Bayesian Persuasion€¦ · Motivational example A prosecutor (Sender) wishes to persuade the judge (Receiver) to take an action. There are two states of the world, !2fguilty;innocentg

Bayesian Persuasion

Emir Kamenica and Matthew Gentzkow (2010)

presented by Johann Caro Burnett and Sabyasachi Das

March 4, 2011

Emir Kamenica and Matthew Gentzkow (2010) (presented by Johann Caro Burnett and Sabyasachi Das)Bayesian Persuasion March 4, 2011 1 / 23

Introduction

Introduction

Study of strategic communication between two persons - a Senderand a Receiver.

The Sender reveals information about the state of the world.

The Receiver observes the information revealed and then takes anaction which affects the Sender.

The authors ask, can the Sender persuade the Receiver to take anaction that benefits him? If yes, when?


Introduction

Motivational example

A prosecutor (Sender) wishes to persuade the judge (Receiver) totake an action.

There are two states of the world, ω ∈ guilty , innocent.Both players share a prior about the defendant, Pr(guilty) = 0.3.

The judge get a payoff of one if they take the correct action, and zerootherwise.

The prosecutor gets a payoff of one if the judge convicts and zerootherwise, regardless of the state.

The prosecutor conducts an investigation (signal) and is required, bylaw, to report its full outcome.


Introduction


A prosecutor (Sender) wishes to persuade the judge (Receiver) totake an action.

There are two states of the world, ω ∈ guilty , innocent.Both players share a prior about the defendant, Pr(guilty) = 0.3.

The judge get a payoff of one if they take the correct action, and zerootherwise.

The prosecutor gets a payoff of one if the judge convicts and zerootherwise, regardless of the state.

The prosecutor conducts an investigation (signal) and is required, bylaw, to report its full outcome.


Introduction


The investigation consists of:

Choosing a realization space S , and conditional distributions π(s|ω).

And after choosing the signal (S , π), reporting the true realizations ∈ S .

Because of the prior, Pr(guilty) = 0.3, the judge would acquit if theprosecutor does not send a signal ( a pair (S , π)). So, the payoff of theprosecutor would be zero surely. But we will show that he can get astrictly higher payoff.


Introduction


Consider, S ≡ i , g and π(i |innocent) = 4/7 and π(i |guilty) = 0. Thatis, the realization i is totally informative, but the realization g is not.

With this signal (S , π), µ(innocent|i) = 1, and µ(innocent|g) = 1/2. So,if the judge sees i they will acquit. In the case that they see g , they areindifferent; so (we can suggest that) they take the action convict.Moreover, g happens with a probability of 6/10; which is the expectedpayoff of the prosecutor.

More impressive is the fact that the judge convicts 60% of the time, whilethe prior of guilty is only 30%.


The Model

The Model

There are two players, Sender and Receiver.

There is a finite number of states of the world, ω ∈ Ω, and bothplayers share the same prior µ0 ∈ int∆(Ω).

The Receiver can take an action from a compact set, a ∈ A, whichaffects both players payoff.

Preferences are u(a, ω) for the Receiver and v(a, ω) for the Sender,and they are both continuous.


The Model

Strategies

Sender only knows the prior and the preferences. He chooses arealization space S , and conditional probabilities π(s|ω). Afterchoosing the signal, he has to report the true realization s ∈ S .

Receiver observes the signal (S , π) and the realization s ∈ S . Sheupdates her beliefs and takes an action.


The Model

Equilibrium Concept

The equilibrium concept is the Sender-preferred subgame perfectequilibrium. That is, if the Receiver is indifferent between two actions, shechooses the one the favors the Sender.

When both players hold some belief µ, we denote the (unique) optimalaction of Receiver by a(µ), and the Sender’s payoff by:

v(µ) = Eµv(a(µ), ω)


The Model

Consistency of beliefs

Each realization induces a distribution over states:

µs(ω) =π(s|ω)µ0(ω)∑

ω′∈Ω π(s|ω′)µ0(ω′)for all s and ω

Moreover, a signal (S , π) induces a distribution τ over posteriors s.t.

