Constrained Persuasion with Private Information

Andrew Kosenko

doi:10.1515/bejte-2021-0017

Published by De Gruyter June 13, 2022

Constrained Persuasion with Private Information

Andrew Kosenko

From the journal The B.E. Journal of Theoretical Economics

https://doi.org/10.1515/bejte-2021-0017

Showing a limited preview of this publication:

Abstract

I study a model of strategic communication between a privately informed sender who can persuade a receiver using Blackwell experiments. Hedlund (2017). “Bayesian Persuasion by a Privately Informed Sender.” Journal of Economic Theory 167 (January): 229–68, shows that private information in such a setting results in extremely informative equilibria. I make three points: first, the informativeness of equilibria relies crucially on two features – the mere availability of a fully revealing experiment, and a compact action space for the receiver. I show by examples that absent these features, equilibria may be uninformative. Secondly, I characterize equilibria in a simple model with constraints for the sender (only two experiments available, none are fully revealing) and the receiver (discrete action space). I argue that noisy experiments and discrete actions are the norm rather than the exception (and therefore, private information need not result in information revelation). Thirdly, I define a novel refinement that selects the most informative equilibria in most cases.

Keywords: persuasion; communication; information provision; signaling; information transmission; information design

JEL Classification: D82; D83; C72

Corresponding author: Andrew Kosenko, Assistant Professor of Economics, Department of Economics, Accounting, and Finance, School of Management, Marist College, Poughkeepsie, USA, E-mail: kosenko.andrew@gmail.com

Appendix

Proof of Proposition 7

First, it is immediate that SEP is a BPM equilibrium, since there are no out-of-equilibrium beliefs to consider, and thus criterion BPM is trivially satisfied. The reason that PNT-LL(a _H) and PNT-HH(a _H) survive criterion BPM is that deviations from those equilibria do not yield a strictly higher payoff for either type. The computation that eliminates FNT-L and PNT-LH(a _L) goes as follows: Take any pooling equilibrium where both types choose the experiment Π_L and the receiver takes different actions on the equilibrium path. In that equilibrium, u*(θ _H) =

(6) v ̂ ( Π L , π , θ H ) = ρ L P ( ω H | θ H ) 1 β ( Π L , σ H , π ) ≥ 1 2 + P ( ω L | θ H ) 1 β ( Π L , σ L , π ) ≥ 1 2 + ( 1 − ρ L ) P ( ω H | θ H ) 1 β ( Π L , σ L , π ) ≥ 1 2 + P ( ω L | θ H ) 1 β ( Π L , σ H , π ) ≥ 1 2

and u*(θ _L) =

(7) v ̂ ( Π L , π , θ L ) = ρ L P ( ω H | θ L ) 1 β ( Π L , σ H , π ) ≥ 1 2 + P ( ω L | θ L ) 1 β ( Π L , σ L , π ) ≥ 1 2 + ( 1 − ρ L ) P ( ω H | θ L ) 1 β ( Π L , σ L , π ) ≥ 1 2 + P ( ω L | θ L ) 1 β ( Π L , σ H , π ) ≥ 1 2

Fix a μ and consider the utility of deviating to Π_H for both types:

Now let μ ̲ solve ρ H μ ̲ ρ H μ ̲ + ( 1 − ρ H ) ( 1 − μ ̲ ) = 1 2 , (i.e. μ ̲ = 1 − ρ H ) and let μ ̄ solve ρ L μ ̄ ρ L μ ̄ + ( 1 − ρ L ) ( 1 − μ ̄ ) = 1 2 (i.e. μ ̄ = 1 − ρ L ) and note that since ρ _H > ρ _L, μ ̲ < μ ̄ . Also let † μ solve ( 1 − ρ L ) † μ ( 1 − ρ L † μ + ρ L ( 1 − † μ ) ) = 1 2 (i.e. † μ = ρ _L) and μ † = ( 1 − ρ H ) μ † ( 1 − ρ H ) μ † + ρ H ( 1 − μ † ) = 1 2 (i.e. μ† = ρ _H) and note that † μ < μ†. As before, I focus on nontrivial equilibria (so that I disregard the terms that involve observing the low signal/action). Now compute directly:

