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)

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

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

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

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

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?

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

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

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

Example Messages R action R beliefs

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

``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

(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

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

Example Outcomes Quadratic preferences with biases Message space Equilibrium

Proof

Proof

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

Extensions Preferences (quasiconcavity in outcomes) Dimensionality

Proof

Proof cn

Proof cn

Proof cn

Proof cn

Proof cn

Proof cn

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

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

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

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

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

Heuristic argument consider