1. Regime change
Ec 517
October 2, 2017

2. 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

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

4. 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)

5. 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…

6. 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*

7. 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

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

9. 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

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

11. 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

12. 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

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

14. 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

15. 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

16. 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

46. 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 ?

47. 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).

48. 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.

49. 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

50. 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.

51. Convergence of a bounded martingale: representation of the trading strategyΜ
Sell
Sell
Hold
Buy
Buy
Buy
Sell Τ

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

53. 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.

54. Geometric representation5 6 7
CASCADE 2 4  1 3 2 t
Two consecutive identical signals induce a cascade
Exponential convergence to a cascade

55. 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.

56. 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.

57. 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.

58. 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.

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

60. 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.

61. 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.