1. Many Senders (part 2) L9

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. 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

4. Previous class In the previous class we assumed
Revelation principle adapted to FR equilibrium
FR equilibrium exists under mild conditionse
Example

5. Plausibility of equilibri
Fully revealing equilibium exists under mild assumptions
Are fully revealing equilibria plausible?
Ad hoc off-equilibrium beliefs
Discontinuity: negligible discrepancy results in dramatic changes in beliefs
Introspection
A reasonable restriction on off equilibrium beliefs
No widely accepted refinement criterion for continuous types

6. ``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’s mistakes
``More continuous’’ beliefs

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

8. Heuristic argument Complication:: revelation principle does not apply
Full revelation
Large sets may support robust beliefs that are not robust with truthtelling

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

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

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

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

13. Heuristic argument
consider

14. Two alternative solutions to the problem
With mistakes, expected value cannot be different from combination of reports
This rules our extreme actions as punishments
Solution to nonexistence problem
- Battaglini: multidimensional type spaces
- Ambrus and Lue: Almost fully revealing equilibrium

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

16. Example Outcomes Quadratic preferences with biases Message space
Equilibrium

17. Proof

18. Proof cn

19. Proof cn

20. Extensions
Preferences (quasiconcavity in outcomes)
Dimensionality