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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions Attention, Coordination, and Bounded Recall Alessandro Pavan Northwestern University Chicago FED, February 2016 Motivation Model Equilibrium Efficiency Bounded Recall

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Attention, Coordination, and Bounded Recall

Alessandro Pavan

Northwestern University

Chicago FED, February 2016

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Motivation

Many socioeconomic environments

  • large group of agents
  • actions under dispersed information
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

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

...

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

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

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

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

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

This paper

Flexible (yet rich) framework

  • complementarity or substitutability in actions
  • rich set of payoff interdependencies
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

This paper

Flexible (yet rich) framework

  • complementarity or substitutability in actions
  • rich set of payoff interdependencies

Equilibrium and efficient allocation of attention

  • perfect recall
  • bounded recall
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Questions

What payoff interdependencies create inefficiency in eq. allocation of attention? How does inefficiency in attention relate to inefficiency in use of information? What is the effect of bounded recall? What policies can alleviate such inefficiencies? (related work)

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Related literature (incomplete)

Efficient use of information and social value of information

Radner (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 settings

Vives and Van Zandt (2007) Hellwig and Veldkamp (Restud, 2009) Amir and Lazzati (2014) Ma´ ckowiak 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)

... Memory

Benabou Tirole (JPE 2004) Wilson (2004), Kocer (2010)

... Analogy-based equilibrium

Jehiel (JET 2005)

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Plan

1

Model (perfect recall)

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Plan

1

Model (perfect recall)

2

Equilibrium allocation of attention

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Plan

1

Model (perfect recall)

2

Equilibrium allocation of attention

3

Efficient allocation of attention

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Plan

1

Model (perfect recall)

2

Equilibrium allocation of attention

3

Efficient allocation of attention

4

Bounded recall

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Model

Actions and gross payoffs

ui

  • ki , {kj}j=i, θ
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Model

Actions and gross payoffs

Continuum of agents with payoffs: u

  • k , K , θ, σ2

k

  • where:

k ∈ R — individual action K =

kdΨ(k) — aggregate action

σ2

k =

(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

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

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 2 dΨ(k) = (ki −K)2+σ2

k

and ¯ L =

  • L(k)dΨ(k) = 2σ2
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Examples

Monetary economies (Woodford 2005, Colombo, Femminis and Pavan, 2014, Llosa and Venkateswaran, 2015) u(θ,Ci,Ni) ≡ V (Ci)−Ni Ci =

  • [0,1] c

v−1 v

hi dh

  • v

v−1

Yi = θαNi

  • [0,1] phchidh ≤ piYi −T

Cournot and Bertrand games (Vives JET 1984) ui = (a−θK)·ki − 1 2k2

i

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Model

Information and attention

Common prior: θ ∼ N(0,π−1

θ )

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Model

Information and attention

Common prior: θ ∼ N(0,π−1

θ )

N = 1,234,576 sources of information: yl = θ +εl with εl ∼ N(0,η−1

l

) l = 1,...,N

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Model

Information and attention

Common prior: θ ∼ N(0,π−1

θ )

N = 1,234,576 sources of information: yl = θ +εl with εl ∼ N(0,η−1

l

) l = 1,...,N Agent i’s "impressions" xi = (xi

l)N l=1 with

xi

l = yl +ξ i l

with ξ i

l ∼ N

  • 0,
  • zi

l ·tl

−1 l = 1,...,N where ηl : accuracy tl : transparency/clarity zi

l

: attention

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Model

Attention cost and net payoffs

Attention cost: C(zi) where zi = (zi

l)N l=1

· C

  • n
  • zi

> 0, all zi = 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,σ2

k,θ

  • −C(zi)
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Model

Timing

agents allocate attention zi update their beliefs based on xi commit their actions ki payoffs realized

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Plan

1

Model (perfect recall)

