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Social Networks 101 P ROF. J ASON H ARTLINE AND P ROF. N ICOLE I MMORLICA

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Announcement Lecture is in L211 this Friday (4/17), not Pancoe!

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Guessing game Experiment: There are three balls in this urn, either or

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Guessing game Experiment: This is called a blue urn. This is called a yellow urn.

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Guessing game Experiment: When I call your name, 1. You (and only you) will see a random ball. 2. You must then guess if the urn is a blue urn or a yellow urn, and tell the class your guess. If you guess correctly, you will earn one point.

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What should the 1 st student do? ? ?? Guess that urn is same color as ball.

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What should the 2 nd student do? ? ?? 1 st guess was blue.

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What should the 2 nd student do? ? ?? 1 st guess was blue. Guess that urn is same color as ball.

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What should the 3 rd student do? ? ?? 1 st and 2 nd guesses were blue. Guess that urn is blue no matter what she sees!

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What should the n th student do? ? ?? First (n-1) guesses were blue. First 2 students told the truth.

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If the first two guesses are blue, everyone should guess blue.

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Lecture Eight: Information cascades

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Staring up at the sky. Choosing a restaurant in a strange town. Self-reinforcing success of best-selling books. Voting for popular candidates.

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Information cascades When: 1. People make decisions sequentially, 2. and observe actions of earlier people. Information cascade: People abandon own info. in favor of inferences based on others actions.

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ImitationCascade Rational (not simply peer pressure)

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Observations 1.Cascades are easy to start. (every student makes same guess so long as first two students make same guess)

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Observations 2. Cascades can lead to bad outcomes. (given blue urn, chance of seeing red ball is 1/3, so first two students guess red with prob. (1/3) 2 = 1/9)

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Observations 3. Cascades are fragile. (if a few students show the color of the ball they picked to the entire class, the cascade can be reversed)

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Conditional probability How likely is it that the urn is yellow given what youve seen and heard?

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Time for

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Strategy for urns A player should guess yellow if, Pr [ yellow urn | what she saw and heard ] > ½ and blue otherwise.

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Analysis From setup of experiment, Pr [ ] = 1/2

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Analysis From composition of urns, Pr [ | ] = 2/3

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Analysis Suppose first student draws : Pr [ | ] = Pr [ | ] x Pr [ ] Pr [ ]

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Analysis Suppose first student draws : Pr [ | ] = x Pr [ ] Pr [ ] Pr [ ] = Pr [ | ] (2/3) x (1/2) Pr [ | ] xPr [ ] Pr [ | ] xPr [ ] + (2/3) x(1/2) (1/3) x(1/2)= (1/2)

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Analysis Suppose first student draws : then he should guess yellow. Pr [ | ] = (2/3) > (1/2)

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Analysis Suppose second student draws too: Pr [ |, ] 1 st students guess 2 nd students draw

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Analysis Suppose second student draws too: by a similar analysis, so she also guesses yellow. Pr [ |, ] > (1/2)

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Analysis Suppose third student draws : Pr [ |,, ] 1 st students guess 2 nd students guess 3 rd students draw

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Analysis Suppose third student draws : Pr [ |,, ] = Pr [,, | ] x Pr [ ] Pr [,, ] (1/3) x(2/3) x(2/3)(1/2)

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Analysis Pr [,, | ] x Pr [ ] Pr [,, ] = Pr [,, | ] x Pr [ ] + (1/3) x(2/3) x(2/3)(1/2) (2/3) x(1/3) x(1/3)(1/2)

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Analysis Suppose third student draws : Pr [ |,, ] = (1/3) x(2/3) x(2/3)x (1/2) (1/3) x(2/3) x(2/3)x (1/2) + (2/3) x(1/3) x(1/3)x (1/2) = (2/3) > (1/2)

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Analysis The best strategy for the third student is to guess yellow no matter what he draws.

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Other sequences # yellow guesses - # blue guesses Blue cascade starts. Yellow cascade starts student

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Announcement Lecture is in L211 this Friday (4/17), not Pancoe!

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Next time TBA

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