Many Senders (part 2) L9

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

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

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

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

``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

(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

Heuristic argument 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

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

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

Example Outcomes Quadratic preferences with biases Message space Equilibrium

Proof

Proof cn

Proof cn

Extensions Preferences (quasiconcavity in outcomes) Dimensionality