Persuasion of Heterogenous Audiences and Optimal Media Censorship

Transcription

1 Persuasion of Heterogenous Audiences and Optimal Media Censorship Tymofiy Mylovanov joint work with Anton Kolotilin, Ming Li, Andriy Zapechelnyuk October 6,

2 Environment Sender Receiver Receiver makes a binary decision (action or inaction) 2

3 To convince Receiver to act, Sender controls disclosure of decision relevant information: Receiver s preferences are private Sender fully controls persuasion 3

4 Timing 1. Sender commits to an information disclosure mechanism 2. Sender observes the state & Receiver observes his type 3. Communication between Sender and Receiver 4. Receiver updates his beliefs and decides whether to act 4

5 Examples Ratings given by credit agency Grading policies designed by school Media controlled by government 5

6 Questions What can be achieved purely by means of persuasion? What can persuasion mechanisms gain over experiments? What persuasion mechanisms are optimally chosen? 6

7 Model Receiver must choose between inaction and action, a {0, 1}. Receiver s private type r G on R = [0, 1] State of the world ω F on Ω = [0, 1] Random variables r and ω are independent Distributions of F and G are common knowledge 7

8 Payoffs Receiver s payoff from actions (a = 0, 1): u(ω, r, a) = ω r, a = 1 0, a = 0 Sender s payoff from actions (a = 0, 1): v(ω, r, a) = 1 + ρ(r) (ω r), a = 1 0, a = 0 where ρ(r) R 8

9 Receiver s Private Information: Interpretation One Receiver Incomplete information about Receiver s type A continuum of heterogeneous Receivers an audience 9

10 Persuasion mechanism A direct mechanism π : R Ω A asks Receiver to report r recommends â = 1 w/pr π ( r, ω) and â = 0 o/w WLOG restrict attention to mechanisms in which Receiver is honest and obedient 10

11 Experiment An experiment σ : Ω (M) sends to Receiver a (stochastic) message in M for each realized state ω Unlike persuasion mechanisms, experiments inform all types of Receiver identically Any experiment can be represented as a persuasion mechanism Reverse? 11

12 Bayesian persuasion: Interpretation Commitment Sender is informed Ex-ante can commit to the way of disclosing information Test Sender is uninformed Can design a test whose results are publicly revealed 12

13 Incentive-compatible mechanisms 13

14 Notation For a mechanism π, define probability that r takes the action q π (r) = π (r, ω) df (ω) expected value of ω given r and a = 1 p π (r) = 1 q π (r) expected payoff of Receiver r U π (r) = ωπ (r, ω) df (ω) (ω r)π (r, ω) df (ω) = q π (r)(p π (r) r) analogy: standard linear trading environment 14

15 Incentive compatible mechanisms Lemma. A (feasible) mechanism π is incentive compatible iff q π is non-increasing (1) U π (r) = U π (0) = 1 r q π(s)ds, (2) 1 0 ωdf (ω) = E[ω], (3) there are obedience instead of individual rationality constraints there are also no transfers 15

16 Incentive compatible mechanisms Lemma. A mechanism π is incentive compatible if and only if q π is non-increasing (4) U π (r) = U π (0) = 1 r q π(s)ds, (5) 1 0 q π(s)ds = E[ω], (6) Differences from Mirrlees: There are two boundary conditions, for r = 1 and r = 0 Not all q π s that satisfy (??) (??) are implementable The set of implementable q π s depends on F not only through E F [ω] 16

17 Lemma is not too useful: What is the set of implementable (q π,u π,v π )? What can persuasion mechanisms gain over experiments? 17

18 Implementable utility schedules 18

19 Utility schedules A pair (q π,v π ) is pinned down by U π : q π (r) = U π(r), V π (r) = U π(r) + ρ(r)u π (r). We can focus on implementable utility profiles U π 19

20 Necessary condition 1 No information U (r) = max {E[ω] r, 0} Full information U (r) = 1 r (ω r) df (ω) Thus, under any mechanism U(r) U π (r) U(r) 20

21 Necessary condition 2 Incentive compatibility U π is convex (the standard Envelope Theorem argument) 21

22 Necessary conditions U(r) U(r) U(r) r 0 1 U (r) U (r) U (r) U is convex 22

23 Main Result Theorem. The following statements are equivalent: (a) U is a convex function such that U(r) U(r) U(r) for all r (b) U is implementable by a persuasion mechanism (c) U is implementable by an experiment Sketch of proof: (c) (b) trivially (b) (a) by monotonicity of q (a) (c) Mirrlees meets Blackwell 23

24 Sketch of Proof: (a) (c) Every experiment σ can be equivalently described by a c.d.f. H of the posterior mean state m = E σ [ω m]. E.g., fully informative experiment (m = ω): H(m) = F (m). 24

25 Sketch of Proof: (a) (c) Fix U convex s.t. U(r) U(r) U(r). Construction: Let H(r) = 1 + U (r) H is a c.d.f: H(0) = 0 because U (0) = U (0) = 1 H(1) = 1 because U (1) = U (1) = 0 H(r) is increasing because U is convex 25

