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, [email protected]
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:
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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:
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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, [email protected]
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]
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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
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Remark: Sparse networks
3 No significant herding towards a or b.
Why?
Sparse random graph contains (with ) isolated vertices (independent decisions)
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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
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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
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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.
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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
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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
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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
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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
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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.)