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Subjective Expected Utility in Games

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  • Alfredo Di Tillio

Abstract

This paper extends Savage’s subjective approach to probability and utility from decision problems under exogenous uncertainty to choice in strategic environments. Interactive uncertainty is modeled both explicitly — using hierarchies of preference relations, the analogue of beliefs hierarchies — and implicitly — using preference structures, the analogue of type spaces à la Harsanyi — and it is shown that the two approaches are equivalent. Preference structures can be seen as those sets of hierarchies arising when certain restrictions on preferences, along with the players’ common certainty of the restrictions, are imposed. Preferences are a priori assumed to satisfy only very mild properties (reflexivity, transitivity, and monotone continuity). Thus, the results provide a framework for the analysis of behavior in games under essentially any axiomatic structure. An explicit characterization is given for Savage’s axioms, and it is shown that a hierarchy of relatively simple preference relations uniquely identifies the decision maker’s utilities and beliefs of all orders. Connections with the literature on beliefs hierarchies and correlated equilibria are discussed.

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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 311.

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Date of creation: 2006
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Handle: RePEc:igi:igierp:311

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  1. Aumann, Robert J, 1987. "Correlated Equilibrium as an Expression of Bayesian Rationality," Econometrica, Econometric Society, Econometric Society, vol. 55(1), pages 1-18, January.
  2. Harsanyi, John C., 1994. "Games with Incomplete Information," Nobel Prize in Economics documents, Nobel Prize Committee 1994-1, Nobel Prize Committee.
  3. Aumann, Robert J., 1974. "Subjectivity and correlation in randomized strategies," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 67-96, March.
  4. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, Elsevier, vol. 1(2), pages 170-190, June.
  5. Heifetz, Aviad & Samet, Dov, 1998. "Topology-Free Typology of Beliefs," Journal of Economic Theory, Elsevier, Elsevier, vol. 82(2), pages 324-341, October.
  6. Joseph B. Kadane & Patrick D. Larkey, 1982. "Subjective Probability and the Theory of Games," Management Science, INFORMS, INFORMS, vol. 28(2), pages 113-120, February.
  7. Battigalli, Pierpaolo & Bonanno, Giacomo, 1999. "Editors' note," Research in Economics, Elsevier, Elsevier, vol. 53(3), pages 243-246, September.
  8. FORGES , Françoise, 1993. "Five Legitimate Definitions of Correlated Equilibrium in Games with Incomplete Information," CORE Discussion Papers, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) 1993009, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  9. Battigalli, Pierpaolo & Siniscalchi, Marciano, 2002. "Strong Belief and Forward Induction Reasoning," Journal of Economic Theory, Elsevier, Elsevier, vol. 106(2), pages 356-391, October.
  10. Brandenburger Adam & Dekel Eddie, 1993. "Hierarchies of Beliefs and Common Knowledge," Journal of Economic Theory, Elsevier, Elsevier, vol. 59(1), pages 189-198, February.
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Cited by:
  1. Willemien Kets, 2012. "Bounded Reasoning and Higher-Order Uncertainty," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 1547, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  2. Jayant V. Ganguli & Aviad Heifetz, 2012. "Universal interactive preferences," Economics Discussion Papers, University of Essex, Department of Economics 722, University of Essex, Department of Economics.
  3. Michael Trost, 2013. "Epistemic characterizations of iterated deletion of inferior strategy profiles in preference-based type spaces," International Journal of Game Theory, Springer, Springer, vol. 42(3), pages 755-776, August.
  4. Dirk Bergemann & Stephen Morris & Satoru Takahashi, 2010. "Interdependent Preferences and Strategic Distinguishability," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1772RR, Cowles Foundation for Research in Economics, Yale University, revised Jul 2014.
  5. Aviad Heifetz & Willemien Kets, 2012. "All Types Naive and Canny," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 1550, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  6. Friedenberg, Amanda, 2010. "When do type structures contain all hierarchies of beliefs?," Games and Economic Behavior, Elsevier, Elsevier, vol. 68(1), pages 108-129, January.
  7. 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, University of Queensland, School of Economics 151501, University of Queensland, School of Economics.

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