Social Learning

A Guessing Game Why are Wolfgang Puck restaurants so crowded? Why do employers turn down promising job candidates on the basis of rejections by previous employers? “Nobody goes there anymore, it’s too crowded.” -- Yogi Berra

The Experiment Consider two urns: Urn A contains two blue marbles and one white marble. Urn B contains two whites and one blue One of the urns will be chosen randomly Your job is to guess which urn is chosen

Timing of Moves A randomly chosen individual will be shown a single draw from the urn (with replacement) The individual must then guess which urn was chosen The guess is revealed to the rest of the class, but not the ball drawn from the urn! Repeat with another individual.

Payoffs A correct guess is worth $1, an incorrect guess is worth 0.

How to guess correctly? First player: Let p be the probability that A is chosen. Prior to seeing any marbles, p =.5 Suppose a blue is chosen: Pr(A|Blue) = Pr(Blue|A)Pr(A)/Pr(Blue) Pr(A|Blue) =.667 x.5 /.667 x x.5 Pr(A|Blue) =.667 So if you observe blue, choose A. Observe white, choose B.

Second Player Suppose second player observes blue also. Since second person has heard pl 1’s prediction of A, then, she knows pl 1 has seen a blue marble. Hence, prior probabilities are Pr(A)=.667 Updating: Pr(A|Blue) =.667 x.667 / Pr(Blue) Pr(A|Blue) =.8 Suppose that pl 2 saw white, then 50/50 chance of urn A.

Third player Suppose that when 2 sees a white marble after 1 sees a blue, 2 always predicts B. Then if 1 & 2 predict A, pl 3 should always predict A! Why: if 3 sees a blue marble, it’s A is obvious If 3 sees a white, then his observations are: 1 white + 2 blue (inferred) and blue observations dominate; hence choose A. All other players will also always choose A!

Information Cascade The game illustrates the notion of an information cascade -- a subset of information dominates all subsequent info. Economist offers this as an explanation for widespread use of Prozac. “Yes, ten million people can be wrong,” The Economist, Feb 18, 1994, p 81.

Game (and Decision) Theory How did we arrive at this conclusion: Bayes’ rule Inferring observation from action Sequential reasoning We can then hypothesize frequency of incorrect information cascades how to avoid cascades

Investment Decisions N individuals each of whom makes a once in a lifetime retirement savings decision Two actions 0 or 1 Each person receives a signal s, which might be positive or negative Action 0 is the safe action and pays off 0 Action 1 is risky and pays off according to the sum of signals received by the N people

Simple Case Suppose that s is uniformly distributed on [- 1,1] Suppose that an individual observes the entire past history of actions What strategy should a player choose?

Strategies Given her information, the problem is to determine the sum of the signals. Consider player 1’s problem: 1 only knows her own signal and no others She also has no actions on which to base her choice. Player 1 reasons as follows: All the rest of the players will, on average, have signals that equal zero. Therefore, if my signal is positive action 1 is good Otherwise, action 0 is good

Cutoff Rule Therefore, player 1 follows a cutoff rule: Choose action 1 if s > 0, else choose 0

Player 2 Now consider player 2’s problem: Player 2 gets to observe 1’s action Player 2 also knows her own signal. Hence, player 2 knows that if player 1 chose action 1, her signal was, on average 1/2 If 1 chose action 0 her signal was, on average, -1/2 Hence, player 2’s “cutoff” depends on what 1 did: If player one chose the risky action, then 2 chooses risky if her signal is more than -1/2 If player one chose the safe action, player 2 chooses the risky action if her signal is more than 1/2.

Key insight Notice that the action of player 1 influences that of player 2 There’s inertia built in: If player 1 chose action 1, 2 is more likely to choose it If player one chose action 0, 2 is more likely to choose that action.

Simulations To see how this situation plays out, we do some simulations.

Real Life Government procurement decisions Technology adoption Experimentation with drugs Waves of corporate takeovers Jury verdicts