15. 05. 2007 Observational Learning in Random Networks Julian Lorenz, Martin Marciniszyn, Angelika Steger Institute of Theoretical Computer Science, ETH.

15. 05. 2007

Observational Learning in Random Networks

Julian Lorenz, Martin Marciniszyn, Angelika Steger

Institute of Theoretical Computer Science, ETH Zürich

215.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Observational Learning

Examples: Brand choice, fashion, bestseller list … Stock market bubbles (Animal) Mating: Females choose males they observed being selected by other females (Gibson/Hoglund ´92, “Copying and Sexual Selection“)

When people make a decision, they typically look around how others have decided.

Decision process of a group where each individual combines own opinion and observation of others

„Word of mouth“ learning, social learning:

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Model of Sequential Observational Learning

Agents are Bayes-rational and decide using Stochastic private signal (correct with prob. >0.5) Observation of other agents‘ actions

Macro-behavior of such learning processes?How well does population as a whole?

(Bikhchandani, Hirshleifer, Welch 1998)

Population of n agents makes one-time decision between two alternatives (a and b) sequentially

a or b is aposteriori superior choice for all agents (unknown during decision process)

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Model of Sequential Observational Learning

Agents can observe actions of all predecessors

= Predecessors that chose option= Predecessors that chose option

If tie, follow private signal.

“Majority voting of observed actions & private signal“

In total votes.

Bayes optimal local decision rule in [BHW98]:

Can show: Bayes optimal strategy for each agent

(optimizes probability of correct choice)

Information externality Imitation rational

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Sequential Observational Learning [BHW98]

a

b

ba

a

…

Example:

615.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Informational Cascades in [BHW98]:

Agent chooses a if ¸ 2, b if · -2 and follows private signal if -1· ·+1.

Equivalent version of decision rule

Obviously, key variable is .

Eventually, hit “absorbing state“ or

In the long run, almost all agents make same decision Incorrect informational cascades quite likely!

???????? ????

Globally inefficient use of information

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Informational Cascades in [BHW98]:

[correct cascade]

[incorrect cascade]

Confidence of private signal

[correct cascade]

Even in cascade imitation is rationalLocally rational vs. globally beneficial

Remark:

815.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Wisdom of Crowds

Actions observable of subset only

What would improve global behavior?

“Why the Many Are Smarter Than the Few and How Collective Wisdom Shapes Business, Economies, Societies and Nations” (2004)

… vs. incorrect informational cascades ?

915.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Learning in Random Networks

Random graph on n vertices, each edge present with probability p.

Agents can only observe actions of their acquaintancesmodeled by random graph :

Then agent

chooses

Agent‘s local decision rule: Same as [BHW98]

be #acquaintances that chose optionLet

and and

For p=1: Recover [BHW98]

1015.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Theorem (L., Marciniszyn, Steger ’07)

= # correct agents in

1

a.a.s. almost all agents correct

Network of agents is random graph , > 0.5

Result: Macro-behavior of process depends on p=p(n)

2 constant:

with constant probability almost all agents incorrect

1115.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Remark: Sparse networks

3 No significant herding towards a or b.

Why?

Sparse random graph contains (with ) isolated vertices (independent decisions)

1215.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Discussion

2 constant: with constant probability almost all agents incorrect

Generalization of [BHW98]

Entire population benefits from learning and imitation

Intuition: Agents make independent decisions in the beginning, information accumulates locally first

Less information for each individual Entire population better off

1 a.a.s. almost all agents correct

1315.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

whp next agent pk1 À 1 neighbors & majority correct whp correct decision!

Idea of Proof (I)

Suppose correct bias among first k1 À p-1 agents

However, technical difficulties: Need to establish correct “critical mass” Almost all subsequent agents must be correct … and everything must be with high probability

1 a.a.s. almost all agents correct

Proof uses Chernoff type bounds and techniques from random graph theory

1415.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Idea of Proof (II)

2 const.: const prob almost all agents incorrect

With constant probability, an incorrect criticial mass will emerge

1Herding as in

Because of high density of network, no local accumulation of information.

1515.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

1 a.a.s. almost all agents correct

Phase I : whp fraction correct

Phase II : whp fraction correct

Phase III : whp almost all agents correct

We show:

Proof

Phase I Phase II Phase III

Early adoptors Critical phase

Herding

Choose „suitable“ and .

Then: Because of follows.1

1615.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

1 a.a.s. almost all agents correct

Phase II :During Phase II, increases to

Lemma :

More and more agents disregard private signal

But:

Proof

Consider groups of agents who are „almost independent“.

But: Conditional probabilities & dependencies between agents in Phase II …

Critical phase

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1 a.a.s. almost all agents correct

ProofPhase II (cont)

w.h.p. edge in each Wi

…

… …

& sharp concentration

Iteratively, w.h.p. fraction correct in Phase II

correct agents

1815.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

1 a.a.s. almost all agents correct

Proof

Phase III :

Whp almost all agents correct in Phase III.

… w.h.p next agent has À 1 neighbors & follows majority

… again technical difficulties (consider groups of agents), but finally ….

Herding

1915.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

p=1/log n, correct cascade

Numerical Experiments

Population size n

Rela

tive

Frequ

ency

p= , correct cascade

p=0.5, correct cascade

p=0.5, incorrect cascade

Signal confidence: =0.75

2015.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Conclusion

Macro-behavior of observational learning depends on density of random network

Intuition

Future work

Critical mass of independent decisions in beginning (information accumulates)

Correct herding of almost all subsequent agents

Dense: incorrect informational cascades possible Moderately linked: whp correct informational cascade

Other types of random networks (scale-free networks etc.)

2115.05.2007 Julian Lorenz, jlorenz@inf.ethz.ch

Thank you very much for your attention!

Questions?