2

Equilibrium allocation of attention

3

Efficient allocation of attention

4

Bounded Recall

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

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

Optimality: kj = E[ κ +α(K −κ) | x j ; zj] where κ = κ0 +κ1θ (complete-info. equilibrium action) α ≡ ukK

|ukk|

− → equilibrium degree of coordination

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Equilibrium allocation of attention

Theorem There exists a unique symmetric equilibrium. In this eq., the attention ˆ z that each agent assigns to the various sources of information is s.t., for any source n = 1,...,N that receives strictly positive attention, ˆ zn = κ1γn

  • |ukk|

2C

n(ˆ

z)tn where γn ≡

(1−α)πn 1−αρn

πθ +∑N

s=1 (1−α)πs 1−αρs

is "influence" of the source and where πs = ηsˆ zsts ˆ zsts +ηs is endogenous precision and ρs = πs ηs is endogenous "publicity" Given equilibrium allocation of attention ˆ z, equilibrium actions are given by ki = κ0 +κ1

  • ∑N

n=1 γnxi n

  • all i ∈ [0,1], almost all xi ∈ RN.
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Private value of attention

Envelope reasoning: hold k(·; ˆ z) fixed

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Private value of attention

Envelope reasoning: hold k(·; ˆ z) fixed Agent’s eq. continuation payoff (fixing k(·; ˆ z)): Ui(zi; ˆ z) = E[u(K,K,σk,θ)]+ ukk 2 Var[ki −K | zi, ˆ z,k(·; ˆ z)]−C(zi)

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Private value of attention

Envelope reasoning: hold k(·; ˆ z) fixed Agent’s eq. continuation payoff (fixing k(·; ˆ z)): Ui(zi; ˆ z) = E[u(K,K,σk,θ)]+ ukk 2 Var[ki −K | zi, ˆ z,k(·; ˆ z)]−C(zi) Private value of attention −|ukk| 2 · ∂Var[k −K | z,k(·;z)] ∂zn private aversion to dispersion · reduction in dispersion (fixing eq. strategy k(·;z))

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Private value of attention

Envelope reasoning: hold k(·; ˆ z) fixed Agent’s eq. continuation payoff (fixing k(·; ˆ z)): Ui(zi; ˆ z) = E[u(K,K,σk,θ)]+ ukk 2 Var[ki −K | zi, ˆ z,k(·; ˆ z)]−C(zi) Private value of attention −|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)

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Plan

1

Model (perfect recall)

2

Equilibrium allocation of attention

3

Efficient allocation of attention

4

Bounded Recall

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Efficiency

Welfare : ex-ante utility of representative agent Definition Efficient allocation consists of attention z∗ along with action rule k∗(·;z∗) that jointly maximize E[u(k,K,σ2

k,θ) | z]−C(z)

Team problem Planner’s problem: control incentives but cannot transfer information

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

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

Given attention z, efficiency in actions requires that k∗(·;z) solves k∗(x;z) = E[κ∗ +α∗(K −κ∗) | x ; z] ∀x, where κ∗ = κ∗

0 +κ∗ 1θ

− → FB α∗ ≡ uσσ −2ukK −uKK ukk +uσσ = 1− aversion to volatility

aversion to dispersion

socially optimal degree of coordination

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Efficient allocation of attention

Theorem Efficiency 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 +∑N

s=1 (1−α∗)πs 1−α∗ρs

is efficient "influence" of the source πs = ηsz∗

sts

z∗

sts +ηs

is endogenous precision and ρs = π∗

s

ηs is endogenous publicity

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Efficient allocation of attention

Theorem Efficiency 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 +∑N

s=1 (1−α∗)πs 1−α∗ρs

is efficient "influence" of the source πs = ηsz∗

sts

z∗

sts +ηs

is endogenous precision and ρs = π∗

s

ηs is endogenous publicity Recall that eq. ˆ zn = κ1γn

  • |ukk|

2C

n(ˆ

z)tn

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Efficient allocation of attention

Envelope reasoning

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Efficient allocation of attention

