Learning to Coordinate Very preliminary - Comments welcome Edouard Schaal 1 Mathieu Taschereau-Dumouchel 2 1 New York University 2 Wharton School University of Pennsylvania 1 / 30
Introduction • We want to understand how agents learn to coordinate in a dynamic environment • In the global game approach to coordination, information determines how agents coordinate ◮ In most models, information comes from various exogenous signals ◮ In reality, agents learn from endogenous sources (prices, aggregates, social interactions, ...) • Informativeness of endogenous sources depends on agents’ decisions • We find that the interaction of coordination and learning generates interesting dynamics ◮ The mechanism dampens the impact of small shocks... ◮ ...but amplifies and propagates large shocks 2 / 30
Overview of the Mechanism • Dynamic coordination game ◮ Payoff of action depends on actions of others and on unobserved fundamental θ ◮ Agents use private and public information about θ ◮ Observables (output,...) aggregate individual decisions • These observables are non-linear aggregators of private information ◮ When public information is very good or very bad, agents rely less on their private information ◮ The observables becomes less informative ◮ Learning is impeded and the economy can deviate from fundamental for a long time 3 / 30
Roadmap • Stylized game-theoretic framework ◮ Characterize equilibria and derive conditions for uniqueness ◮ Explore relationship between decisions and information ◮ Study the planner’s problem ◮ Provide numerical examples and simulations along the way 4 / 30
Abbreviated Literature Review • Learning from endogenous variables ◮ Angeletos and Werning (2004); Hellwig, Mukherji and Tsyvinksi (2005): static, linear-Gaussian framework (constant informativeness) ◮ Angeletos, Hellwig and Pavan (2007): dynamic environment, non-linear learning, fixed fundamental, stylized cannot be generalized ◮ Chamley (1999): stylized model with cycles, learning from actions of others, public signal is fully revealing upon regime change and uninformative otherwise 5 / 30
Model • Infinite horizon model in discrete time • Mass 1 of risk-neutral agents indexed by i ∈ [0 , 1] • Agents live for one period and are then replaced by new entrant • Each agent has a project that can either be undertaken or not 6 / 30
Model Realizing the project pays π it = (1 − β ) θ t + β m t − c where: • θ t is the fundamental of the economy ◮ Two-state Markov process θ t ∈ { θ l , θ h } , θ h > θ l with P ( θ t = θ j | θ t − 1 = θ i ) = P ij and P ii > 1 2 • m t is the mass of undertaken projects plus some noise • β determines the degree of complementarity in the agents payoff • c > 0 is a fixed cost of undertaking the project 7 / 30
Information Agents do not observe θ directly but have access to several sources of information 1 A private signal v it ◮ Drawn from cdf G θ for θ ∈ { θ l , θ h } with support v ∈ [ a , b ] ◮ G θ are continuously differentiable with pdf g θ ◮ Monotone likelihood ratio property: g h ( v ) / g l ( v ) is increasing 2 An exogenous public signal z t drawn from cdf F z θ and pdf f z θ 3 An endogenous public signal m t ◮ Agents observe the mass of projects realized with some additive noise ν t m t ( θ, ˆ v ) = mass of projects realized + ν t ◮ ν t ∼ iid cdf F ν with associated pdf f ν ◮ Assume without loss of generality that F ν has mean 0 8 / 30
Timing Agents start with the knowledge of past public signals z t and m t 1 θ t is realized 2 Private signals v it are observed 3 Decisions are made 4 Public signals m t and z t are observed 9 / 30
Information Information sets: • At time t , the public information is � m t − 1 , z t − 1 � F t = • Agent i ’s information is F it = { v it } ∪ F t Beliefs: • Beliefs of agent i about the state of the world p it = P ( θ = θ h |F it ) • Beliefs of an outside observer without private information p t = P ( θ = θ h |F t ) 10 / 30
Agent’s Problem Agents i realizes the project if its expected value is positive E [(1 − β ) θ t + β m t − c |F it ] > 0 For now, restrict attention to monotone strategy equilibria : • There is a threshold ˆ v t such that Agent i undertakes his project ⇔ v it ≥ ˆ v t • Later, we show that all equilibria have this form • With this threshold strategy, the endogenous public signal is m t = 1 − G θ (ˆ v t ) + ν t ���� � �� � noise signal 11 / 30