Suppτ = µs : s ∈ S and

τ(µ) =∑

s:µs=µ

∑ω′∈Ω

π(s|ω′)µ0(ω′) for all µ


The Model

Consistency of beliefs

A distribution of posterior is Bayes-plausible if the expected posteriorequals the prior: ∫

µdτ(µ) = µ0


The Model

Proposition 1

A signal is called straightforward if S ⊂ A.

Proposition

The following are equivalent:

(i) There exists a signal that attains value v∗.

(ii) There exists a straightforward signal that attains value v∗.

(iii) There is a Bayes-plausible distribution of posteriors τ such thatEτ v(µ) = v∗.


The Model

Definition

Sender benefits from persuasion if there exists a signal (S , π) that yieldsan expected payoff strictly higher than the payoff with no signal.


The Model

Corollary

Sender benefits from persuasion if and only if there exists aBayes-plausible distribution over measures µ on Ω such that:

Eτ v(µ) > v(µ0)

Moreover, the value of an optimal signal is given by:

supτ

Eτ v(µ)

s.t.

∫µdτ(µ) = µ0


The Model

Value of Optimal Signal

We define V as the concave closure of v :

V (µ) ≡ supz |(µ, z) ∈ co(v)

where co(v) denotes the convex hull of the graph of v .

Note that V is the smallest concave function that is everywhereweakly greater than v .

Corollary

The value of an optimal signal is V (µ0). Sender benefits from persuasionif and only if V (µ0) > v(µ0))


The Model



V (µ) ≡ supz |(µ, z) ∈ co(v)



Corollary



The Model



V (µ) ≡ supz |(µ, z) ∈ co(v)



Corollary



The Model

Value in Motivating Example


The Model

When Does Sender Benefit from Persuasion?

Or, for what values of prior µ0 we get V (µ0) > v(µ0)?

For the motivating example this is the case when µ0 ∈ (0, 12 )

In general we require a τ such that Eτ v(µ) > v(Eτ (µ))

If v is concave, Sender does not benefit from persuasion for any prior.If v is convex and not concave, Sender benefits from persuasion forevery prior.

Many times v is neither concave nor convex, as in the example. Sowhat do we do?


The Model








The Model








The Model








The Model








The Model

Conditions for Benefit from Persuasion

There is information sender would share if ∃ µ s.t.

v(µ) > Eµv(a(µ0), ω)

Receiver’s preference is discrete at belief µ if ∃ ε > 0 s.t.

∀a 6= a(µ), Eµu(a(µ), ω) > Eµu(a, ω) + ε

Proposition

If there is no information Sender would share, Sender does not benefitfrom persuasion. If there is information Sender would share and Receiver’spreference is discrete at the prior, Sender benefits from persuasion. If A isfinite, Receiver’s preference is discrete at the prior generically.


The Model






Proposition



The Model






Proposition



The Model

Proof

Claim

If Receiver’s belief is discrete at µ then a(µ) doesn’t change in theneighborhood of µ.

Proof.

Suppose this is false.Let µ

′ u µ and µ′ 6= µ. Then a(µ

′) 6= a(µ).

Now we have Eµu(a(µ), ω) > Eµu(a(µ′), ω) + ε.

But Eµu(a(µ), ω) is continuous in µ by the Maximum theorem.

Hence Eµu(a(µ), ω) u Eµ′ u(a(µ), ω) and Eµu(a(µ′), ω) u Eµ′ u(a(µ), ω)

Hence if µ′

is chosen to be close enough the inequality would hold for µ′

as well.But that would contradict the definition of a(µ).


The Model

Proof

Claim


Proof.



′) 6= a(µ).




Hence if µ′




The Model

Proof

Claim


Proof.



′) 6= a(µ).




Hence if µ′




The Model

Proof

Claim


Proof.



′) 6= a(µ).




Hence if µ′




The Model

Proof

Claim


Proof.



′) 6= a(µ).




Hence if µ′




The Model

Proof

Claim


Proof.



′) 6= a(µ).




Hence if µ′




The Model

Proof Contd.

Now there exists µl such that v(µl) > Eµl v(a(µ0), ω).

There is µh in the neighborhood of µ0 such that a(µh) = a(µ0).

