1. Bayesian Persuasion cn
L18
Kamienica and Genzkow (AER 2011)

2. Basic Bayesian Persuasion
Two agents: Sender (S) and Receiver (R)
Type space Action space Message space
Preferences
S sends message, , R responds with an action
S ex ante commits to
Solution concept
Important tool

3. Concavification Aumann and Mashler (1995)
set of all Bayes plausible distributions of posteriors
Proof (straightforward)
Implication: Concavification of value function

4. Example 2: KG example Story: prosecutor S and judge R Binary model
Preferences
For beliefs R optimal choice
Expected S utility given common beliefs (no persuasion)

5. Obfuscation of information
KG example: function is neither concave nor concave

6. Lessons from the example
Partial transmission may be optimal (obfuscation of information)
Information transmission iff default action
(beliefs ) reveal low type, (worst) action is optimal ex post
(beliefs ) does not reveal high type,
No action is optimal ex post
Actions are indifferent
KG generalize these observations

7. Transmission of information (necessity)
Transmission of information=benefits from persuasion
Fix prior and let
D: S would share beliefs with R if
Example:
P: S has incentives to transmit information only if exist that S would share
Remark: verification of the condition requires graph of

8. Heuristic proof

9. Transmission of information (sufficiency)
D: R preferences are discrete at if
Remarks:
preferences are discrete, generically
and continuous no
is optimal for posterior in the neighborhood of
P: Suppose at R preferences are discrete and there is that S would share. Then S benefits from transmission of information

10. Heuristic proof

11. Ex post optimality of worst action
In KG example for message (posterior ) r optimal ex post
D: is worst action if
D: Action is ex post optimal for if
Suppose one of optimal posteriors leads to worst action,
P: Action is ex post optimal at
Proof
Does worst action always exist?
Indifference condition generalizes under stronger conditions

12. Expected state model hard to work with when Suppose Expected S utility
Exists
Let be a concave closure of
Can be used to make predictions

13. Expected state model For any prior
is not a value function of the persuasion problem but its upper bound
P: S benefits from transmission of information iff
Not helpful in finding optimal message strategy

14. Summary Value function in the persuasion problem is concave
A geometric tool to find optimal message strategy in 2 state example
Some results characterizing persuasion
Not clear if they this help to find optimal mechanism in a general model
Limitations:
settings with more than two states?
Settings with two or more agents?
Solutions:
Settings with two or more agents
2-stage approach