Dynamics of Information • For a given signal s t , beliefs are updated using the likelihood ratio LR it = P ( s t | θ h , F it ) P ( s t | θ l , F it ) • Using Bayes’ rule, we have the following updating rule 1 P ( θ h | p it , s t ) = := L ( p it , LR it ) 1 + 1 − p it p it LR − 1 it 12 / 30
Dynamics of Information • At the beginning of every period, the individual beliefs are given by � � p t , g h ( v it ) p it ( p t , v it ) = L g l ( v it ) • By the end of the period, public beliefs p t are updated according to � � p t , f z h ( z t ) P ( m t | θ h , F t ) p end = L t f z l ( z t ) P ( m t | θ l , F t ) • Moving to the next period, � � p t +1 = p end 1 − p end P hh + P lh t t Full expression for dynamic of p 13 / 30
Dynamics of Information Lemma 1 The distribution of individual beliefs is entirely described by ( θ, p ) : 1 ˆ dG θ ( v i ) . P ( p i ≤ ˜ p | θ, p ) = 1 ≤ ˜ p I g l ( v i ) 1 + 1 − p p g h ( v i ) • Conditional on θ agents know that all signals come from G θ • From G θ and p they can construct the distribution of beliefs • Rich structure of higher-order beliefs in the background 14 / 30
Monotone Strategy Equilibrium Definition A monotone strategy equilibrium is a threshold function ˆ v ( p ) and an endogenous public signal m such that 1 Agent i realizes his project if and only if his v i is higher than ˆ v ( p ) 2 The public signal m is defined by m = 1 − G θ (ˆ v ( p )) + ν 3 Public and private beliefs are consistent with Bayesian learning Given the payoff function π ( v i ; ˆ v , p ) = E [(1 − β ) θ + β (1 − G θ (ˆ v )) − c | p , v i ] the threshold function ˆ v ( p ) satisfies π (ˆ v ( p ); ˆ v ( p ) , p ) = 0 for every p . 15 / 30
Equilibrium Characterization: Complete Information Lemma 2 (Complete info) If β ≥ c − (1 − β ) θ ≥ 0 , the economy admits multiple equilibria under complete information. In particular, there is an equilibrium in which all projects are undertaken and one equilibrium in which no projects are undertaken. 16 / 30
Equilibrium Characterization: Incomplete Information Assumption 1 The likelihood ratio g h g l is differentiable and there exists ρ > 0 such that � � ′ � � g h � � � � � ≥ ρ. � � g l � Proposition 1 (Incomplete info) Under assumption 1, β 1 If 1 − β ≤ θ h − θ l , all equilibria are monotone, ρ P hl P lh β 1 − β ≤ 2 If max {� g h � , � g l �} 3 , there exists a unique equilibrium. Uniqueness requires: 1 an upper bound on β ; Role of β 2 enough beliefs dispersion. Role of dispersion 17 / 30
Endogenous vs Exogenous Information Sample path with only exogenous information: 1.0 0.8 0.6 0.4 0.2 1 − G θ (ˆ v ) θ 0.0 0 200 400 600 800 1000 Sample path with only endogenous information: 1.0 0.8 0.6 0.4 0.2 1 − G θ (ˆ v ) 0.0 θ 0 200 400 600 800 1000 From now on, focus on endogenous public signal only: Var( z t ) → ∞ 18 / 30
Endogenous Information Lemma 3 If F ν ∼ N (0 , σ 2 ν ) , then the mutual information between θ and m is I ( θ ; m ) = p (1 − p ) ∆ 2 � ∆ 3 � + O 2 σ 2 ν v ) − G h (ˆ v ) ≥ 0 . where ∆ = G l (ˆ Version of the Lemma with general F ν : General Lemma The informativeness of the public signal depends on: 1 The current beliefs p 2 The amount of noise σ ν added to the signal 3 The difference between G l (ˆ v ) and G h (ˆ v ) Point 3 is the source of endogenous information. Definition of mutual information 19 / 30
Signal vs. Noise Example 1: Normal case with different means µ h > µ l Distributions 4 3 g l g h 2 g θ 1 0 0 0.2 0.4 0.6 0.8 1 v i Signal distance ∆ = G l (ˆ v ) − G h (ˆ v ) 1 G l − G h 0 0 0.2 0.4 0.6 0.8 1 ˆ v v = µ h + µ l Result: more information when ˆ , i.e., 0 ≪ m ≪ 1. Alt. signals 2 20 / 30
Inference from Endogenous Signal m t = 1 − G θ (ˆ v t ) + ν t ���� � �� � noise signal Example 1: Normal case with different means µ h > µ l 4 3 g l g h 2 g θ 1 0 0 0.2 0.4 0.6 0.8 1 v ˆ 1 1 − G θ 1 − G l (ˆ v ) ± σ ν 1 − G h (ˆ v ) ± σ ν 0 0 0.2 0.4 0.6 0.8 1 21 / 30
Signal vs. Noise Example 2: Information contained in m under the equilibrium ˆ v 0.7 0.6 0.5 Mutual information 0.4 0.3 0.2 0.1 0.0 -0.1 0.0 0.2 0.4 0.6 0.8 1.0 Current beliefs p Result : in the extremes of the state-space, the endogenous signal reveals no information Parameters 22 / 30
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