Information Design, Bayesian Persuasion and Bayes ... · Public Persuasion I agent observes a type...

Information Design, Bayesian Persuasionand

Bayes Correlated Equilibrium

Dirk Bergemann and Stephen Morris

February 2016

Introduction

I in decision problems / games with imperfect informationadditional information can influence / change behavior

I what is (maximal) extent by which a principal by providinginformation alone can change the behavior of agent?

I Bayesian persuasion: a single agent with zero informationbeyond common prior:

I how does private information by (many) agents limit thescope of persuasion?

Information Design and Bayes Correlated Equilibrium

I how to optimal provide additional information to many,privately informed agents?

I an immediate diffi culty: space of possible informationstructures, especially in many agent settings, is very large

I an apparent detour: first describe the set of all possibleoutcome across all possible information structures

I the solution concept, Bayes correlated equilibrium,characterizes outcomes independent of any specificinformation structure

I robust prediction: independent of details of any specificinformation structure

I information design: find the optimal information structure fora given objective function

Information Design and Mechanism Design

I Mechanism Design:I Fix an economic environment and information structureI Design the rules of the game to get a desirable outcome

I Information DesignI Fix an economic environment and rules of the gameI Design an information structure to get a desirable outcome

Main Lessons

I information by the agents, public or private, limits the set ofimplementable outcomes by tightening the best responseconditions

I yet, persuasion, with or without, elicitation of privateinformation still substantially influences equilibrium outcomes

I suggests a ranking of (many agents) information structures interms of equilibrium outcomes

I by contrast, information expands the set of feasible outcomes(before best response conditions)

"Basic Game"

I players i = 1, ..., II (payoff) states θ ∈ ΘI actions (Ai )

ii=1

I utility functions (ui )ii=1, each ui : A×Θ→ R

I state distribution ψ ∈ ∆ (Θ)

I G =((Ai , ui )

ii=1 ,ψ

)I games with information removedI "decision problem" in the one player case

Information Structure

I signals (types) (Ti )ii=1

I signal distribution π : Θ→ ∆ (T1 × T2 × ...× Ti )I S =

((Ti )

ii=1 ,π

)I "experiment" in the one player case

Expanding Information StructuresI S∗ =

((T ∗i )

ii=1 ,π

∗)is a combined information structure for

S1 =((T 1i)ii=1 ,π

1)and S2 =

((T 2i)ii=1 ,π

2)if

T ∗i = T1i ∪ T 2i for each i ,

∑t2∈T 2

π∗((t1, t2

)|θ)= π1

(t1|θ

)for each t1 ∈ T 1 and θ ∈ Θ and

∑t1∈T 1

π∗((t1, t2

)|θ)= π2

(t2|θ

)for each t2 ∈ T 2 and θ ∈ Θ.

I there are many combined information structures for any pairof information structures as only marginals have to match

I if S∗ is a combination of S and another information structure,say that S∗ is an expansion of S .

Games with Incomplete Information

I a pair (G ,S) is a standard game of incomplete informationI a decision rule is a mapping σ : T ×Θ→ ∆ (A)

I decision rule as a recommendation

I an outcome is a mapping ν : Θ→ ∆ (A)I outcome describes relationship between states and actionsindependent of information

Obedience

DEFINITION. A decision rule σ : T ×Θ→ ∆ (A) is obedientfor (G ,S) if

∑a−i ,t−i ,θ

ψ (θ)π ((ti , t−i ) , θ) σ ((ai , a−i ) |(ti , t−i ) , θ ) ui ((ai , a−i ) , θ)

≥∑

a−i ,t−i ,θ

ψ (θ)π ((ti , t−i ) , θ) σ ((ai , a−i ) |(ti , t−i ) , θ ) ui((a′i , a−i

), θ)

for each i , ai ∈ Ai , ti ∈ Ti and a′i ∈ Ai .

Bayes correlated equilibrium

DEFINITION. A decision rule σ : T ×Θ→ ∆ (A) is a Bayescorrelated equilibrium if it is obedient.

