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Mixed Extensions of Decision Problems under Uncertainty

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  • Pierpaolo Battigalli
  • Simone Cerreia-Vioglio
  • Fabio Maccheroni
  • Massimo Marinacci

Abstract

In a decision problem under uncertainty, a decision maker considers a set of alternative actions whose consequences depend on uncertain factors outside his control. Following Luce and Raiffa (1957), we adopt a natural representation of such situation that takes as primitives a set of conceivable actions A, a set of states S and a consequence function from actions and states to consequences in C. With this, each action induces a map from states to consequences, or Savage act, and each mixed action induces a map from states to probability distributions over consequences, or Anscombe-Aumann act. Under a consequentialist axiom, preferences over pure or mixed actions yield corresponding preferences over the induced acts. The most common approach to the theory of choice under uncertainty takes instead as primitive a preference relation over the set of all Anscombe-Aumann acts (functions from states to distributions over consequences). This allows to apply powerful convex analysis techniques, as in the seminal work of Schmeidler (1989) and the vast descending literature. This paper shows that we can maintain the mathematical convenience of the Anscombe-Aumann framework within a description of decision problems which is closer to applications and experiments. We argue that our framework is more expressive, it allows to be explicit and parsimonious about the assumed richness of the set of conceivable actions, and to directly capture preference for randomization as an expression of uncertainty aversion.

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Bibliographic Info

Paper provided by IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University in its series Working Papers with number 485.

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Date of creation: 2013
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Handle: RePEc:igi:igierp:485

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  1. David Schmeidler, 1989. "Subjective Probability and Expected Utility without Additivity," Levine's Working Paper Archive 7662, David K. Levine.
  2. Eddie Dekel, 1997. "A Unique Subjective State Space for Unforeseen Contingencies," Discussion Papers 1202, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  3. Swinkels, J., 1989. "Subgames And The Reduced Normal Form," Papers 344, Princeton, Department of Economics - Econometric Research Program.
  4. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Econometrica, Econometric Society, vol. 74(6), pages 1447-1498, November.
  5. Itzhak Gilboa & Fabio Maccheroni & Massimo Marinacci & David Schmeidler, 2010. "Objective and Subjective Rationality in a Multiple Prior Model," Econometrica, Econometric Society, vol. 78(2), pages 755-770, 03.
  6. Paolo Ghirardato & Fabio Maccheroni & Massimo Marinacci, 2002. "Certainty Independence and the Separation of Utility and Beliefs," ICER Working Papers - Applied Mathematics Series 40-2002, ICER - International Centre for Economic Research.
  7. Sarin, R. & Wakker, P.P., 1996. "A Single-Stage Approach to Anscombe and Aumann's Expected Utility," Discussion Paper 1996-45, Tilburg University, Center for Economic Research.
  8. Simone Cerreia-Vioglio & Paolo Ghirardato & Fabio Maccheroni & Massimo Marinacci & Marciano Siniscalchi, 2010. "Rational Preferences under Ambiguity," Carlo Alberto Notebooks 169, Collegio Carlo Alberto.
  9. Kreps, David M, 1979. "A Representation Theorem for "Preference for Flexibility"," Econometrica, Econometric Society, vol. 47(3), pages 565-77, May.
  10. Glazer, Jacob & Rubinstein, Ariel, 1996. "An Extensive Game as a Guide for Solving a Normal Game," Journal of Economic Theory, Elsevier, vol. 70(1), pages 32-42, July.
  11. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2008. "Uncertainty Averse Preferences," Carlo Alberto Notebooks 77, Collegio Carlo Alberto.
  12. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
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