Persuasion of Heterogenous Audiences and Optimal Media Censorship
Tymofiy Mylovanov
joint work with
Anton Kolotilin, Ming Li, Andriy Zapechelnyuk
October 6, 2017
1 Environment
Sender Receiver
Receiver makes a binary decision (action or inaction)
2 • To convince Receiver to act, Sender controls disclosure of decision relevant information:
• Receiver’s preferences are private
• Sender fully controls persuasion
3 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 Examples
• Ratings given by credit agency
• Grading policies designed by school
• Media controlled by government
5 Questions
• What can be achieved purely by means of persuasion?
• What can persuasion mechanisms gain over experiments?
• What persuasion mechanisms are optimally chosen?
6 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 Payoffs
• Receiver’s payoff from actions (a = 0, 1): ω − r, a = 1 u(ω, r, a) = 0, a = 0
• Sender’s payoff from actions (a = 0, 1):
1 + ρ(r) (ω − r) , a = 1 v(ω, r, a) = 0, a = 0
where ρ(r) ∈ R
8 Receiver’s Private Information: Interpretation
• One Receiver
– Incomplete information about Receiver’s type
• A continuum of heterogeneous Receivers – an audience
9 Persuasion mechanism
• A direct mechanism π : R × Ω → ∆A
– asks Receiver to report rb
– recommends ab = 1 w/pr π (r,b ω) and ab = 0 o/w
• WLOG restrict attention to mechanisms in which
– Receiver is honest and obedient
10 Experiment
• An experiment σ :Ω → ∆(M) sends to Receiver a (stochas- tic) 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 mecha- nism
• Reverse?
11 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 Incentive-compatible mechanisms
13 Notation
• For a mechanism π, define
– probability that r takes the action Z qπ (r) = π (r, ω) dF (ω)
– expected value of ω given r and a = 1 1 Z pπ (r) = ωπ (r, ω) dF (ω) qπ(r)
– expected payoff of Receiver r Z Uπ (r) = (ω − r)π (r, ω) dF (ω) = qπ(r)(pπ (r) − r)
∗ analogy: standard linear trading environment
14 Incentive compatible mechanisms
Lemma. A (feasible) mechanism π is incentive compatible iff
qπ is non-increasing (1) Z 1 Uπ (r) = qπ(s)ds, (2) r Z 1 Uπ (0) = ωdF (ω) = E[ω], (3) 0
• there are obedience instead of individual rationality constraints
• there are also no transfers
15 Incentive compatible mechanisms
Lemma. A mechanism π is incentive compatible if and only if
qπ is non-increasing (4) Z 1 Uπ (r) = qπ(s)ds, (5) r Z 1 Uπ (0) = qπ(s)ds = E[ω], (6) 0
Differences from Mirrlees:
• There are two boundary conditions, for r = 1 and r = 0
• Not all qπ’s that satisfy (??)–(??) are implementable
– The set of implementable qπ’s depends on F not only through EF [ω]
16 Lemma is not too useful:
• What is the set of implementable (qπ,Uπ,Vπ)?
• What can persuasion mechanisms gain over experiments?
17 Implementable utility schedules
18 Utility schedules
• A pair (qπ,Vπ) is pinned down by Uπ: 0 qπ(r) = −Uπ(r), 0 Vπ(r) = −Uπ(r) + ρ(r)Uπ(r).
• We can focus on implementable utility profiles Uπ
19 Necessary condition 1
• No information
U (r) = max {E[ω] − r, 0}
• Full information
Z 1 U (r) = (ω − r) dF (ω) r
• Thus, under any mechanism
U(r) ≤ Uπ(r) ≤ U(r)
20 Necessary condition 2
Incentive compatibility ⇒ Uπ is convex
(the standard Envelope Theorem argument)
21 Necessary conditions
U(r) U(r) U(r) r
0 1
U (r) ≤ U (r) ≤ U (r) U is convex
22 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 Sketch of Proof: (a) ⇒ (c)
• Every experiment σ can be equivalently described by a c.d.f. H of the posterior mean state m = Eσ[ω|m].
• E.g., fully informative experiment (m = ω): H(m) = F (m).
24 Sketch of Proof: (a) ⇒ (c)
• Fix U convex s.t. U(r) ≤ U(r) ≤ U(r).
• Construction: Let H(r) = 1 + U0(r)
• H is a c.d.f:
– H(0) = 0 because U0(0) = U0(0) = −1
– H(1) = 1 because U0(1) = U0(1) = 0
– H(r) is increasing because U is convex
25 Sketch of Proof: (a) ⇒ (c)
• Observe that H(r) = 1 + U0(r) satisfies Z 1 Z 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 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 Optimal Mechanisms
28 Sender’s Problem
Lemma 1 For every incentive-compatible mechanism π, Z Z Vπ(r)dG(r) = g(0)E[ω] + Uπ(r)I(r)dr, R R where I(r) = g0(r) + ρ(r)g(r) for all r ∈ R.
29 Sender’s Problem Z max U(r)I(r)dr. U(r) R subject to
U(r) is convex such that U(r) ≤ U(r) ≤ U(r)
30 Example
• I(r) = g0(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 Media Control
32 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 en- dorses the government (sends message ms = 1) if ω > s and criticizes it (sends message ms = 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 empir- ically measured as the frequency with which the outlet uses anti-government language.
33 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, ¯a) = a(ω − r) + ¯aξ(r), where ¯a denotes the average action in the society, and ¯aξ(r) is a type-specific externality term that contributes to the reader’s utility but does not affect the reader’s optimal action.
34 Model
• The government’s utility is equal to Z Z u(ω, r, ar, ¯a)dG(r) +¯aγ = ar(ω − r) +¯a(ξ(r) + γ) dG(r), R R where ar denotes an action of type r, and ¯aγ is the govern- ment’s intrinsic benefit from the average action.
– We assume that Z −1 ρ = (ξ(r) + γ)dG(r) > 0, R meaning that the government is biased towards a greater average action in the society.
35 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 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 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 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 gov- ernment be better off if it could create more complex mecha- nisms that aggregate information from multiple media outlets and add noise.
39 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 Other applications
• AAA-rated tranches constituted more than half of the out- standing securitized products in 2008
• A is the most commonly awarded grade at Harvard
41 Comparative statics
42
Comparative statics:
Corollary. Let the distribution of types be Gt (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 Proof:
• By the revealed preference argument: Z Z (q2 (r) + ρ2U2 (r)) dG (r) ≥ (q1 (r) + ρ2U1 (r)) dG (r) Z Z (q1 (r) + ρ1U1 (r)) dG (r) ≥ (q2 (r) + ρ1U2 (r)) dG (r)
• Summing up these conditions gives: Z (ρ2 − ρ1) (U2 (r) − U1 (r)) dG (r) ≥ 0
• Thus, more information is disclosed under higher ρ:
ω∗ (ρ) is increasing
44 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