information design, bayesian persuasion and bayes ... · public persuasion i agent observes a type...
TRANSCRIPT
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Information Design, Bayesian Persuasionand
Bayes Correlated Equilibrium
Dirk Bergemann and Stephen Morris
February 2016
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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?
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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
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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
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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)
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"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
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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
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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 .
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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
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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 .
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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Bank Run: Equilibrium SetI set of BCE described by outcome (νB , νG )
I σB = 1, σG = 0, is complete information BCE
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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
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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 ′
).
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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 ):
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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 ):
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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
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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
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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
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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.
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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
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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
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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
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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...
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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
I Probability of run is 56
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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:
Payoff θG , tG θG , t0 θB , t0Stay 1
6 0 0Run 0 1
623
I Probability of run is 56
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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:
Payoff θG , tG θG , t0 θB , t0Stay 1
6 0 0Run 0 1
623
I Probability of run is 56
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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.....
Payoff θG , tG θG , t0 θB , t0Stay 1
616
16
Run 0 0 12
I Probability of run is 12
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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.....
Payoff θG , tG θG , t0 θB , t0Stay 1
616
16
Run 0 0 12
I Probability of run is 12
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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
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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
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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
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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
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Lessons
1. Without loss of generality, can restrict attention toinformation structures where each player’s signal space isequal to his action space
2. Prior information limits the scope for information design
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Lessons
1. Without loss of generality, can restrict attention toinformation structures where each player’s signal space isequal to his action space
2. Prior information limits the scope for information design
3. Public signals optimal if strategic complementarities; privatesignals optimal if strategic substitutes
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Bank Run: two depositors with initial information
have also analyzed elsewhere....
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Bank Run with Imprecise Information 2