attention, coordination, and bounded recallapa522/slides...is endogenous precision and r s = p s h s...

Motivation Model Equilibrium Effi ciency Bounded Recall Conclusions

Attention, Coordination, and Bounded Recall

Alessandro Pavan

Northwestern University

Chicago FED, February 2016


Motivation

Many socioeconomic environments

- large group of agents

- actions under dispersed information

Useful modelization for:

- production or network externalities

- incomplete markets

- business cycles

- large Cournot-Bertrand games

- elections

...

Most of the literature: exogenous information structure

Many phenomena of interest: attention (info. acquisition) is central


This paper

Flexible (yet rich) framework

- complementarity or substitutability in actions

- rich set of payoff interdependencies

Equilibrium and effi cient allocation of attention

- perfect recall

- bounded recall


Questions

What payoff interdependencies create ineffi ciency in eq. allocation ofattention?

How does ineffi ciency in attention relate to ineffi ciency in use ofinformation?

What is the effect of bounded recall?

What policies can alleviate such ineffi ciencies? (related work)


Related literature (incomplete)

Effi cient use of information and social value of informationRadner (1977), Vives (JET 1984, 2013)

Morris and Shin (AER 2002)

Angeletos and Pavan (AER, 2004, Ecma 2007, Jeea, 2009)

...Information acquisition/(in)attention in coordination settingsVives and Van Zandt (2007)

Hellwig and Veldkamp (Restud, 2009)

Amir and Lazzati (2014)

Mackowiak and Wiederholt (AER, 2009, 2012)

→ Myatt and Wallace (Restud 2012)

Szkup and Trevino (2013), Yang (2013)

→ Colombo, Femminis and Pavan (Restud 2014)

Tirole (2014), Denti (2016)

...MemoryBenabou Tirole (JPE 2004)

Wilson (2004), Kocer (2010)

...Analogy-based equilibriumJehiel (JET 2005)


Plan

1 Model (perfect recall)

2 Equilibrium allocation of attention

3 Effi cient allocation of attention

4 Bounded recall


ModelActions and gross payoffs

ui

(ki , {k j} j 6=i, θ

)


ModelActions and gross payoffs

Continuum of agents with payoffs:

u

(k , K , θ , σ

2k

)where:k ∈ R — individual action

K =∫

k′dΨ(k′) —aggregate action

σ2k =

∫(k′−K)2dΨ(k′) —dispersion

θ ∈ R —underlying uncertainty ("fundamentals")

Assumptions:u(·) quadratic in (k,K,θ), linear in σ2

k

u(·) s.t. equilibrium and first-best unique and bounded


Examples

Investment spillovers (Angeletos and Pavan AER 2004)

ui = Rki− c(ki)

R= (1−a)θ +aK and c(ki) =1

2k2

i

Beauty contest (Morris and Shin AER 2002)

ui =−(1− r) · (ki−θ)2− r · (L(ki)− L)

L(ki)≡∫ (

k′− ki

)2dΨ(k′)= (ki−K)2+σ

2k and L=

∫L(k)dΨ(k)= 2σ

2


Examples

Monetary economies (Woodford 2005, Colombo, Femminis and Pavan,2014, Llosa and Venkateswaran, 2015)

u(θ ,Ci,Ni)≡V (Ci)−Ni

Ci =

(∫[0,1]

cv−1

v

hidh

) vv−1

Yi = θα Ni

∫[0,1]

phchidh≤ piYi−T

Cournot and Bertrand games (Vives JET 1984)

ui = (a−θK) · ki−1

2k2

i


ModelInformation and attention

Common prior:θ ∼ N(0,π−1

θ)

N = 1,234,576 sources of information:

yl = θ + ε l with ε l ∼ N(0,η−1l) l = 1, ...,N

Agent i’s "impressions" xi = (xil)Nl=1 with

xil = yl +ξ

il with ξ

il ∼ N

(0,(

zil · tl)−1

)l = 1, ...,N

where

η l : accuracy

tl : transparency/clarity

zil : attention


ModelAttention cost and net payoffs

Attention cost: C(zi) where zi = (zil)Nl=1

· C′n(zi)> 0, all zi 6= 0

· limzn→∞C′n(zi) = ∞

· convex (results extend to concave, e.g., entropy reduction)

