1. Competition in Persuasion
L22
Gentzkow and Kamenica (Restud 2017)

2. Research questions
Competition among senders fosters information transmission
(Krishna and Morgan 2001, Battaglini 2002, Ambrus and Lu 2009)
Robust fully revealing BNE with 2 senders
Bubbling equilibrium
Does competition among senders also improve transmission when
senders maximize ex ante welfare
commit to revealing signals
In particular:
Informativeness of equilibrium relative to a collusive benchmark
Simple characterization of equilibrium set

3. Motivating example
Senders: 2 pharmaceutical firms, maximize market share for their drugs
Receivers: consumers (unit mass)
for each consumer efficacy of drug is
mass 0.5 buys drug with higher expected efficacy
mass 0.5 buys better drug only if
that reveals efficacy of a drug
Two variants of a game
A firm can commit to revealing test results for own drug (Game 1)
A firm can commit to revealing results for arbitrary tests (Game 2)

4. Game 1 (own drug tests) Firm can chose two ``strategies’’
Demands (payoffs)
Nash: Collusive outcome
Observations:
Competition: less than full revelation information
Competitive outcome less informative than collusive outcome

5. Game 2 (testing both drugs)
Each firm has four strategies
Demands (payoffs)
Nash vs collusive outcome
Nash outcome as informative as collusive one

6. Lessons What structure drives the predictions in both games?
In Game 2 none of the firms has a monopoly on a particular piece of information
This feature of information structure is called Blackwell connectedness
The game becomes of a provision of free public good
In what follows:
We define a game that captures competition in persuasion
Introduce a notion of Blackwell connectedness
Demonstrate information aggregation result
Give some other results

7. Simultaneous move (persuasion) game
State space , common prior
Signal
Game:
Players: senders
Strategies: , available signals
For its outcome is a distribution over posteriors
Payoffs:
Example

8. Important outcomes Feasible outcome Set of feasible outcomes
Equilibrium outcome
Collusive outcome
We assume that is unique

9. Blackwell connectedness
Blackwell partial order on set feasible outcomes
Two signals are equivalent if
D: Set is Blackwell connected if for any
Interpretation:
Each sender can provide as much information as other players combined
No player has exclusivity in providing particular type of information
Each player has to be able to exactly match the benchmark

10. Blackwell Connected S: examples
Independent draws: each player choses a number of i.i.d. draws from some fixed distribution Aggregate information
Facts: Each player choses which facts to uncover , aggregate information
Precision: each player choses
All or nothing:
In these examples Blackwell order is complete on

11. Lower bound on equilibrium informativeness
P: (for all preferences) iff is Blackwell connected
Remark:
For BC strategy space either or not comparable
In the examples Blackwell order is complete
In the latter competition is good for information transmission
Independent tests of drugs: collusion hurts information transmission
Other examples

12. Proof If
Only if

13. Equilibrium set For the remaining results assume:
Identical strategy sets
Blackwell connectedness

14. Characterization of equilibrium set
D: For sender outcome is unimprovable if for any feasible
one has
P: A feasible outcome I. is an equilibrium outcome iff it is unimprovable for all
Remark: Equilibrium set can be determined as an intersection of unimprovable sets for all players

16. Other results Suppose there exists feasible. such that
P: is an equilibrium outcome
Proof: is trivially unimprovable
Consider least informative equilibrium (minimal equilibrium)
Informativess of minimal equilibrium is non-decreasing in
number of players
misalignment of players preferences