Attention, Coordination, and Bounded Recall

Transcription

1 Attention, Coordination, and Bounded Recall Alessandro Pavan Northwestern University Chicago FED, February 2016

2 Motivation Many socioeconomic environments - large group of agents - actions under dispersed information

3 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...

4 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

5 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

6 This paper Flexible (yet rich) framework - complementarity or substitutability in actions - rich set of payoff interdependencies

7 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

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

9 Related literature (incomplete) Effi cient 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) Maćkowiak 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)

10 Plan 1 Model (perfect recall)

11 Plan 1 Model (perfect recall) 2 Equilibrium allocation of attention

12 Plan 1 Model (perfect recall) 2 Equilibrium allocation of attention 3 Effi cient allocation of attention

13 Plan 1 Model (perfect recall) 2 Equilibrium allocation of attention 3 Effi cient allocation of attention 4 Bounded recall

14 Model Actions and gross payoffs u i ( ki, {k j } j i, θ )

15 Model Actions and gross payoffs Continuum of agents with payoffs: ( ) u k, K, θ, σ 2 k where: k R individual action K = k dψ(k ) aggregate action σ 2 k = (k K) 2 dψ(k ) dispersion θ R underlying uncertainty ("fundamentals") Assumptions: u( ) quadratic in (k,k,θ), linear in σ 2 k u( ) s.t. equilibrium and first-best unique and bounded

16 Examples Investment spillovers (Angeletos and Pavan AER 2004) u i = Rk i c(k i ) R = (1 a)θ + ak and c(k i ) = 1 2 k2 i Beauty contest (Morris and Shin AER 2002) u i = (1 r) (k i θ) 2 r (L(k i ) L) (k L(k i ) ) 2 k i dψ(k ) = (k i K) 2 +σ 2 k and L = L(k)dΨ(k) = 2σ 2

17 Examples Monetary economies (Woodford 2005, Colombo, Femminis and Pavan, 2014, Llosa and Venkateswaran, 2015) u(θ,c i,n i ) V (C i ) N i ( ) C i = c v 1 v v 1 v hi dh [0,1] Y i = θ α N i p h c hi dh p i Y i T [0,1] Cournot and Bertrand games (Vives JET 1984) u i = (a θk) k i 1 2 k2 i

18 Model Information and attention Common prior: θ N(0,π 1 θ )

19 Model Information and attention Common prior: θ N(0,π 1 θ ) N = 1,234,576 sources of information: y l = θ + ε l with ε l N(0,η 1 l ) l = 1,...,N

20 Model Information and attention Common prior: θ N(0,π 1 θ ) N = 1,234,576 sources of information: y l = θ + ε l with ε l N(0,η 1 l ) l = 1,...,N Agent i s "impressions" x i = (xl i)n l=1 with ( ( ) ) 1 xl i = y l + ξ i l with ξ i l N 0, z i l t l l = 1,...,N where η l : accuracy t l : transparency/clarity z i l : attention

21 Model Attention cost and net payoffs Attention cost: C(z i ) where z i = (z i l )N l=1 C n ( z i ) > 0, all z i 0 lim zn C n(z i ) = convex (results extend to concave, e.g., entropy reduction) E.g. C(z i ) = c ( l z i ) l E.g. C(z i ) = l g(z i l )...but also C(z i ) = µ(z i ;y) (entropy reduction) Net payoff ) u (k i,k,σ 2 k,θ C(z i )

22 Model Timing agents allocate attention z i update their beliefs based on x i commit their actions k i payoffs realized

23 Plan 1 Model (perfect recall) 2 Equilibrium allocation of attention 3 Effi cient allocation of attention 4 Bounded Recall

24 Equilibrium use of information (Angeletos and Pavan, Ecma 2007) Optimality: where k j = E[ κ + α(k κ) x j ; z j ] κ = κ 0 + κ 1 θ (complete-info. equilibrium action) α u kk u kk equilibrium degree of coordination

25 Equilibrium allocation of attention Theorem There exists a unique symmetric equilibrium. In this eq., the attention ẑ that each agent assigns to the various sources of information is s.t., for any source n = 1,..., N that receives strictly positive attention, u ẑ n = κ 1 γ kk n 2C n(ẑ)t n where and where (1 α)π n 1 αρ γ n n π θ + N (1 α)π s s=1 1 αρ s is "influence" of the source π s = η sẑ s t s ẑ s t s + η s is endogenous precision and ρ s = π s η s is endogenous "publicity" Given equilibrium allocation of attention ẑ, equilibrium actions are given by ) k i = κ 0 + κ 1 ( N n=1 γ n xi n all i [0,1], almost all x i R N.

26 Private value of attention Envelope reasoning: hold k( ; ẑ) fixed

27 Private value of attention Envelope reasoning: hold k( ; ẑ) fixed Agent s eq. continuation payoff (fixing k( ; ẑ)): U i (z i ;ẑ) = E[u(K,K,σ k,θ)] + u kk 2 Var[k i K z i,ẑ,k( ;ẑ)] C(z i )

28 Private value of attention Envelope reasoning: hold k( ; ẑ) fixed Agent s eq. continuation payoff (fixing k( ; ẑ)): U i (z i ;ẑ) = E[u(K,K,σ k,θ)] + u kk 2 Var[k i K z i,ẑ,k( ;ẑ)] C(z i ) Private value of attention u kk Var[k K z,k( ;z)] 2 z n private aversion to dispersion reduction in dispersion (fixing eq. strategy k( ; z))