E.g. C(zi) = c(∑l zi

l

)E.g. C(zi) = ∑l g(zi

l)

...but also C(zi) = µ(zi;y) (entropy reduction)

Net payoff

u

(ki,K,σ

2k ,θ)−C(zi)


ModelTiming

agents allocate attention zi

update their beliefs based on xi

commit their actions ki

payoffs realized


Plan




4 Bounded Recall


Equilibrium use of information (Angeletos and Pavan, Ecma 2007)

Optimality:k j = E[ κ+α(K−κ) | x j ; z j]

where

κ = κ0+κ1θ (complete-info. equilibrium action)

α ≡ ukK

|ukk | −→ equilibrium degree of coordination


Equilibrium allocation of attention

Theorem

There exists a unique symmetric equilibrium. In this eq., the attention z thateach agent assigns to the various sources of information is s.t., for any sourcen= 1, ...,N that receives strictly positive attention,

zn = κ1γn

√|ukk|

2C′n(z)tn

where

γn ≡(1−α)πn

1−αρn

πθ +∑Ns=1

(1−α)πs

1−αρs

is "influence" of the source

and where

πs =ηszsts

zsts+ηs

is endogenous precision and ρs =πs

ηs

is endogenous "publicity"

Given equilibrium allocation of attention z, equilibrium actions are given by

ki = κ0+κ1

(∑

Nn=1 γnxi

n

)all i ∈ [0,1], almost all xi ∈ RN .

alessandro pavan

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alessandro pavan

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Private value of attention

Envelope reasoning: hold k(·; z) fixed

Agent’s eq. continuation payoff (fixing k(·; z)):

Ui(zi; z) = E[u(K,K,σ k,θ)]+

ukk

2Var[ki−K | zi, z,k(·; z)]−C(zi)


−|ukk|2· ∂Var[k−K | z,k(·;z)]

∂ zn

private aversion to dispersion · reduction in dispersion(fixing eq. strategy k(·;z))

Result generalizes Colombo, Femminis, Pavan (Restud 2014)






ukk



−|ukk|2· ∂Var[k−K | z,k(·;z)]

∂ zn



alessandro pavan

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ukk



−|ukk|2· ∂Var[k−K | z,k(·;z)]

∂ zn




Plan




4 Bounded Recall


Effi ciency

Welfare : ex-ante utility of representative agent

Definition

Effi cient allocation consists of attention z∗ along with action rule k∗(·;z∗) thatjointly maximize

E[u(k,K,σ2k ,θ) | z]−C(z)

Team problem

Planner’s problem: control incentives but cannot transfer information


Effi cient use of information (Angeletos and Pavan, Ecma 2007)

Given attention z, effi ciency in actions requires that k∗(·;z) solves

k∗(x;z) = E [κ∗+α∗(K−κ

∗) | x ; z ] ∀x,

whereκ∗ = κ

∗0+κ

∗1θ −→ FB

α∗ ≡ uσσ −2ukK −uKK

ukk+uσσ

= 1− aversion to volatilityaversion to dispersion

socially optimal degree of coordination


Effi cient allocation of attention

Theorem

Effi ciency in attention requires that, for any n for which z∗n > 0,

z∗n = κ∗1γ∗n

√|ukk+uσσ |2C′n(z∗)tn

where

γ∗n ≡

(1−α∗)πn

1−αρn

πθ +∑Ns=1

(1−α∗)πs

1−α∗ρs

is effi cient "influence" of the source

πs =ηsz∗s ts

z∗s ts+ηs

is endogenous precision and ρs =π∗sηs

is endogenous publicity

Recall that eq.

zn = κ1γn

√|ukk|

2C′n(z)tn

alessandro pavan

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alessandro pavan

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alessandro pavan

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Theorem

Effi ciency in attention requires that, for any n for which z∗n > 0,

z∗n = κ∗1γ∗n

√|ukk+uσσ |2C′n(z∗)tn

where

γ∗n ≡

(1−α∗)πn

1−αρn

πθ +∑Ns=1

(1−α∗)πs

1−α∗ρs

is effi cient "influence" of the source

πs =ηsz∗s ts

z∗s ts+ηs

is endogenous precision and ρs =π∗sηs

is endogenous publicity

Recall that eq.

