Many Senders L8

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

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

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

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

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

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

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

Proof cn

Proof cn

Proof cn

Proof cn

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

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

Proof

Proof

Example Messages R prior and action

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

``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

(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

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

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