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Distribution-Valued Solution Concepts

Author

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  • David H. Wolpert
  • James Bono

Abstract

Under its conventional positive interpretation, game theory predicts that the mixed strategy pro?le of players in a noncooperative game will satisfy some setvalued solution concept. Relative probabilities of pro?les in that set are unspeci?ed, and all pro?les not satisfying it are implicitly assigned probability zero. However the axioms underlying Bayesian rationality say that we should reason about player behavior using a probability density over all mixed strategy pro?les, not using a subset of all such pro?les. Such a density over pro?les can be viewed as a solution concept that is distribution-valued rather than set-valued. A distribution-valued concept provides a best single prediction for any noncooperative game, i.e., a universal re?nement. In addition, regulators can use a distribution-valued solution concept to make Bayes optimal choices of a mechanism, as required by Savage's axioms. In particular, they can do this in strategic situations where conventional mechanism design cannot provide advice. We illustrate all of this on a Cournot duopoly game.

Suggested Citation

  • David H. Wolpert & James Bono, 2010. "Distribution-Valued Solution Concepts," Working Papers 2010-13, American University, Department of Economics.
  • Handle: RePEc:amu:wpaper:2010-13
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    File URL: https://drive.google.com/open?id=1V5Nl--hKBQkp9xjZwoTKxKmOz43OuczH
    File Function: First version, 2010
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    References listed on IDEAS

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    1. Colin F. Camerer & Teck-Hua Ho & Juin-Kuan Chong, 2004. "A Cognitive Hierarchy Model of Games," The Quarterly Journal of Economics, Oxford University Press, vol. 119(3), pages 861-898.
    2. Vincent P. Crawford & Miguel A. Costa-Gomes, 2006. "Cognition and Behavior in Two-Person Guessing Games: An Experimental Study," American Economic Review, American Economic Association, vol. 96(5), pages 1737-1768, December.
    3. Aguirregabiria, Victor & Mira, Pedro, 2010. "Dynamic discrete choice structural models: A survey," Journal of Econometrics, Elsevier, vol. 156(1), pages 38-67, May.
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    Cited by:

    1. David H. Wolpert & James Bono, 2010. "A theory of unstructured bargaining using distribution-valued solution concepts," Working Papers 2010-14, American University, Department of Economics.

    More about this item

    Keywords

    Quantal Response Equilibrium; Bayesian Statistics; Entropic prior; Maximum entropy JEL Codes: C02; C11; C70; C72;

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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