1. Many Senders L8 Gilligan and Khrehbiel (AJPS 1989)
Krishna and Morgan (APSR 2001)
Battaglini (ECMA 2002)
Ambrus and Takahashi (TE 2008)
Ambrus and Lu (GEB 2014)

2. Observations Two benefits from consulting multiple senders
Confirming facts and punishing conflicting information
Eliciting information along common interests, aggregating
Battaglini (2002) : a thriller with a happy end
General message: (almost) full revelation is robustly feasible under mild conditions even without off equilibrium punishment
Conceptual contributions
Revelation principle for games
Robustness (Trembling hand ) criterion

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

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

5. 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

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

7. 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

8. Observation Consider messages Does enforce truth telling
Penalty has to be message dependent
Can we always find extreme action penalizing a liar?
Can we support extreme actions with beliefs?

9. Proof
Consider report
What enforces truth telling (assuming truth telling of for

10. Proof
L: Fully revealing equilibrium exists iff for any pair set is nonempty

11. Proof
L: Fully revealing equilibrium exists iff for any pair set is nonempty

12. Example
Messages
R action
R beliefs

13. Substantive insight
Fully revealing equilibium exists under mild assumptions
Is such equilibrium plausible?
Ad hoc off-equilibrium beliefs
Discontinuity: negligible discrepancy results in dramatic changes in beliefs
Introspection: FR equilibrium is just a theoretical peculiarity
No widely accepted belief refinement for continuous types

14. ``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
``All reports on equilibrium path

15. (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
``Battaglini’s’’ trembling hand
Heuristic argument
With mistakes, expected value cannot be different from combination of reports
This rules our extreme actions as punishments

16. Two alternative solutions to the problem
Solution to nonexistence problem
- Battaglini: multidimensional type spaces
- Ambrus and Lue: Almost fully revealing equilibrium

17. Example Outcomes Quadratic preferences with biases Message space
Equilibrium

18. Proof

19. Proof

20. Battaglini Assume dimensional state space
Quadratic preferences with arbitrarily large biases
independent bias vectors
P: Robust fully revealing equilibrium exists

21. Extensions
Preferences (quasiconcavity in outcomes)
Dimensionality

22. Proof

23. Proof cn

24. Proof cn

25. Proof cn

26. Proof cn

27. Proof cn

28. Proof cn

29. Non-existence of robust equilibira
Complication:: revelation principle does not apply
Full revelation
Large sets may support robust beliefs that are not robust with truthtelling

30. Heuristic argument Complication:: revelation principle does not apply
Full revelation
Large sets may support robust beliefs that are not supportable with truth telling

31. Heuristic argument Suppose Any two elements of must be apart.
With sufficiently large bias is one to one

32. Heuristic argument Suppose By analogous argument one to one
Three possible events (on equilibrium path)

33. Heuristic argument Suppose By analogous argument one to one
Three possible events (on equilibrium path)

34. Heuristic argument
consider