic Non-expected Utility Theories: Correction and Comment," Journal of Economic Theory, Elsevier, vol. 59(1), pages 183-188, February.
  5. 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.
  6. Edward SchleeE, 1997. "The sure thing principle and the value of information," Theory and Decision, Springer, vol. 42(1), pages 21-36, January.
  7. Peter A. Diamond, 1967. "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparison of Utility: Comment," Journal of Political Economy, University of Chicago Press, vol. 75, pages 765.
  8. Costis Skiadas, 1997. "Conditioning and Aggregation of Preferences," Econometrica, Econometric Society, vol. 65(2), pages 347-368, March.
  9. Vind, Karl, 1991. "Independent preferences," Journal of Mathematical Economics, Elsevier, vol. 20(1), pages 119-135.
  10. Gul, Faruk, 1991. "A Theory of Disappointment Aversion," Econometrica, Econometric Society, vol. 59(3), pages 667-86, May.
  11. Grant, Simon, 1995. "Subjective Probability without Monotonicity: Or How Machina's Mom May Also Be Probabilistically Sophisticated," Econometrica, Econometric Society, vol. 63(1), pages 159-89, January.
  12. 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.
  13. Machina, Mark J & Schmeidler, David, 1992. "A More Robust Definition of Subjective Probability," Econometrica, Econometric Society, vol. 60(4), pages 745-80, July.
  14. Chew, S. H. & Epstein, L. G., 1989. "A unifying approach to axiomatic non-expected utility theories," Journal of Economic Theory, Elsevier, vol. 49(2), pages 207-240, December.
  15. Segal, Uzi, 1992. "Additively separable representations on non-convex sets," Journal of Economic Theory, Elsevier, vol. 56(1), pages 89-99, February.
  16. Epstein Larry G. & Le Breton Michel, 1993. "Dynamically Consistent Beliefs Must Be Bayesian," Journal of Economic Theory, Elsevier, vol. 61(1), pages 1-22, October.
  17. Sarin, Rakesh & Wakker, Peter P, 1998. "Dynamic Choice and NonExpected Utility," Journal of Risk and Uncertainty, Springer, vol. 17(2), pages 87-119, November.
  18. Fishburn, Peter C., 1990. "Continuous nontransitive additive conjoint measurement," Mathematical Social Sciences, Elsevier, vol. 20(2), pages 165-193, October.
  19. Gul, Faruk & Lantto, Outi, 1990. "Betweenness satisfying preferences and dynamic choice," Journal of Economic Theory, Elsevier, vol. 52(1), pages 162-177, October.
  20. Gerard Debreu, 1959. "Topological Methods in Cardinal Utility Theory," Cowles Foundation Discussion Papers 76, Cowles Foundation for Research in Economics, Yale University.
  21. Daniel Ellsberg, 2000. "Risk, Ambiguity and the Savage Axioms," Levine's Working Paper Archive 7605, David K. Levine.
  22. Epstein, Larry G., 1986. "Implicitly additive utility and the nature of optimal economic growth," Journal of Mathematical Economics, Elsevier, vol. 15(2), pages 111-128, April.
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Citations

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Cited by:
  1. Halevy, Yoram, 2004. "The possibility of speculative trade between dynamically consistent agents," Games and Economic Behavior, Elsevier, vol. 46(1), pages 189-198, January.
  2. Grant, Simon & Meneghel, Idione & Tourky, Rabee, 2013. "Savage Games: A Theory of Strategic Interaction with Purely Subjective Uncertainty," Risk and Sustainable Management Group Working Papers 151501, University of Queensland, School of Economics.
  3. Atsushi Kajii & Takashi Ui, 2007. "Interim Efficient Allocations under Uncertainty," KIER Working Papers 642, Kyoto University, Institute of Economic Research.
  4. Atsushi Kajii & Takashi Ui, 2004. "Trade with Heterogeneous Multiple Priors," KIER Working Papers 582, Kyoto University, Institute of Economic Research.
  5. Simon Grant & Atsushi Kajii & Ben Polak, 1998. "On the Skiadas 'Conditional Preference Approach' to Choice Under Uncertainty," Cowles Foundation Discussion Papers 1178, Cowles Foundation for Research in Economics, Yale University.
  6. Ozaki, Hiroyuki, 2009. "Conditional implicit mean and the law of iterated integrals," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 1-15, January.

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