1. Many Senders L8

2. Papers Gilligan and Khrehbiel (AJPS 1989)
Krishna and Morgan (APSR 2001)
Battaglini (ECMA 2002)
Ambrus and Takahashi (TE 2008)
Ambrus and Lu (GEB 2014)

3. Observations Two benefits from consulting multiple senders
Confirming facts and punishing conflicting information
Eliciting information along common interests, aggregating
Battaglini (2002) : a thriller with happy end
General message: (almost) full revelation is feasible under mild conditions even without off equilibrium punishment

4. Multidimensional Cheap Talk
Agents: Two senders and Receiver
Timing and actions:
State
Each senders observe signal
Senders simultaneously send
Receiver observes messages , choses action
Preferences
We first assume

5. Notation Each of the papers assumes its own notation
Battaglini vs ``our’’ notation

6. PBN Equilibrium Strategies: Senders Receives Posterior
D: Equilibrium s.t. 1. 2. 3.

7. Revelation principle Fully revealing equilibrium: Truthful revelation
Message space
Equilibrium strategies
L:Suppose fully revealing equilibrium exists. Then there exists a truthfully revealing equilibrium with degenerate beliefs (in and out of equilibrium).
Nonexistence of fully revealing e can be established in a simple setting
Revelation principle stronger than in MD

8. Proof
First we show revelation principle and then degeneracy of beliefs
Let be a PBN equilibrium

9. Proof cn

10. Proof cn

11. Proof cn

12. Proof cn

13. Existence of a fully revealing equilibrium (d=1)
Krishna and Morgan (APSR 2001)
Battaglini (ECMA 2002): Necessary and sufficient condition
Assume
one dimensional state space, (hard case)
Opposite biases
P: Fully revealing equilibrium exists if and only if
Idea:
Discrepancies penalized with extreme action (off equilibrium)
Problem: sequential rationality and existence of extreme action

14. Proof Consider messages Does enforce truth telling
Can we always find extreme action penalizing a liar?

15. Proof

16. Proof

17. Example
Messages
R prior and action

18. Substantive insight
Fully revealing equilibium exists under very mild assumptions
Are fully revealing equilibria plausible?
Ad hoc off-equilibrium beliefs
Discontinuity: negligible discrepancy results in dramatic changes in beliefs
Introspection: FR equilibrium is just a theoretical peculiarity
A reasonable restriction on off equilibrium beliefs
No widely accepted refinement criterion for continuous types

19. ``Battaglini’s’’ trembling hand
Robust equilibrium
Consider a game with signals
For each game find equilibrium
Limit of a sequence of equilibria as is a robust equilibrium
Game specific analog of ``trembling hand’’
Restrictions
Discrepancies interpreted as expert mistakes
``Continuous’’ beliefs

20. (Non)Existence of fully revealing equilibrium
Assume
P: For biases large enough there does not exist robust fully revealing equilibrium for any W
Robust equilibrium refines away all equilibra
Implication: full revelation should not be observed in reasonable settings

21. Heuristic argument Problem: revelation principle does not apply
Set might be large ( )
With biases large enough
Assume

22. Next class Quite negative result
Two ways to salvage the full revelation result:
Battaglini: multidimensional type space
Ambrus and Lue: nearly robust equilibrium