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Weighted Sets of Probabilities and MinimaxWeighted Expected Regret: New Approaches for Representing Uncertainty and Making Decisions

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  • Joseph Y. Halpern
  • Samantha Leung

Abstract

We consider a setting where an agent's uncertainty is represented by a set of probability measures, rather than a single measure. Measure-bymeasure updating of such a set of measures upon acquiring new information is well-known to suffer from problems; agents are not always able to learn appropriately. To deal with these problems, we propose using weighted sets of probabilities: a representation where each measure is associated with a weight, which denotes its significance. We describe a natural approach to updating in such a situation and a natural approach to determining the weights. We then show how this representation can be used in decision-making, by modifying a standard approach to decision making-minimizing expected regret-to obtain minimax weighted expected regret (MWER).We provide an axiomatization that characterizes preferences induced by MWER both in the static and dynamic case.

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File URL: http://arxiv.org/pdf/1210.4853
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Paper provided by arXiv.org in its series Papers with number 1210.4853.

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Date of creation: Oct 2012
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Handle: RePEc:arx:papers:1210.4853

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Web page: http://arxiv.org/

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  1. Larry Epstein & Martin Schneider, 2006. "Learning Under Ambiguity," RCER Working Papers 527, University of Rochester - Center for Economic Research (RCER).
  2. Stoye, Jörg, 2011. "Axioms for minimax regret choice correspondences," Journal of Economic Theory, Elsevier, vol. 146(6), pages 2226-2251.
  3. Todd Sarver, 2008. "Anticipating Regret: Why Fewer Options May Be Better," Econometrica, Econometric Society, Econometric Society, vol. 76(2), pages 263-305, 03.
  4. Marciano Siniscalchi, 2006. "Dynamic Choice Under Ambiguity," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 1430, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  5. 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.
  6. Jörg Stoye, 2011. "Statistical decisions under ambiguity," Theory and Decision, Springer, vol. 70(2), pages 129-148, February.
  7. 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, Econometric Society, vol. 51(4), pages 1065-92, July.
  8. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
  9. Hayashi, Takashi, 2008. "Regret aversion and opportunity dependence," Journal of Economic Theory, Elsevier, vol. 139(1), pages 242-268, March.
  10. Chateauneuf, Alain & Faro, José Heleno, 2009. "Ambiguity through confidence functions," Journal of Mathematical Economics, Elsevier, vol. 45(9-10), pages 535-558, September.
  11. Cesaltina Pacheco Pires, 2002. "A Rule For Updating Ambiguous Beliefs," Theory and Decision, Springer, vol. 53(2), pages 137-152, September.
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