Regime change Ec 517 October 2, 2017

Dynamics of complementarity and learning Boom and busts in business cycles Switch of political regimes Eastern Europe Arab world Two structural properties Strategic complementarity implies that the marginal agents are in the tail of the distribution The observation of the tail of the distribution provides little information on the entire distribution

Interactions with externalities and complexity Boom and busts in business cycles Financial crises Switch of political regimes or general opinion Eastern Europe Herding behavior

Structural models of interactions “Canonical” model Minimalist, abstract Interactions are caused by externalities Two types of externalities Payoff externality (coordination) Information externality Imperfect observation of the state Learning from others’ actions (social learning, informational cascades, herding)

Coordination Positive externality: strategic complementarity The action of others increases my incentive to act Bank runs Speculative attacks against a fixed exchange regime Change of political regime, etc…

Stylized model Large number of individuals, each chooses, 0-1 action with an individual cost Gross payoff of action: mass of actions Net payoff Distribution of individuals with costs Each individual knows his own c and has an estimate of the others’ costs If agents with cost small than c* invests, and F is the cdf, the payoff is F(c*)-c Under perfect information, Nash equilibrium is F(c*)=c*

Stylized model Large number of individuals, each chooses, 0-1 action with an individual cost Gross payoff of action: mass of actions Net payoff Distribution of individual costs

Static equilibrium Strategy Equilibrium With sufficient strat. compl., Multiple equilibria

Problem of the “choice” of equilibrium Global games (Carlsson and Van Damme, Morris & Shin) (very) Small uncertainty on the structure of the economy No common knowledge

Evolution (dynamics), learning Global games (Carlsson and Van Damme, Morris & Shin) (very) Small uncertainty on the structure of the economy No common knowledge

Imperfect information Individuals have a strategy They observe only Impact of the shift of the c.d.f Aggregate activity shows the tail of the distribution Property inherent to strategic complementarity

Model of regime choice Kuran: “Public lies private truths”. Continuum of people, total mass normalized to one. Y is the mass of people choosing action 0 (speak against) Mass that supports the regime is 1-Y Individual is defined by his preference for the regime Payoff of individual with parameter c : The difference between the payoffs is

Representation Two (stable) equilibria L1 and H1 The structure (cdf) evolves randomly and slowly

Evolution of the structure of costs The distribution of costs is “placed” by a parameter that evolves by a random walk (with regression to the mean) Unique equilibrium with low activity Multiple equilibria (under perfect information) Unique equilibrium with high activity T=60

Evolution of beliefs and actions Individuals form their beliefs from Their private information (cost c) The history of aggregate activities (mass of total actions in each past period) Individuals have a monotone strategy (invest in period t iff the own cost is smaller than ct* (can be proven to be iteratively dominant under the structure of information) Unique equilibrium

Under imperfect information On the left, the true state of nature and the public belief (from history) On the right, the payoff of agent with cost c if every agent (uncertain mass) with cost lower than c chooses to invest

Delays Assumption: individuals can delay Delay has a cost But with delay, more information Tradeoff Impact of the cost of delay (equivalent to longer period) Does a lower cost (shorter period) facilitate coordination ?

Impact of the cost of delay No under the following argument: Because of strategic complementarity, the marginal agent is in the tail of the distribution. Little information If more agents act, more information that reduces the incentive to act: more incentive to delay Waiting for information reduces information Under some plausible assumption, delay prevents an equilibrium with high activity (work with Lucia Esposito).

Social learning Definition for economists: learning for the observation of what others Do Say Learning about what? State of nature that affects the payoff of the observed and me (possibly in different ways.

Rational (Bayesian) learning Requirements Common knowledge on The prior distribution of the states of nature The mechanical properties of the private signals The payoff functions of the others (how they take action as a function of the information: the common prior and their private signal) What is not known is the private information (signal) of others. One’s action is a signal (noisy) on one’s private information

General properties Because learning is rational (Bayesian), the beliefs (probability distributions about the state) converge (in probability). Consequence of the Martingale Convergence Theorem But beliefs may not converge to the truth More “pathologies” in social learning when individuals are selfishly rational.

Convergence of a bounded martingale: representation of the trading strategy Μ Sell Sell Hold Buy Buy Buy Sell Τ

Framework of social learning history (common knowledge): ht = { x1, x2,... xt-1}

Informational cascades Private actions are the messages through which individuals communicate. When the actions are in a discrete set, the vocabulary is limited… Textbook case of two actions, no invest or invest (fixed size) When the private signals are bounded, the belief update by the individual is bounded The combination of the two assumption leads to a cascade: after some finite (random ) time, all individuals take the same action (herd) and nothing is learned anymore.

Geometric representation 5 6 7 CASCADE 2 4  1 3 2 t Two consecutive identical signals induce a cascade Exponential convergence to a cascade

Conditions for a cascade Two necessary and sufficient conditions (model of observation of actions): Discrete actions Bounded private information Intuition Quantum of information is necessary to jump from one action to another. History can be sufficiently powerful to overwhelm any private information.

Mechanism of Informational cascades An individual’s updating combines the private signal and the history of learning. With more history, the weight of the private signal decreases. If there is asymptotic perfect learning, the weight of the individual signal tends to zero. Individual signals have vanishing impact on the action. If the action is in a discrete, set, at some point, the private signal is not sufficiently “powerful” to shift from one action to another. At this point, whatever the signal, the individual takes the same action. Individuals herd. But the reverse is not true: a herd is not a cascade Actually with discrete actions, and unbounded private signals, there is never a cascade, but a herd eventually begins, with probability one.

Herds Cascade: ex ante all agents choose the same action whatever their private signal (which is ignored) Herd: there exists a finite and random T such that all agents after T take the same action. Property is remarkable when private beliefs are unbounded and there is no cascade. Proposition 1: The (public) belief converges to a limit (which is a random variable) Proposition 2: A herd must take place eventually. Cascade implies herd, but the converse is not true.

Existence of herds Intuition: A “deviating” agents creates a large change of the public belief “Despite the strong public, the agent deviates: he must have a very strong private signal” Infinite number of large changes of beliefs are impossible. Hence, herds must take place.

Applications Juanjuan Zhang “The sound of silence”: observational learning in the kidney market Impact of house price when they stay long on the market

Delays in learning Trade-off between the cost of delay and gaining information Slow learning When people more optimistic, the cost of delay is higher In equilibrium, delay must bring more information Hence more activity.

Properties Convergence of belief is very fast (exponential). Beliefs (and actions) may converge to a wrong value Questions Are these properties robust? (small changes of the model keep the properties) Analysis by authors (Vives, Smith&Sorensen) show that, yes, the properties are robust in some sense: In general, social learning is inefficiently slow Why? agents provide an information externalities to others (who make decisions later) but they are not “paid” for this externality agents are (rationally) influenced by others when they take an action and thus reduce the weight of their own information on their action. This reduces the information that their action provide to others.