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The Persuasion Duality

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  • Piotr Dworczak
  • Anton Kolotilin

Abstract

We present a unified duality approach to Bayesian persuasion. The optimal dual variable, interpreted as a price function on the state space, is shown to be a supergradient of the concave closure of the objective function at the prior belief. Strong duality holds when the objective function is Lipschitz continuous. When the objective depends on the posterior belief through a set of moments, the price function induces prices for posterior moments that solve the corresponding dual problem. Thus, our general approach unifies known results for one-dimensional moment persuasion, while yielding new results for the multi-dimensional case. In particular, we provide a necessary and sufficient condition for the optimality of convex-partitional signals, derive structural properties of solutions, and characterize the optimal persuasion scheme in the case when the state is two-dimensional and the objective is quadratic.

Suggested Citation

  • Piotr Dworczak & Anton Kolotilin, 2019. "The Persuasion Duality," Papers 1910.11392, arXiv.org, revised Sep 2022.
    • Handle: RePEc:arx:papers:1910.11392
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    References listed on IDEAS

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    1. Robert J. Aumann, 1995. "Repeated Games with Incomplete Information," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262011476, December.
    2. Emir Kamenica & Matthew Gentzkow, 2011. "Bayesian Persuasion," American Economic Review, American Economic Association, vol. 101(6), pages 2590-2615, October.
    3. Anton Kolotilin & Tymofiy Mylovanov & Andriy Zapechelnyuk & Ming Li, 2017. "Persuasion of a Privately Informed Receiver," Econometrica, Econometric Society, vol. 85(6), pages 1949-1964, November.
    4. Kolotilin, Anton, 2018. "Optimal information disclosure: a linear programming approach," Theoretical Economics, Econometric Society, vol. 13(2), May.
    5. Piotr Dworczak & Giorgio Martini, 2019. "The Simple Economics of Optimal Persuasion," Journal of Political Economy, University of Chicago Press, vol. 127(5), pages 1993-2048.
    6. D. Gale, 1967. "A Geometric Duality Theorem with Economic Applications," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 34(1), pages 19-24.
    7. Dizdar, Deniz & Kováč, Eugen, 2020. "A simple proof of strong duality in the linear persuasion problem," Games and Economic Behavior, Elsevier, vol. 122(C), pages 407-412.
    8. Luis Rayo & Ilya Segal, 2010. "Optimal Information Disclosure," Journal of Political Economy, University of Chicago Press, vol. 118(5), pages 949-987.
    9. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, September.
    10. Nikandrova, Arina & Pancs, Romans, 2017. "Conjugate information disclosure in an auction with learning," Journal of Economic Theory, Elsevier, vol. 171(C), pages 174-212.
    11. Laura Doval & Vasiliki Skreta, 2018. "Constrained Information Design," Papers 1811.03588, arXiv.org, revised Aug 2022.
    12. Matthew Gentzkow & Emir Kamenica, 2016. "A Rothschild-Stiglitz Approach to Bayesian Persuasion," American Economic Review, American Economic Association, vol. 106(5), pages 597-601, May.
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    Cited by:

    1. Emiliano Catonini & Sergey Stepanov, 2022. "On the optimality of full disclosure," Papers 2202.07944, arXiv.org, revised Feb 2023.
    2. Anton Kolotilin & Alexander Wolitzky, 2020. "Assortative Information Disclosure," Discussion Papers 2020-08, School of Economics, The University of New South Wales.
    3. Yamashita, Takuro & Smolin, Alex, 2022. "Information Design in Concave Games," TSE Working Papers 22-1313, Toulouse School of Economics (TSE).
    4. Zeng, Yishu, 2023. "Derandomization of persuasion mechanisms," Journal of Economic Theory, Elsevier, vol. 212(C).
    5. Alex Smolin & Takuro Yamashita, 2022. "Information Design in Smooth Games," Papers 2202.10883, arXiv.org, revised Dec 2023.

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