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Mixed Extensions of Decision Problems under Uncertainty

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

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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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. Dekel, Eddie & Lipman, Barton L & Rustichini, Aldo, 2001. "Representing Preferences with a Unique Subjective State Space," Econometrica, Econometric Society, vol. 69(4), pages 891-934, July.
  2. Kreps, David M, 1979. "A Representation Theorem for "Preference for Flexibility"," Econometrica, Econometric Society, vol. 47(3), pages 565-77, May.
  3. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
  4. Itzhak Gilboa & Fabio Maccheroni & Massimo Marinacci & David Schmeidler, 2008. "Objective and Subjective Rationality in a Multiple Prior Model," Carlo Alberto Notebooks 73, Collegio Carlo Alberto, revised 2008.
  5. David Schmeidler, 1989. "Subjective Probability and Expected Utility without Additivity," Levine's Working Paper Archive 7662, David K. Levine.
  6. 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.
  7. Ghirardato, Paolo & Maccheroni, Fabio & Marinacci, Massimo, 2005. "Certainty Independence and the Separation of Utility and Beliefs," Journal of Economic Theory, Elsevier, vol. 120(1), pages 129-136, January.
  8. 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.
  9. Sarin, Rakesh & Wakker, Peter, 1997. "A Single-Stage Approach to Anscombe and Aumann's Expected Utility," Review of Economic Studies, Wiley Blackwell, vol. 64(3), pages 399-409, July.
  10. Simone Cerreia-Vioglio & Paolo Ghirardato & Fabio Maccheroni & Massimo Marinacci & Marciano Siniscalchi, 2010. "Rational Preferences under Ambiguity," Carlo Alberto Notebooks 169, Collegio Carlo Alberto.
  11. Swinkels, J., 1989. "Subgames And The Reduced Normal Form," Papers 344, Princeton, Department of Economics - Econometric Research Program.
  12. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2008. "Uncertainty Averse Preferences," Carlo Alberto Notebooks 77, Collegio Carlo Alberto.
  13. J├╝rgen Eichberger & Simon Grant & David Kelsey, 2014. "Randomization and Dynamic Consistency," Discussion Papers 1409, Exeter University, Department of Economics.
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