29 Private value of attention Envelope reasoning: hold k( ; ẑ) fixed Agent s eq. continuation payoff (fixing k( ; ẑ)): U i (z i ;ẑ) = E[u(K,K,σ k,θ)] + u kk 2 Var[k i K z i,ẑ,k( ;ẑ)] C(z i ) Private value of attention u kk Var[k K z,k( ;z)] 2 z n private aversion to dispersion reduction in dispersion (fixing eq. strategy k( ; z)) Result generalizes Colombo, Femminis, Pavan (Restud 2014)

30 Plan 1 Model (perfect recall) 2 Equilibrium allocation of attention 3 Effi cient allocation of attention 4 Bounded Recall

31 Effi ciency Welfare : ex-ante utility of representative agent Definition Effi cient allocation consists of attention z along with action rule k ( ;z ) that jointly maximize E[u(k,K,σ 2 k,θ) z] C(z) Team problem Planner s problem: control incentives but cannot transfer information

32 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, where κ = κ 0 + κ 1θ FB α u σσ 2u kk u KK u kk + u σσ aversion to volatility = 1 aversion to dispersion socially optimal degree of coordination

33 Effi cient allocation of attention Theorem Effi ciency in attention requires that, for any n for which z n > 0, z n = κ u kk + u σσ 1 γ n 2C n(z )t n where (1 α )π n γ n 1 αρ n π θ + N (1 α )π s s=1 1 α ρ s is effi cient "influence" of the source π s = η sz st s z st s + η s is endogenous precision and ρ s = π s η s is endogenous publicity

34 Effi cient allocation of attention Theorem Effi ciency in attention requires that, for any n for which z n > 0, z n = κ u kk + u σσ 1 γ n 2C n(z )t n where (1 α )π n γ n 1 αρ n π θ + N (1 α )π s s=1 1 α ρ s is effi cient "influence" of the source π s = η sz st s z st s + η s is endogenous precision and ρ s = π s η s is endogenous publicity Recall that eq. ẑ n = κ 1 γ n u kk 2C n(ẑ)t n

35 Effi cient allocation of attention Envelope reasoning

36 Effi cient allocation of attention 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 (π x,π z ) u kk + 2u kk + u KK Var[K κ k ( ;z),z] 2 + u kk + u σσ Var[k K k ( ;z),z] 2

37 Effi cient allocation of attention 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 (π x,π z ) u kk + 2u kk + u KK Var[K κ k ( ;z),z] 2 + u kk + u σσ Var[k K k ( ;z),z] 2 Holding k ( ;z), Var[K κ k ( ;z),z] independent of z

38 Effi cient allocation of attention 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 (π x,π z ) u kk + 2u kk + u KK Var[K κ k ( ;z),z] 2 + u kk + u σσ Var[k K k ( ;z),z] 2 Holding k ( ;z), Var[K κ k ( ;z),z] independent of z Social value of attention u kk + u σσ Var[k K z,k ( ;z)] 2 z n social aversion to dispersion reduction in dispersion (fixing eff. strategy k ( ;z) )

39 Equilibrium vs effi cient allocation of attention Private value of attention u kk Var[k K z,k( ;z)] 2 z n private aversion to dispersion reduction in dispersion (fixing eq. strategy k( ; z))

40 Equilibrium vs effi cient allocation of attention Private value of attention u kk Var[k K z,k( ;z)] 2 z n private aversion to dispersion reduction in dispersion (fixing eq. strategy k( ; z)) Social value of attention u kk + u σσ Var[k K z,k ( ;z)] 2 z n social aversion to dispersion reduction in dispersion (fixing eff. strategy k ( ;z) )

41 Effi cient allocation of attention Effi ciency in attention requires - effi ciency in use of information: k( ;z) = k ( ;z) - private = social aversion to dispersion u σσ = 0

42 Plan 1 Model (perfect recall) 2 Equilibrium allocation of attention 3 Effi cient allocation of attention 4 Bounded Recall

43 Bounded Recall Idea: posteriors correct, but agents cannot recall influence of individual sources Given attention z j, posterior beliefs about θ continues to be Normal with mean x j = N n=1 δ nxn, i π n with δ n π θ + N s=1 π s and π s η sz s t s z s t s + η s and precision π θ + N s=1 π s However, agent is unable to decompose x j into various impressions x j (x j 1,...,xj N ). Equivalently, unable to decompose his posteriors into Measurability constraint on k(x j ;z z ) θ x i n Distinction relevant only in strategic setting

44 Bounded Recall For simplicity: π θ = 0 Theorem In unique symmetric equilibrium, given allocation z #, actions given by k i = κ 0 + κ 1 x i For any source that receives strictly positive attention in eq., C n(z # ) = u kk Var [ k K;z #,k( ;z # ) ] u [ kk Var K κ;z #,k( ;z # ) ] (1 α) 2 z n 2 z n Novel effect: u [ kk Var K κ;z #,k( ;z # ) ] (1 α) 2 z n private aversion to volatility of own s average action reduction in volatility (fixing eq. strategy k # ( ;z))

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

46 Bounded vs Perfect Recall Reallocation of attention towards sources of average (endogenous) publicity ρ n = z st s z s t s + η s Sources of low publicity: useful to forecast θ Sources of high publicity: useful to forecast K Sources of intermediate transparency: good compromises

47 Bounded vs Perfect Recall Previous result about best responses extends to equilibrium Suppose C(z) = c ( N s=1 z ) s Theorem Let ẑ be eq. attention with perfect recall and z # eq. attention with bounded recall. There exist thresholds t,t R ++ s.t. z # n > ẑ n only if t n [t,t ]. Furthermore for any n for which t n [t,t ], z # n < ẑ n only if z # n = 0.

48 Effi ciency under Bounded Recall...see paper!

49 Conclusions Attention in large economies with - complementarity / substitutability in actions - rich set of payoff interdependencies - rich information structure

50 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

51 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 with intermediate transparency

52 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.)

53 Thank You!