26 Sketch of Proof: (a) (c) Observe that H(r) = 1 + U (r) satisfies 1 1 (1 H(s))ds = U(r) U(r) = (1 F (s))ds. (7) r r where 1 F (r) is probability that fully informed Receiver r chooses to act. Therefore, F is a mean-preserving spread of H Thus required experiment σ with c.d.f. H can be obtained by garbling of the fully informative experiment QED 26

27 Questions What can be achieved purely by means of persuasion? Any convex utility profile U between U and U What can persuasion mechanisms gain over experiments? Nothing 27

28 Optimal Mechanisms 28

29 Sender s Problem Lemma 1 For every incentive-compatible mechanism π, V π(r)dg(r) = g(0)e[ω] + U π(r)i(r)dr, R R where I(r) = g (r) + ρ(r)g(r) for all r R. 29

30 Sender s Problem subject to max U(r) R U(r)I(r)dr. U(r) is convex such that U(r) U(r) U(r) 30

31 Example I(r) = g (r) + ρ(r)g(r) is single-crossing from above e.g., density g(r) is single-peaked and ρ(r) = 0 There exists ω such that states ω < ω are separated and states ω > ω are pooled 31

32 Media Control 32

33 Model The government s state of affairs is a random variable ω drawn from [0, 1] according to the distribution F that admits a density f A continuum of media outlets [0, 1] A media outlet s [0, 1] has an editorial policy that endorses the government (sends message m s = 1) if ω > s and criticizes it (sends message m s = 0) if ω < s. The cutoff s can be interpreted as a slant or political bias of the outlet against the government and can be empirically measured as the frequency with which the outlet uses anti-government language. 33

34 Model A continuum of heterogeneous readers indexed by r [0, 1] distributed with G that admits a log-concave density g Each reader observes endorsements of all available media outlets and chooses between action (a = 1) and inaction (a = 0) A reader s utility is equal to u(ω, r, a, ā) = a(ω r) + āξ(r), where ā denotes the average action in the society, and āξ(r) is a type-specific externality term that contributes to the reader s utility but does not affect the reader s optimal action. 34

35 Model The government s utility is equal to ( ) u(ω, r, a r, ā)dg(r) + āγ = a r (ω r) + ā(ξ(r) + γ) dg(r), R R where a r denotes an action of type r, and āγ is the government s intrinsic benefit from the average action. We assume that ρ = ( R ) 1 (ξ(r) + γ)dg(r) > 0, meaning that the government is biased towards a greater average action in the society. 35

36 Model The government s censorship policy is a measurable set of the media outlets S [0, 1] that are prohibited to broadcast. Readers observe messages only from the permitted media outlets in [0, 1]\S. 36

37 Model The timing is as follows. the government chooses a set of prohibited media outlets. the state of affairs is realized, and every permitted media outlet endorses or criticizes the government, according to its editorial policy. readers observe messages of the permitted media outlets and decide whether to act or not. 37

38 Results Theorem 2 The government s optimal censorship policy is to prohibit all media outlets s > s and permit all media outlets s < s. 38

39 Results A media outlet with a higher editorial policy cutoff is more disloyal to the government, in the sense that it criticizes the government on a larger set of states. Theorem says that it is optimal for the government to prohibit all sufficiently disloyal media outlets from broadcasting. This government s censorship policy is optimal among all persuasion mechanisms. In particular, the government would not be better off if it could restrict each reader to follow a single media outlet of his choice and ban readers from communicating with one another, as in Chan and Suen (2008). Nor would the government be better off if it could create more complex mechanisms that aggregate information from multiple media outlets and add noise. 39

40 Results Enikolopov et al (2011) study the effect of voters access to NTV, the only independent national TV channel, on the regional results of the 1999 Russian parliamentary elections. They show that local access to NTV substantially decreased the regional aggregate vote for the government party. Our paper suggests a different interpretation of their findings: The government optimally permits access to NTV only in the regions with low initial support of the government. Thus, access to NTV in these regions may be a consequence, not a cause, of low electoral support of the government. 40

41 Other applications AAA-rated tranches constituted more than half of the outstanding securitized products in 2008 A is the most commonly awarded grade at Harvard 41

42 Comparative statics 42

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44 Comparative statics: Corollary. Let the distribution of types be G t (r) = G (r t) and g be logconcave. For all ρ and t such that ω (ρ, t) (0, 1): ω (ρ, t) is strictly increasing in ρ. ω (ρ, t) is strictly increasing in t. 43

45 Proof: By the revealed preference argument: (q 2 (r) + ρ 2 U 2 (r)) dg (r) (q 1 (r) + ρ 1 U 1 (r)) dg (r) (q 1 (r) + ρ 2 U 1 (r)) dg (r) (q 2 (r) + ρ 1 U 2 (r)) dg (r) Summing up these conditions gives: (ρ 2 ρ 1 ) (U 2 (r) U 1 (r)) dg (r) 0 Thus, more information is disclosed under higher ρ: ω (ρ) is increasing 44

46 Conclusions What can be achieved purely by means of persuasion? Any convex utility profile U between U and U What can be gained by private over public persuasion? Nothing When are simple censorship policies optimal? When the density of Listener s type is regular When is it optimal to disclose more information? When Persuader is less biased and Listener is more skeptical 45