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Iterated Admissibility as Solution Concept in Game Theory

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  • Mario Gilli

    (Department of Economics, University of Milan-Bicocca)

Abstract

Admissibility, i.e. the deletion of weakly dominated strategies, is a highly controversia1 solution concept for non cooperative games. This paper proposes a complete theory of weak dominance and, contrary to almost al1 the literature on this topic, it provides positive results on foundations of iterated admissibility. The main contribution of this work is to show that (iterated) admissibility can be justified once payoffs' ties are seriously taken into considerations and players optimise taking into consideration the information provided by these ties, i.e. using strategic independent sets (Mailath at al. 1993) and conditional dominance (Shimoji and Watson 1998). In particular we prove that (iterated) maximal simultaneous deletion of weakly dominated strategies endogenously emerges as axiomatic characterization of iterated admissibility. As a consequence of this result, the paper provides axiomatic and Bayesian foundations of iterated admissibility, proves the logical consistency of (iterated) admissibility as solution concept in game theory, and that common knowledge of admissibility leads to iterated admissibility, showing why previous attempts using cautious behaviour were ineffective.

Suggested Citation

  • Mario Gilli, 2002. "Iterated Admissibility as Solution Concept in Game Theory," Working Papers 47, University of Milano-Bicocca, Department of Economics, revised Mar 2002.
  • Handle: RePEc:mib:wpaper:47
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    File URL: http://repec.dems.unimib.it/repec/pdf/mibwpaper47.pdf
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    References listed on IDEAS

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    1. Samuelson, L., 1989. "Evolutionnary Stability In Asymmetric Games," Papers 11-8-2, Pennsylvania State - Department of Economics.
    2. Marx, Leslie M. & Swinkels, Jeroen M., 2000. "Order Independence for Iterated Weak Dominance," Games and Economic Behavior, Elsevier, vol. 31(2), pages 324-329, May.
    3. Tan, Tommy Chin-Chiu & da Costa Werlang, Sergio Ribeiro, 1988. "The Bayesian foundations of solution concepts of games," Journal of Economic Theory, Elsevier, vol. 45(2), pages 370-391, August.
    4. Kreps,David M. & Wallis,Kenneth F. (ed.), 1997. "Advances in Economics and Econometrics: Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521589819, November.
    5. Mailath, George J & Samuelson, Larry & Swinkels, Jeroen M, 1993. "Extensive Form Reasoning in Normal Form Games," Econometrica, Econometric Society, vol. 61(2), pages 273-302, March.
    6. van Damme, E.E.C., 1991. "Refinements of Nash equilibrium," Other publications TiSEM 5d736b2e-6d58-4aef-b78c-4, Tilburg University, School of Economics and Management.
    7. Borgers, Tilman & Samuelson, Larry, 1992. ""Cautious" Utility Maximization and Iterated Weak Dominance," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(1), pages 13-25.
    8. Vincent J. Vannetelbosch & P. Jean-Jacques Herings, 1999. "Refinements of rationalizability for normal-form games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(1), pages 53-68.
    9. Jackson Matthew O. & Palfrey Thomas R. & Srivastava Sanjay, 1994. "Undominated Nash Implementation in Bounded Mechanisms," Games and Economic Behavior, Elsevier, vol. 6(3), pages 474-501, May.
    10. Bernheim, B Douglas, 1984. "Rationalizable Strategic Behavior," Econometrica, Econometric Society, vol. 52(4), pages 1007-1028, July.
    11. Hillas, John, 1990. "On the Definition of the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 58(6), pages 1365-1390, November.
    12. Abreu Dilip & Matsushima Hitoshi, 1994. "Exact Implementation," Journal of Economic Theory, Elsevier, vol. 64(1), pages 1-19, October.
    13. Ho, Teck-Hua & Camerer, Colin & Weigelt, Keith, 1998. "Iterated Dominance and Iterated Best Response in Experimental "p-Beauty Contests."," American Economic Review, American Economic Association, vol. 88(4), pages 947-969, September.
    14. Stahl, Dale O., 1995. "Lexicographic rationalizability and iterated admissibility," Economics Letters, Elsevier, vol. 47(2), pages 155-159, February.
    15. Samuelson, Larry & Zhang, Jianbo, 1992. "Evolutionary stability in asymmetric games," Journal of Economic Theory, Elsevier, vol. 57(2), pages 363-391, August.
    16. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    17. Moulin, Herve, 1979. "Dominance Solvable Voting Schemes," Econometrica, Econometric Society, vol. 47(6), pages 1137-1151, November.
    18. Milgrom, Paul & Roberts, John, 1991. "Adaptive and sophisticated learning in normal form games," Games and Economic Behavior, Elsevier, vol. 3(1), pages 82-100, February.
    19. Kreps,David M. & Wallis,Kenneth F. (ed.), 1997. "Advances in Economics and Econometrics: Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521589833, November.
    20. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-1050, July.
    21. Marx, Leslie M., 1999. "Adaptive Learning and Iterated Weak Dominance," Games and Economic Behavior, Elsevier, vol. 26(2), pages 253-278, January.
    22. Battigalli, Pierpaolo & Siniscalchi, Marciano, 1999. "Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games," Journal of Economic Theory, Elsevier, vol. 88(1), pages 188-230, September.
    23. Moulin, Herve, 1994. "Social choice," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 31, pages 1091-1125, Elsevier.
    24. Shimoji, Makoto & Watson, Joel, 1998. "Conditional Dominance, Rationalizability, and Game Forms," Journal of Economic Theory, Elsevier, vol. 83(2), pages 161-195, December.
    25. Abreu, Dilip & Pearce, David G., 1984. "On the inconsistency of certain axioms on solution concepts for non-cooperative games," Journal of Economic Theory, Elsevier, vol. 34(1), pages 169-174, October.
    26. Kreps,David M. & Wallis,Kenneth F. (ed.), 1997. "Advances in Economics and Econometrics: Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521589826, November.
    27. Samuelson, Larry, 1992. "Dominated strategies and common knowledge," Games and Economic Behavior, Elsevier, vol. 4(2), pages 284-313, April.
    28. Bernheim, B Douglas, 1986. "Axiomatic Characterizations of Rational Choice in Strategic Environme nts," Scandinavian Journal of Economics, Wiley Blackwell, vol. 88(3), pages 473-488.
    29. Dufwenberg, Martin & Stegeman, Mark, 1999. "When Order matters for Iterated Strict Dominance," Research Papers in Economics 1999:2, Stockholm University, Department of Economics.
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    Cited by:

    1. Lucia Buenrostro & Amrita Dhillon & Peter Vida, 2013. "Scoring rule voting games and dominance solvability," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(2), pages 329-352, February.

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    More about this item

    Keywords

    admissibility; axiomatic and Bayesian foundations; logical consistency; cautious behaviour;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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