1. Many Senders (part 3)
L10

2. Previous classes
Fully revealing (FR) equilibrium exists under mild assumptions
FR equilibria not very plausible:
Ad hoc off-equilibrium beliefs
Discontinuity: negligible discrepancy results in dramatic changes in beliefs
Rational emotions?
Robust FR equilibrium (Battaglini’s trembling hand)
Robust FR equilibrium exists iff
Today: , almost fully revealing equilibria

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

5. Noiseless cheap talk Noiseless setting, fixed biases
P: For every there exists large enough such that in equilibrium
for any
Constructive proof
2 ideas:
Safe margin
partition with sets identified by 2 indices
Creating two dimensions within one dimension

6. Safe margin Suppose true state is and satisfies For what we know that
If bound holds for then

7. Partition Partition with labels Each cell uniquely identified by
Message space
Truth telling
Unilateral deviations

8. General Partition Suppose there are blocks and cells
cell in block is uniquely identified by
Message space
Truth telling
Misreporting

9. Proof P: For every there exists large enough such that in equilibrium
for any
Suffices to show that for large enough exists a partition such that

10. Block and cell size

11. Noise How about small noise?
Suppose in the noiseless setting W s.t. strict preference for truth telling.
P:There exists s.t. for all the following profile is a PBN

12. Heuristic argument How about small compact set
Expected payoff continuous in
Maximum theorem
Strict preference for truth telling with

13. Implications For PBN Equilibrium is robust
For condition is satisfied with probability as
With , for outcome converges to a fully revealing one

14. Extensions Unbounded state space Continuous noise
Multidimensional state space
Commitment

15. Summary: Framework and demonstration of FR equilibrium
Gilligan and Khrehbiel (AJPS 1989)
Krishna and Morgan (APSR 2001)
Necessary and sufficient condition for existence of FR equilibrium
Battaglini (ECMA 2002) d=1
Ambrus and Takahashi (TE 2008) d=2
Arguments rely on a revelation principle
Robust (Almost) FR equilibrium
- Ambrus and Lu (GEB 2014)

16. Limitations and Open Questions
Strong existence FR equilibrium results
The equilibrium set very large
Theoretical questions:
Characterization of full set of eqilibria
Plausibility of different equilibria (refinements)
Welfare rankings
Empirically motivated questions
Benefits from consulting more than 2 experts (robust to collusion?)
Sequential communication,
Conferences