Constrained Persuasion with Private Information

• Andrew Kosenko

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.

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:

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:

(8) v ̂ ( Π H , μ , θ H ) u * ( θ H ) = ρ H P ( ω H | θ H ) 1 μ | β ( Π i , σ H , μ ) 1 2 + P ( ω L | θ H ) 1 | μ | β ( Π i , σ L , μ ) 1 2 + ( 1 ρ H ) P ( ω H | θ H ) 1 μ | β ( Π i , σ L , μ ) 1 2 + P ( ω L | θ H ) 1 μ | β ( Π i , σ H , μ ) 1 2 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 = P ( ω H | θ H ) ρ H 1 μ | β ( Π i , σ H , μ ) 1 2 ρ L 1 β ( Π L , σ H , π ) 1 2 + ( 1 ρ H ) 1 μ | β ( Π i , σ L , μ ) 1 2 ( 1 ρ L ) 1 β ( Π L , σ L , π ) 1 2 + ( P ( ω L | θ H ) ) ρ H 1 | μ | β ( Π i , σ L , μ ) 1 2 ρ L 1 β ( Π L , σ L , π ) 1 2 + ( 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

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. □

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