Cascade Principles, Bayes Rule and Wisdom of the Crowds Lecture 6 (Largely drawn from Kleinberg book) Following the crowd We are often influenced by others Opinions Political positions Fashion Technologies to use
Why do we sometimes imitate the choices of others even if information suggests otherwise? Why do you smoke? Why did you vote for a particular party? Why did you guess a particular color? Following the crowd It could be rational to do so:
You pick some restaurant A in an unfamiliar part of town Nobody there, but many others sitting at a restaurant B Maybe they have more information than you! You join them regardless of your own private information This is called herding, or an information cascade Following the crowd Milgram,
Bickman, Berkowitz in1960 x number of people stare up How many passers by will also look up? Increasing social force for conformity? Or expect those looking up to have more information? Information cascades partly explain many imitations in social settings Fashion, fads, voting for popular candidates Self-reinforcing success of books on highseller lists
Herding There is a decision to be made People make the decision sequentially Each person has some private information that helps guide the decision You cant directly observe the private information of others Can make inferences about their private information
Rational reasons Informational effects Wisdom of the crowds Direct-benefit effects Different set of reasons for imitation Maybe aligning yourself with others directly benefits you Consider the first fax machine Operating systems
Facebook We will consider the first one today Modeling information cascades Pr[A] where A is some event What is the probability this is the better restaurant? Pr[A | B] where A and B are events What is the probability this is the
better restaurant, given the reviews I read? Probability of A given B. Modeling information cascades Def: So: Notation P[A] = prior probability of A P[A | B] = posterior probability of A given B
Using Bayes rule Applies when assessing the probability that a particular choice is the best one, given the event that we received certain private information Lets take an example Bayes rule, example Crime
in a city involving a taxi 80% of taxis are black 20% of taxis are yellow Eyewitness testimony 80% accurate What is the probability that a taxi is yellow if the witness said it was? True = actual color of vehicle Report = color stated by witness
Want: Pr[true = Y | report = Y] Bayes rule, example We can compute this: If report is yellow, two possibilities: Cab is truly yellow
Cab is actually black So Bayes rule, example Putting it together Conclusion: Even though witness said taxi was yellow, it is equally likely to be truly yellow or black! Second example
Spam filtering Suppose: 40% of your e-mail is spam 1% of spam has the phrase check this out 0.4% of non-spam contain the phrase Apply Bayes rule!
Second example Numerator is easy 0.4 * 0.01 = 0.004 Denominator: So General cascade model Group of people sequentially making decisions
Choice between accepting or rejecting some option Wear a new fashion Buy new technology (I) State of the world Randomly in one of two states: The option is a good idea (G) The option is a bad idea (B) General cascade model Everyone
knows probability of the state World is in state G with probability p World is in state B with probability 1p (II) Payoffs Reject: payoff of 0 Accept a good option: vg > 0 Accept a bad option: vb < 0 General cascade model
(III) Signals Model the effect of private information High signal (H): Suggests that accepting is a good idea Low signal (L): Suggests that accepting is a bad idea Make this precise: General cascade model
Three main ingredients (I) State of the world (II) Payoffs (III) Signals Herding fits this framework General cascade model Consider an individual
Suppose he only uses private information If he gets high signal: Shifts To: What is this probability? General cascade model So
high signal = should accept Makes intuitive sense since option more likely to to be good than bad Analogous for low signal (should reject) What about multiple signals? Information from all the other people Can use Bayes rule for this
Suppose I see a sequence S with a high signals and b low ones General cascade model So what does a person decide given a sequence S? Want the following facts Accept if more high signals than low ones Lets derive this
General cascade model How does this compare to p? General cascade model Suppose we changed the term Whole expression becomes p Does this replacement make the
denominator smaller or larger? Herding experiment Using the model we can derive: People >3 will ignore own signal Cascades - lessons Cascades can be wrong
Accepting an option may be a bad idea But if first two people get high signals cascade of acceptances Cascades can be based on very little information People ignore private information once cascade starts Cascades are fragile
Adding even a little bit more information can stop even a longrunning cascade
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