I set of BCE describes everything what could happen under anyexpansion of information structure T as a Bayes Nashequilibrium

I properties that hold across all BCE are robust predictions

Information Design

I an information designer has utility function w : A×Θ→ R

I his ex ante utility from decision rule σ is

W (σ) = ∑a,t ,θ

ψ (θ)π (t|θ) σ (a |t, θ )w (a, θ)

I information designer to choose σ to maximize WI argmax describes designer’s preferred information structure

Bank Run: one depositor and no initial information

I a bank depositor is deciding whether to run from the bank:I binary state: θ ∈ {B,G}, insolvent or solventI binary action: a ∈ {r , s}, run or stayI payoffs

u B Gr 0 0s −1 y

with 0 < y < 1I prior probability of each state is 12I designer (regulator) seeks to minimize probability of bank run(independent of state)

Outcome Distribution

I outcome ν : Θ→ ∆ (A) with no information:

outcome ν B G

r 12

12

s 0 0

I probability of run is 1

Bank Run: Common Prior Only

I suppose we have the prior information only - the nullinformation structure:

S◦ = (T ◦,π◦) , T ◦ = {t◦}

I parameterized decision rule:

σ (θ) B Gr σB σG

n 1− σB 1− σG

I σB = σ (r |B ) : probability of running if insolventI σG = σ (r |G ) : probability of running if solvent

Bank Run: Obedience

I agent may not necessarily know state θ but makes choicesaccording to σ (·)

I if "advised" to run, run has to be a best response:

0 ≥ σG y − σB ⇔σB ≥ σG y

I if "advised" to stay, stay has to be a best response

(1− σG ) y − (1− σB ) ≥ 0⇔σB ≥ (1− y) + σG y

I here, stay provides binding constraint:

σB ≥ (1− y) + σG y

I never to run, σB = 0, σG = 0, cannot be a BCE

Bank Run: Equilibrium SetI set of BCE described by outcome (σB , σG ) = (νB , νG )

I never to run, σB = 0, σG = 0, is not be a BCE

Bank Run: Information Design

I BCE minimizing the probability of runs has:

σB = 1− y , σG = 0

I noisy stress test S = {sB , sG } implements BNE viainformative signals:

π (s |θ ) B GsB 1− y 0sG y 1

I the bank is said to be healthy if it is solvent (always) and if itis insolvent (sometimes)

I solvent and insolvent banks are bundled

Bank Run: Private Information

I suppose agent observes conditionally independent signal(type) of the state with accuracy:

q >12

I S = (T ,π) where T = {b, g}:

T / Θ B Gb q 1− qg 1− q q

I strictly more information than null information q = 12

Bank Run: Additional Obedience Constraints

I conditional probability of running now depends on the signal:t ∈ {b, g}

I (σB , σG ) becomes((

σbB , σbG

), (σgB , σ

gG ))

I conditional obedience constraints, say for t = g :

r : 0 ≥ qσgG y − (1− q) σgBs : q (1− σgG ) y − (1− q) (1− σgB ) ≥ 0

or

r : σgB ≥q

1− qσgG y

s : σgB ≥ 1−q

1− q y +q

1− qσgG y

I reversal of relevant constraint with precise private information

Bank Run: Equilibrium SetI set of BCE described by outcome (νB , νG )

I σB = 1, σG = 0, is complete information BCE

Incentive Compatibility Ordering

I Write BCE (G , S) for the set of BCE (random) choices of(G , S)

DefinitionExperiment S is more incentive constrained than experiment S ′ if,for all decision problems G ,

BCE (G , S) ⊆ BCE(G ,S ′

).

I Note that "more incentive constrained" corresponds,intuitively, to having more information

Feasibility

Definition (Feasible Decision Rule)

A decision rule σ is feasible for (G , S) if it is induced by a decisionrule σ which is consistent (measurable) with S .

I write F (G ,S) for the set of feasible (random) choices of(G , S)

Definition (More Informative)

Experiment S is more informative than experiment S ′ if, for alldecision problems G ,

F (G ,S) ⊇ F(G ,S ′

).

Back to the Example: FeasibilityI suppose we have the prior information only - the nullinformation structure: S0 = (T0,π), T0 = {t0}

I feasible (random) choices ν (θ) can be described by (σB , σS ):

Back to the Example: FeasibilityI suppose player observes conditionally independent privatebinary signal of the state with accuracy q ≥ 1

2 :I feasible (random) choices σ (θ) can be described by (σB , σS ):

Information Design with Private InformationI agent observes a type (signal) t ∈ TI principal offers recommendation, a probability of running,as a function of reported type t and state θ:(

σtB , σtG

)I the constraints are: (i) each type has to truthfully report;(ii) each type has to be willing to follow the recommendation("private persuasion")

I the truthtelling constraints are given by, say for type g :

q (1− σgG ) y −(1− q) (1− σgB ) ≥ q(1− σbG

)y −(1− q)

(1− σbB

)⇔ qy

(σbG − σgG

)≥ (1− q)

(σbB − σgB

)I the obedience constraints described earlier

Private Persuasion vs Bayes Correlated Equilibrium

Av erage probabil i ty of running in Good s tate0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1

Aver

age

prob

abili

ty o

f run

ning

in B

ad s

tate

0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1

I private persuasion (dark red), BCE (light red)I truthtelling imposes constraint at the low and high end ofrunning probabilities

The Limits of Private Persuasion

Av erage probabil i ty of running in Good s tate0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1

Aver

age

prob

abili

ty o

f run

ning

in B

ad s

tate

0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1

I equilibrium set shrinks with the precision of the privateinformation q

Conclusion

I it is feasible and interesting to see what happens under manyinformation structures at once.

