Competition in Persuasion L22 Gentzkow and Kamenica (Restud 2017)

Research questions Competition among senders fosters information transmission (Krishna and Morgan 2001, Battaglini 2002, Ambrus and Lu 2009) Robust fully revealing BNE with 2 senders Bubbling equilibrium Does competition among senders also improve transmission when senders maximize ex ante welfare commit to revealing signals In particular: Informativeness of equilibrium relative to a collusive benchmark Simple characterization of equilibrium set

Motivating example Senders: 2 pharmaceutical firms, maximize market share for their drugs Receivers: consumers (unit mass) for each consumer efficacy of drug is mass 0.5 buys drug with higher expected efficacy mass 0.5 buys better drug only if that reveals efficacy of a drug Two variants of a game A firm can commit to revealing test results for own drug (Game 1) A firm can commit to revealing results for arbitrary tests (Game 2)

Game 1 (own drug tests) Firm can chose two ``strategies’’ Demands (payoffs) Nash: Collusive outcome Observations: Competition: less than full revelation information Competitive outcome less informative than collusive outcome

Game 2 (testing both drugs) Each firm has four strategies Demands (payoffs) Nash vs collusive outcome Nash outcome as informative as collusive one

Lessons What structure drives the predictions in both games? In Game 2 none of the firms has a monopoly on a particular piece of information This feature of information structure is called Blackwell connectedness The game becomes of a provision of free public good In what follows: We define a game that captures competition in persuasion Introduce a notion of Blackwell connectedness Demonstrate information aggregation result Give some other results

Simultaneous move (persuasion) game State space , common prior Signal Game: Players: senders Strategies: , available signals For its outcome is a distribution over posteriors Payoffs: Example

Important outcomes Feasible outcome Set of feasible outcomes Equilibrium outcome Collusive outcome We assume that is unique

Blackwell connectedness Blackwell partial order on set feasible outcomes Two signals are equivalent if D: Set is Blackwell connected if for any Interpretation: Each sender can provide as much information as other players combined No player has exclusivity in providing particular type of information Each player has to be able to exactly match the benchmark

Blackwell Connected S: examples Independent draws: each player choses a number of i.i.d. draws from some fixed distribution. Aggregate information Facts: Each player choses which facts to uncover. , aggregate information Precision: each player choses All or nothing: In these examples Blackwell order is complete on

Lower bound on equilibrium informativeness P: (for all preferences) iff is Blackwell connected Remark: For BC strategy space either or. not comparable In the examples Blackwell order is complete In the latter competition is good for information transmission Independent tests of drugs: collusion hurts information transmission Other examples

Proof If Only if

Equilibrium set For the remaining results assume: Identical strategy sets Blackwell connectedness

Characterization of equilibrium set D: For sender outcome is unimprovable if for any feasible one has P: A feasible outcome I. is an equilibrium outcome iff it is unimprovable for all Remark: Equilibrium set can be determined as an intersection of unimprovable sets for all players

Other results Suppose there exists feasible. such that P: is an equilibrium outcome Proof: is trivially unimprovable Consider least informative equilibrium (minimal equilibrium) Informativess of minimal equilibrium is non-decreasing in number of players misalignment of players preferences