zn = κ1γn

√|ukk|

2C′n(z)tn



Envelope reasoning

Welfare under effi cient use of information (for given attention z)

w∗(z)≡ E[u(κ∗,κ∗,0,θ)]−L ∗(z)−C(z),

where u(κ∗,κ∗,0,θ) is welfare under FB allocation and

L ∗(πx,πz)≡|ukk+2ukK +uKK |

2Var[K−κ

∗ | k∗(·;z),z]

+|ukk+uσσ |

2Var[k−K | k∗(·;z),z]

Holding k∗(·;z), Var[K−κ∗ | k∗(·;z),z] independent of z

Social value of attention

−|ukk+uσσ |2

· ∂Var[k−K | z,k∗(·;z)]∂ zn

social aversion to dispersion · reduction in dispersion(fixing eff. strategy k∗(·;z) )

alessandro pavan

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Effi ciency in attention requires

- effi ciency in use of information: k(·;z) = k∗(·;z)

- private = social aversion to dispersion ⇔ uσσ = 0

alessandro pavan

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alessandro pavan

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Plan




4 Bounded Recall


Bounded Recall

Idea: posteriors correct, but agents cannot recall influence of individualsources

Given attention z j, posterior beliefs about θ continues to be Normal withmean

x j =∑N

n=1δ nxi

n, with δ n ≡πn

πθ +∑Ns=1 πs

and πs ≡ηszsts

zsts+ηs

and precision πθ +∑Ns=1 πs

However, agent is unable to decompose x j into various impressionsx j ≡ (x j

1, ...,xjN).

Equivalently, unable to decompose his posteriors into

θ | xin

Measurability constraint on k(x j;zz)

Distinction relevant only in strategic setting

alessandro pavan

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Bounded Recall

For simplicity: πθ = 0

Theorem

In unique symmetric equilibrium, given allocation z#, actions given by

ki = κ0+κ1xi

For any source that receives strictly positive attention in eq.,

C′n(z#) =−|ukk|

2

∂Var[k−K;z#,k(·;z#)

]∂ zn

− |ukk|2(1−α)

∂Var[K−κ;z#,k(·;z#)

]∂ zn

Novel effect:

−|ukk|2(1−α)

∂Var[K−κ;z#,k(·;z#)

]∂ zn

private aversion to volatility of own’s average action · reduction in volatility(fixing eq. strategy k#(·;z))

alessandro pavan

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Bounded vs Perfect Recall

Theorem

Let z be eq. allocation of attention with perfect recall. There exist publicitythresholds ρ ′,ρ ′′ ∈ [0,1] s.t., starting from z, any agent with bounded recall isbetter off by

(a) locally increasing attention to sources for which ρn ∈ [ρ ′,ρ ′′];

(b) locally decreasing attention to sources for which ρn /∈ [ρ ′,ρ ′′] .



Reallocation of attention towards sources of average (endogenous)publicity

ρn =zsts

zsts+ηs

Sources of low publicity: useful to forecast θ

Sources of high publicity: useful to forecast K

Sources of intermediate transparency: good compromises

alessandro pavan

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Previous result about best responses extends to equilibrium

Suppose C(z) = c(∑

Ns=1 zs

)Theorem

Let z be eq. attention with perfect recall and z# eq. attention with boundedrecall. There exist thresholds t ′, t ′′ ∈ R++ s.t. z#

n > zn only if tn ∈ [t ′, t ′′] .Furthermore for any n for which tn ∈ [t ′, t ′′], z#

n < zn only if z#n = 0.


Effi ciency under Bounded Recall

...see paper!


Conclusions

Attention in large economies with

- complementarity / substitutability in actions

- rich set of payoff interdependencies

- rich information structure

Effi ciency in allocation of attention requires

(a) absence of externalities from action-dispersion

(b) effi ciency in use of information

Bounded recall: reallocation of attention towards sources withintermediate transparency


Conclusions

Future work

- endogenous sources / social learning(e.g., capital mkts → information aggregation)

- "optimal" recall strategy

- dynamics (optimal stopping)

- fully flexible info. structures (attention-based correlated eq.)


Thank You!

attention, coordination, and bounded recallapa522/slides...is endogenous precision and r s = p s h s...

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