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Social Learning Diversity

• Agents observing diverse sets of past actions facilitate social learning.
• Smaller samples of past actions speed up social learning.

Partial Recall and Sampling

• Agent observation of small random samples from past observations can improve social learning efficiency.
• Sampling restriction does not prevent cascades when private beliefs are bounded.

Average Welfare Function and Welfare Improving Principle

• In the absence of a common memory of the past, social learning through observation sets becomes more complex.
• The average welfare function is a tool for analyzing the convergence of beliefs and actions.

Adequate and Complete Learning

• Adequate learning is defined when all agents take the optimal action in the end.
• In models with sampling, adequate and complete learning are equivalent.

Convergence

• The importance of the convergence speed is emphasized in the learning process.
• The restriction to small observation samples may accelerate convergence.

Sampling in Large Population

• In a large population with chain sampling, beliefs can converge to the truth even without private information.
• The model properties differ sharply from those in sequential models.

Sequential Model with Sampling

• Agents observe a fixed number of randomly selected past actions.
• The process of social learning depends on the number of investments in the history.

Process of Social Learning

• Distribution of investments and the number of investments in history affect the social learning process.
• The distribution of investments in history depends on the state of nature.

Common Knowledge and Signal Computation

• Observed investments y_t have a distribution depending on P(y_t=k|Z_t)P(Z_t| ).
• Value of y_t is a signal on .

Agent's Investment Decision

• Agent t invests if his LLR is greater than a threshold value.
• Investment probability in period t depends on observed investments Z_t and the state.

Information Requirement in Common Knowledge

• When agents observe the entire history of actions h_t, the common knowledge is summarized by the public belief.
• When agents observe only a sample of past actions, the common knowledge is about the distributions of Z_t in the two states.

Herding Phenomenon

• Assuming limited private information and a specific precision parameter, a herding phenomenon can be observed.
• This phenomenon leads to agents copying their first observation regardless of their private signal.

Cascade Effect

• Cascade effect occurs when each agent copies one observation, leading to convergence of actual fractions of investing agents.
• Asymptotically, more agents take the correct action with higher precision of private signals.

Role of Weak Evidence

• Observation of others may provide weak evidence, leading agents to rely more on their private information.
• Weak evidence serves as the channel through which private information enters the social learning process.

Cascades and Learning

• Agents adding private information to the pool of social information enables the reception of weak information from others, contributing to cascades.
• The seed phase can induce a cascade with positive probability if it is sufficiently long and the private beliefs are bounded, but the probability decreases with longer seed phases.

Effect of Sample Size

• A larger sample size initially improves learning efficiency by enabling agents to sample more informative actions, but in the long run, weak evidence becomes more likely with a smaller sample size, improving social learning efficiency.
• The convergence of beliefs and actions can't be analyzed with the MCT due to the absence of a martingale of public beliefs, requiring new tools.

Average Welfare Function

• The average welfare function (AWF) is proportional to the level of social welfare and depends on the average welfare of new agents.
• The AWF is used as a Lyapunov function for the analysis of the evolution of public expected values.

Welfare Improving Principle

• The Welfare Improving Principle (WIP) states that the average welfare function is monotonically increasing over time.
• Copying the first observation in the sample of size N is a strictly dominated decision rule at time t, leading to a strict inequality.

Investment Strategy

• Agent's investment decision depends on sets S and S, with at least one having a positive probability. Signals are independent of the observation x.

Ex-Ante Welfare Level

• The ex-ante welfare level of the noncopying strategy agent is determined by a complicated formula involving signals, sets S and S, and the probability of investment in different states.

Convergence Learning and Adequacy

• Complete learning occurs when the mean belief of agent t tends to the truth. Adequate learning implies the expected probability of investment for an outsider tends to 1 for state 1 and 0 for state 0.

AWF and Convergence of Learning

• The AWF converges to a value W. The upper bound of AWF is (1-c)/2 and is attained under certain conditions.

Learning Completeness and Incompleteness

• If private beliefs are unbounded, learning is complete; if bounded, learning is incomplete with a cascade occurring in finite time with positive probability.

Cascades and Herding

• A cascade may arise in which all agents take the same action, leading to herding behavior influenced by observations.

Sampling in a Very Large Population

• The model introduces a fading effect in the probability of observing past actions, linking the previous model with Banerjee and Fudenberg's model.

Interpretations and Implications

• The model's interpretations and representations in social learning and economics are discussed along with simplifications made for presentation.

Investment Decision

• An agent invests only if state 1 is more likely than state 0, given his information.
• The decision process at time t determines the variation of X=(X0, X1) per unit of time.

Observation of One Agent

• The observation of one other action yields a probability estimate greater than 1/2, and the point X is invariant over time.
• Any point above the 45 line is invariant.

Observation of Two Agents

• The point X converges to some limit on the segment AC.
• Any point in the shaded area is invariant.

Effect of Private Information

• When new agents receive a private signal, the effect of the observation of other rms is reinforced.
• Private information determines the agent's action when the observation of others yields weak evidence.

Convergence Analysis

• The representing point converges to the correct point even if the private beliefs are bounded.
• Banerjee and Fudenberg (1995) invented the AWF and the WIP for the general analysis of the convergence in the model.

No Complete Learning: N=1

• The differential equation (5.12) converges to a limit point X=(X0,X1) in the set Delta1, indicated by the shaded area in Figure 5.1.
• Copying is strictly suboptimal if the limit point X is not in Delta1, leading to the conclusion that the AWF is strictly increasing at the limit point, contradicting the convergence to X.

Complete Learning with Bounded Private Beliefs: N=2

• On the segment AC in Figure 5.4, the use of both observations by an agent strictly dominates the copying of the first observation, while the agent can ignore his second observation elsewhere.
• The WIP and an argument about the limit point lead to the conclusion that the limit point cannot be different from the correct point with coordinates (0,1).

No Private Information Needed: N>3

• If the first observation of an agent is investment, then copying that action is optimal only if X1/X0 > 1, and for any other observations, not investing is not strictly preferred.
• Learning is perfect asymptotically for N>3, where the agents do not have private information, as demonstrated by the argument about the limit point and the convergence to the correct point.

Proposition 5.4

• Learning is complete when N=3 and agents do not have private information, with the case X1=X0 being ruled out almost surely.
• A sample of size larger than two improves the asymptotic properties of social learning, making private information unnecessary.

Social Learning or Sampling in a Large Population?

• In the BF model, a larger number of observations improves the efficiency of learning in the limit, and private information is required for complete learning when N=2, but can be dispensed with entirely if N>3.
• The sequential model analyzed by Smith and Sorensen demonstrates that social learning does not converge to the truth if private beliefs are bounded, with social learning being more efficient when the sample size is smaller.

Exercises

• Exercise 5.1: Discuss cascades in the model of Section 5.1 and determine the public belief at the beginning of period 3.
• Exercise 5.2: Address various scenarios related to invariant points, convergence to the frontier of the invariant set, and the evolution of X=(X0,X1) in the BF model.
• Exercise 5.3: Study learning from the previous action in a scenario with N agents and a uniform distribution on [1,1] for the signals.

Model Analysis

• Agent invests if his signal is greater than the expected value
• Relation between S1t and S0t, limit properties make a presentation on this information