(9) v ̂ ( Π H , θ H , μ ) − u * ( θ H ) − ( v ̂ ( Π H , θ L , μ ) − u * ( θ L ) ) = P ( ω H | θ H ) − P ( ω H | θ L ) ρ H 1 μ | β ( Π i , σ H , μ ) ≥ 1 2 − ρ L 1 β ( Π L , σ H , π ) ≥ 1 2 + [ P ( ω L | θ H ) − P ( ω L | θ L ) ] ( 1 − ρ H ) 1 μ | β ( Π i , σ H , μ ) ≥ 1 2 − ( 1 − ρ L ) 1 β ( Π L , σ H , π ) ≥ 1 2 = u * ( θ L ) − u * ( θ H ) < 0 , for μ ∈ [ 0 , μ ̲ ) 2 ( ρ H − ρ L ) ( P ( ω H θ H ) − P ( ω H | θ L ) ) ) > 0 for μ ∈ [ μ ̲ , † μ ) 2 ρ L [ P ( ω H | θ L ) − P ( ω H | θ H ) ] + P ( ω H | θ H ) − P ( ω H | θ L ) < 0 for μ ∈ [ † μ , 1 ]

Since the difference is negative for first of the three ranges exhibited above, criterion BPM does not apply there. For the second range of beliefs the difference is strictly positive, and hence, beliefs that support PNT-LH(a _L) are ruled out. As for the third range, the difference is negative, but beliefs there are such that they cannot be part of any kind of nontrivial equilibrium at all (cf. the upper bounds on off-path beliefs for equilibria in Propositions 4 through 6 and note that criterion BPM restricts beliefs off the equilibrium path) and we are done. □

References

Alonso, R., and O. Camara . 2016. “Bayesian Persuasion with Heterogeneous Priors.” Journal of Economic Theory 165: 672–706. https://doi.org/10.1016/j.jet.2016.07.006.Search in Google Scholar

Alonso, R., and O. Camara . 2018. “On the Value of Persuasion by Experts.” Journal of Economic Theory 174: 103–23. https://doi.org/10.1016/j.jet.2017.12.001.Search in Google Scholar

Aumann, R. J., and M. B. Maschler . 1995. Repeated Games with Incomplete Information. Cambridge: MIT Press.Search in Google Scholar

Banks, J. S., and J. Sobel . 1987. “Equilibrium Selection in Signaling Games,” Econometrica, Econometric Society 55 (3): 647–61. https://doi.org/10.2307/1913604.Search in Google Scholar

Blackwell, D. 1951. “Comparison of Experiments.” In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 93–102. Berkeley: University of California Press.Search in Google Scholar

Blackwell, D. 1953. “Equivalent Comparisons of Experiments.” The Annals of Mathematical Statistics 24 (2): 265–72. https://doi.org/10.1214/aoms/1177729032.Search in Google Scholar

Cho, I.-K., and D. M. Kreps . 1987. “Signaling Games and Stable Equilibria.” Quarterly Journal of Economics 102: 179–221. https://doi.org/10.2307/1885060.Search in Google Scholar

Cho, I.-K., and J. Sobel . 1990. “Strategic Stability and Uniqueness in Signaling Games.” Journal of Economic Theory 50: 381–413. https://doi.org/10.1016/0022-0531(90)90009-9.Search in Google Scholar

Crawford, V. P., and J. Sobel . 1982. “Strategic Information Transmission.” Econometrica 50 (6): 1431–51. https://doi.org/10.2307/1913390.Search in Google Scholar

Degan, A. and M. Li 2015. “Persuasive Signalling” Working Paper. Available at SSRN: http://ssrn.com/abstract=1595511 or https://doi.org/10.2139/ssrn.1595511.Search in Google Scholar

Farrell, J. 1993. “Meaning and Credibility in Cheap-Talk Games.” Games and Economic Behavior 5 (4): 514–31. https://doi.org/10.1006/game.1993.1029.Search in Google Scholar

