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Rationalizability and Minimal Complexity in Dynamic Games

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  • Perea,Andrés

    (METEOR)

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    Abstract

    This paper presents a formal epistemic framework for dynamic games in which players, during the course of the game, may revise their beliefs about the opponents'' utility functions. We impose three key conditions upon the players'' beliefs: (a) throughout the game, every move by the opponent should be interpreted as being part of a rational strategy, (b) the belief about the opponents'' relative ranking of two strategies should not be revised unless one is certain that the opponent has decided not to choose one of these strategies, and (c) the players'' initial beliefs about the opponents'' utility functions should agree on a given profile u of utility functions. Types that, throughout the game, respect common belief about these three events, are called persistently rationalizable for the profile u of utility functions. It is shown that persistent rationalizability implies the backward induction procedure in generic games with perfect information. We next focus on persistently rationalizable types for u that hold a theory about the opponents of ``minimal complexity'''', resulting in the concept of minimal rationalizability. For two-player simultaneous move games, minimal rationalizability is equivalent to the concept of Nash equilibrium strategy. In every outside option game, as defined by van Damme (1989), minimal rationalizability uniquely selects the forward induction outcome.

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    Bibliographic Info

    Paper provided by Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR) in its series Research Memorandum with number 047.

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    Date of creation: 2003
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    Handle: RePEc:unm:umamet:2003047

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    Keywords: microeconomics ;

    References

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    1. Balkenborg, Dieter & Winter, Eyal, 1997. "A necessary and sufficient epistemic condition for playing backward induction," Journal of Mathematical Economics, Elsevier, vol. 27(3), pages 325-345, April.
    2. van Damme, Eric, 1989. "Stable equilibria and forward induction," Journal of Economic Theory, Elsevier, Elsevier, vol. 48(2), pages 476-496, August.
    3. Levine, David & Kreps, David & Fudenberg, Drew, 1988. "On the Robustness of Equilibrium Refinements," Scholarly Articles 3350444, Harvard University Department of Economics.
    4. D. B. Bernheim, 2010. "Rationalizable Strategic Behavior," Levine's Working Paper Archive 661465000000000381, David K. Levine.
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    8. Stalnaker, Robert, 1998. "Belief revision in games: forward and backward induction1," Mathematical Social Sciences, Elsevier, Elsevier, vol. 36(1), pages 31-56, July.
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    12. Makoto Shimoji, 2002. "On forward induction in money-burning games," Economic Theory, Springer, Springer, vol. 19(3), pages 637-648.
    13. Perea,Andrés, 2003. "Proper Rationalizability and Belief Revision in Dynamic Games," Research Memorandum, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR) 048, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    14. Frank Schuhmacher, 1999. "Proper rationalizability and backward induction," International Journal of Game Theory, Springer, Springer, vol. 28(4), pages 599-615.
    15. Asheim, Geir B. & Perea, Andres, 2005. "Sequential and quasi-perfect rationalizability in extensive games," Games and Economic Behavior, Elsevier, Elsevier, vol. 53(1), pages 15-42, October.
    16. Reny Philip J., 1993. "Common Belief and the Theory of Games with Perfect Information," Journal of Economic Theory, Elsevier, Elsevier, vol. 59(2), pages 257-274, April.
    17. Battigalli, Pierpaolo & Siniscalchi, Marciano, 2002. "Strong Belief and Forward Induction Reasoning," Journal of Economic Theory, Elsevier, Elsevier, vol. 106(2), pages 356-391, October.
    18. Zauner, Klaus G., 2002. "The existence of equilibrium in games with randomly perturbed payoffs and applications to experimental economics," Mathematical Social Sciences, Elsevier, Elsevier, vol. 44(1), pages 115-120, September.
    19. Aumann, Robert J., 1995. "Backward induction and common knowledge of rationality," Games and Economic Behavior, Elsevier, Elsevier, vol. 8(1), pages 6-19.
    20. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, Econometric Society, vol. 59(1), pages 81-98, January.
    21. Samet, Dov, 1996. "Hypothetical Knowledge and Games with Perfect Information," Games and Economic Behavior, Elsevier, Elsevier, vol. 17(2), pages 230-251, December.
    22. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Choice under Uncertainty," Econometrica, Econometric Society, Econometric Society, vol. 59(1), pages 61-79, January.
    23. Battigalli, Pierpaolo, 1997. "On Rationalizability in Extensive Games," Journal of Economic Theory, Elsevier, Elsevier, vol. 74(1), pages 40-61, May.
    24. Aumann, Robert & Brandenburger, Adam, 1995. "Epistemic Conditions for Nash Equilibrium," Econometrica, Econometric Society, Econometric Society, vol. 63(5), pages 1161-80, September.
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    Cited by:
    1. Asheim, Geir B. & Perea, Andres, 2005. "Sequential and quasi-perfect rationalizability in extensive games," Games and Economic Behavior, Elsevier, Elsevier, vol. 53(1), pages 15-42, October.
    2. Perea,Andrés, 2003. "Proper Rationalizability and Belief Revision in Dynamic Games," Research Memorandum, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR) 048, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    3. Perea,Andrés, 2004. "Minimal Belief Revision leads to Backward Induction," Research Memorandum, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR) 032, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    4. Oikonomou, V.K. & Jost, J, 2013. "Periodic strategies and rationalizability in perfect information 2-Player strategic form games," MPRA Paper 48117, University Library of Munich, Germany.

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