Now we can choose µh in such a way to make µ0 a convex combination ofthe two, i.e. ∃γ ∈ [0, 1] s.t. µ0 = γµh + (1− γ)µl

V (µ0) ≥ γEµhv(a(µh), ω) + (1− γ)Eµl v(a(µl), ω)> γEµhv(a(µ0), ω) + (1− γ)Eµl v(a(µ0), ω)=

∑ω∈Ω v(a(µ0), ω)[γµh(ω) + (1− γ)µl(ω)]

= v(µ0)


The Model

Proof Contd.





∑ω∈Ω v(a(µ0), ω)[γµh(ω) + (1− γ)µl(ω)]

= v(µ0)


The Model

Proof Contd.





∑ω∈Ω v(a(µ0), ω)[γµh(ω) + (1− γ)µl(ω)]

= v(µ0)


The Model

Proof Contd.





∑ω∈Ω v(a(µ0), ω)[γµh(ω) + (1− γ)µl(ω)]

= v(µ0)


The Model

Optimal Signal

We say no disclosure is optimal if µs = µ0 for all s realized withpositive probability under the optimal signal.

If µs is degenerate for all s realized with positive probability under theoptimal signal, we say full disclosure is optimal under the optimalsignal.

If v is (strictly) concave, no disclosure is (uniquely) optimal, and if vis (strictly) convex full disclosure is (uniquely) optimal.


Application

Lobbying

We consider a setting where a lobbying group commissions a studywith the goal of influencing a benevolent, but nonetheless rational,politician.

The politician (Receiver) chooses a unidimensional policy a ∈ [0, 1]

The state ω ∈ [0, 1] is the socially optimal policy.

The lobbyist (Sender) is employed by the interest group whosepreferred action is a∗ = αω + (1− α)ω∗, with α ∈ [0, 1] and ω∗ > 1.

Politician’s payoff u = −(a− ω)2

Lobbyist’s payoff v = −(a− a∗)2

Therefore we know that a(µ) = Eµ[ω]


Application

Lobbying

We consider a setting where a lobbying group commissions a studywith the goal of influencing a benevolent, but nonetheless rational,politician.

The politician (Receiver) chooses a unidimensional policy a ∈ [0, 1]

The state ω ∈ [0, 1] is the socially optimal policy.

The lobbyist (Sender) is employed by the interest group whosepreferred action is a∗ = αω + (1− α)ω∗, with α ∈ [0, 1] and ω∗ > 1.

Politician’s payoff u = −(a− ω)2

Lobbyist’s payoff v = −(a− a∗)2

Therefore we know that a(µ) = Eµ[ω]


Application

Lobbying Cntd.

Simple algebra reveals that

v(µ) =−(1− α)2ω∗2+2(1− α)2ω∗Eµ[ω]−α2Eµ[ω2]︸︷︷︸linear in µ

+(2α−1) (Eµ[ω])2︸︷︷︸convex in µ

v is linear in µ if α = 12 , strictly convex if α > 1

2 , and strictly concavewhen α < 1

2 .

Therefore we have full disclosure if α > 12 and no disclosure if α < 1

2 .

Optimal signal is independent of ω∗.

Thus theory suggests that lobbyist either commissions a fullyrevealing study or no study at all.

This contrasts with the real life observation.

May be that benevolent politician is not that rational! Or, there maybe a commitment problem on the part of the lobbyist not to distortthe study results ex post.


Application

Lobbying Cntd.






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Conclusion

The paper studies a scenario of strategic communication where oneagent reveals information to another in a strategic way in order topersuade him to take a preferred action.

It gives us necessary and sufficient conditions for the Persuader tobenefit from persuasion.

As it turns out, the scope for persuasion is, surprisingly, quite large.

The model can be easily extended to Receiver having privateinformation about the state.

Also can be extended to the case of multiple receivers.

How about multiple Persuaders? See Gentzkow and Kamenica(Working Paper, 2011)


Documents

Bayesian Persuasion€¦ · Motivational example A prosecutor (Sender) wishes to persuade the judge (Receiver) to take an action. There are two states of the world, !2fguilty;innocentg