I methodology generates new answers for classical economicquestions

I In mechanism design we design the payoffs of the game,assuming the information structure is fixed

I In information design , we design the information received bythe players, assuming the game is fixed.

Public Persuasion

I agent observes a type (signal) t ∈ TI principal offers a recommendation, a probability of running,independent of the type t, but dependent on the true state θ:

I the constraints are: all types have to be willing to follow therecommendation

I private persuasion restricts information design by incentiveconstraints

I public persuasion restricts information design by uniformrecommendation constraints

Optimal Information Design with one depositor and noinitial information

I The regulator cannot stop the depositor withdrawing....I ... but can choose what information is made available toprevent withdrawals

I Best information structure:I tell the depositor that the state is bad exactly often enough sothat he will stay if he doesn’t get the signal.....

Outcome θG θBStay (intermediate signal) 1

313

Run (bad signal) 0 13

I Think of the regulator as a mediator making an actionrecommendation to the depositor subject to an obedienceconstraint

I Probability of run is 13

Lessons

1. Without loss of generality, can restrict attention toinformation structures where each player’s signal space isequal to his action space

I compare with the revelation principle of mechanism design:

I without loss of generality, we can restrict attention tomechanisms where each player’s message space is equal to histype space

Bayesian Persuasion

I This is the leading example in Kamenica-Gentzkow 2011I We are not exploiting "concavification" logic applicable in thiscase; will return to this later in lectures...

Bank Run: one depositor with initial information

I If the state is good, with probability 12 the depositor will

already have observed a signal tG saying that the state is good

I Outcome distribution with no additional information:

Payoff θG , tG θG , t0 θB , t0Stay 1

6 0 0Run 0 1

623


Bank Run: one depositor with initial information

I If the state is good, with probability 12 the depositor will

already have observed a signal tG saying that the state is goodI Outcome distribution with no additional information:


6 0 0Run 0 1

623


Optimal Information Design with one depositor with initialinformation

I Best information structure:I tell the depositor that the state is bad exactly often enough sothat he will stay if he doesn’t get the signal.....


616

16

Run 0 0 12


Is initially more informed depositor good or bad?

I With no information design....I ...and no initial information, probability of run is 1I ...and initial information, probability of run is 56

I With information design....I ...and no initial information, probability of a run is 13I ...and initial information, probability of a run is 12

Is initially more informed depositor good or bad?

I With no information design....I ...(in this example) more initial information is better for theregulator

I With information design....I ...more initial information is always bad for the regulator

Lessons


2. Prior information limits the scope for information design

Bank Runs: two depositors and no initial information (andstrategic complements)

I A bank depositor would like to run from the bank if he assignsprobability greater than 1

2 to a bad state OR the otherdepositor running

state θG Stay RunStay 1 −1Run 0 0

state θB Stay RunStay −1 −1Run 0 0

I Probability of the bad state is 23

Bank Runs: two depositors and no initial information

I Outcome distribution with no information

outcome θG Stay RunStay 0 0Run 0 1

3

outcome θB Stay RunStay 0 0Run 0 2

3

I Best information structure:I tell the depositors that the state is bad exactly often enough sothat they will stay if they don’t get the signal.....

outcome θG Stay RunStay 1

3 0Run 0 0

outcome θB Stay RunStay 1

3 0Run 0 1

3

I ...with public signals optimal

Bank Runs: two depositors, no initial information andstrategic substitutes

I Previous example had strategic complementsI Strategic substitute example: a bank depositor would like torun from the bank if he assigns probability greater than 1

2 to abad state AND the other depositor staying

state θG Stay RunStay 1 1Run 0 0

state θB Stay RunStay −1 1Run 0 0

I Probability of the bad state is 23

Bank Runs: two depositors and no initial information

I Outcome distribution with no information: mixed strategyequilibrium

I Best information structure:I tell the depositors that the state is bad exactly often enough sothat they will stay if they don’t get the signal.....

outcome θG Stay RunStay 1

3 0Run 0 0

outcome θB Stay RunStay 4

919

Run 19 0

I ....with private signals optimal

Lessons


2. Prior information limits the scope for information design

3. Public signals optimal if strategic complementarities; privatesignals optimal if strategic substitutes

Bank Run: two depositors with initial information

have also analyzed elsewhere....

Bank Run with Imprecise Information 2

Information Design, Bayesian Persuasion and Bayes ... · Public Persuasion I agent observes a type...

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