Gentzkow, M., and E. Kamenica . 2017a. “Competition in Persuasion.” The Review of Economic Studies 84 (4): 300–22. https://doi.org/10.1093/restud/rdw052.Search in Google Scholar

Gentzkow, M., and E. Kamenica . 2017b. “Bayesian Persuasion with Multiple Senders and Rich Signal Spaces.” Games and Economic Behavior 104: 411–29. https://doi.org/10.1016/j.geb.2017.05.004.Search in Google Scholar

Gill, D., and D. Sgroi . 2012. “The optimal Choice of Pre-launch Reviewer.” Journal of Economic Theory 147 (3): 1247–60. https://doi.org/10.1016/j.jet.2012.01.019.Search in Google Scholar

Grossman, S. J., and M. Perry . 1986. “Perfect Sequential Equilibrium.” Journal of Economic Theory 39 (1): 97–119. https://doi.org/10.1016/0022-0531(86)90022-0.Search in Google Scholar

de Groot Ruiz, A., T. Offerman and Sander, O. (2013). “Equilibrium Selection in Cheap Talk Games: ACDC Rocks when Other Criteria Remain Silent.” Working paper.10.2139/ssrn.1761947Search in Google Scholar

Hedlund, J. 2017. “Bayesian Persuasion by a Privately Informed Sender.” Journal of Economic Theory 167 (January): 229–68. https://doi.org/10.1016/j.jet.2016.11.003.Search in Google Scholar

Kamenica, E., and M. Gentzkow . 2011. “Bayesian Persuasion.” The American Economic Review 101 (6): 2590–615. https://doi.org/10.1257/aer.101.6.2590.Search in Google Scholar

Kartik, N., F. X. Lee, and W. Suen . 2021. “Information Validates the Prior: A Theorem on Bayesian Updating and Applications.” American Economic Review: Insights 3 (2): 165–82.10.1257/aeri.20200284Search in Google Scholar

Kohlberg, E., and J.-F. Mertens . 1986. “On the Strategic Stability of Equilibria.” Econometrica 54 (5): 1003–37. https://doi.org/10.2307/1912320.Search in Google Scholar

Kosenko, A. 2021. “Noisy Bayesian Persuasion with Private Information.” mimeo.Search in Google Scholar

Kreps, D. M., and R. Wilson . 1982. “Sequential Equilibria.” Econometrica, Econometric Society 50 (4): 863–94. https://doi.org/10.2307/1912767.Search in Google Scholar

Mailath, G. J., M. Okuno-Fujiwara, and A. Postlewaite . 1993. “Belief-Based Refinements in Signalling Games.” Journal of Economic Theory 60 (2): 241–76. https://doi.org/10.1006/jeth.1993.1043.Search in Google Scholar

McKelvey, R. D., and T. R. Palfrey . 1995. “Quantal Response Equilibria for Normal Form Games.” Games and Economic Behavior 10 (1): 6–38. https://doi.org/10.1006/game.1995.1023.Search in Google Scholar

Myerson, R. 1978. “Refinements of the Nash Equilibrium Concept.” International Journal of Game Theory 7: 73–80. https://doi.org/10.1007/bf01753236.Search in Google Scholar

Perez-Richet, E. 2014. “Interim Bayesian Persuasion: First Steps.” The American Economic Review: Papers & Proceedings 104 (5): 469–74. https://doi.org/10.1257/aer.104.5.469.Search in Google Scholar

Rayo, L., and I. Segal . 2010. “Optimal Information Disclosure.” Journal of Political Economy 118 (5): 949–87. https://doi.org/10.1086/657922.Search in Google Scholar

Selten, R. 1975. “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games.” International Journal of Game Theory 4: 25. https://doi.org/10.1007/BF01766400.Search in Google Scholar

Received: 2021-01-26

Accepted: 2022-03-26

Published Online: 2022-06-13

Constrained Persuasion with Private Information

Abstract

Proof of Proposition 7

References

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