Envelope reasoning Welfare under efficient 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| 2 Var[K −κ∗ | k∗(·;z),z] + |ukk +uσσ| 2 Var[k −K | k∗(·;z),z]

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Efficient allocation of attention

Envelope reasoning Welfare under efficient 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| 2 Var[K −κ∗ | k∗(·;z),z] + |ukk +uσσ| 2 Var[k −K | k∗(·;z),z] Holding k∗(·;z), Var[K −κ∗ | k∗(·;z),z] independent of z

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Efficient allocation of attention

Envelope reasoning Welfare under efficient 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| 2 Var[K −κ∗ | k∗(·;z),z] + |ukk +uσσ| 2 Var[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) )

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Equilibrium vs efficient allocation of attention

Private value of attention −|ukk| 2 · ∂Var[k −K | z,k(·;z)] ∂zn private aversion to dispersion · reduction in dispersion (fixing eq. strategy k(·;z))

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Equilibrium vs efficient allocation of attention

Private value of attention −|ukk| 2 · ∂Var[k −K | z,k(·;z)] ∂zn private aversion to dispersion · reduction in dispersion (fixing eq. strategy k(·;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) )

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Efficient allocation of attention

Efficiency in attention requires

  • efficiency in use of information: k(·;z) = k∗(·;z)
  • private = social aversion to dispersion ⇔ uσσ = 0
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Plan

1

Model (perfect recall)

2

Equilibrium allocation of attention

3

Efficient allocation of attention

4

Bounded Recall

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Bounded Recall

Idea: posteriors correct, but agents cannot recall influence of individual sources Given attention zj, posterior beliefs about θ continues to be Normal with mean ¯ x j = ∑

N n=1 δ nxi n, with δ n ≡

πn πθ +∑N

s=1 πs

and πs ≡ ηszsts zsts +ηs and precision πθ +∑N

s=1 πs

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

1,...,x j N).

Equivalently, unable to decompose his posteriors into ˜ θ | xi

n

Measurability constraint on k(x j;zz) Distinction relevant only in strategic setting

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Bounded Recall

For simplicity: πθ = 0 Theorem In unique symmetric equilibrium, given allocation z#, actions given by ki = κ0 +κ1 ¯ xi 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))

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Bounded vs Perfect Recall

Theorem Let ˆ z be eq. allocation of attention with perfect recall. There exist publicity thresholds ρ,ρ ∈ [0,1] s.t., starting from ˆ z, any agent with bounded recall is better off by (a) locally increasing attention to sources for which ρn ∈ [ρ,ρ]; (b) locally decreasing attention to sources for which ρn / ∈ [ρ,ρ].

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Bounded vs Perfect Recall

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

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Bounded vs Perfect Recall

Previous result about best responses extends to equilibrium Suppose C(z) = c

  • ∑N

s=1 zs

  • Theorem

Let ˆ z be eq. attention with perfect recall and z# eq. attention with bounded

  • recall. There exist thresholds t,t ∈ R++ s.t. z#

n > ˆ

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

n < ˆ

zn only if z#

n = 0.

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Efficiency under Bounded Recall

...see paper!

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Conclusions

Attention in large economies with

  • complementarity / substitutability in actions
  • rich set of payoff interdependencies
  • rich information structure
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Conclusions

Attention in large economies with

  • complementarity / substitutability in actions
  • rich set of payoff interdependencies
  • rich information structure

Efficiency in allocation of attention requires (a) absence of externalities from action-dispersion (b) efficiency in use of information

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Conclusions

Attention in large economies with

  • complementarity / substitutability in actions
  • rich set of payoff interdependencies
  • rich information structure

Efficiency in allocation of attention requires (a) absence of externalities from action-dispersion (b) efficiency in use of information Bounded recall: reallocation of attention towards sources with intermediate transparency

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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

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.)
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Motivation Model Equilibrium Efficiency Bounded Recall Conclusions

Thank You!