108 DOUBLE STITCHES

The Law of Truly Large Numbers

The Law of Truly Large Numbers
With a large enough sample, any outrageous thing is likely to happen. The point is that truly rare events, say events that occur only once in a million [as the mathematician Littlewood (1953) required for an event to be surprising] are bound to be plentiful in a population of 250 million people. If a coincidence occurs to one person in a million each day, then we expect 250 occurrences a day and close to 100,000 such occurrences a year.
Going from a year to a lifetime and from the population of the United States to that of the world (5 billion at this writing), we can be absolutely sure that we will see incredibly remarkable events. When such events occur, they are often noted and recorded. If they happen to us or someone we know, it is hard to escape that spooky feeling.
http://www.wired.com/wiredscience/2012/05/methods-for-studying-coincidences/

26 February 2013

Herding And Polarization

by jeff

There are two actions, A and B, and there are two observable types of people L and R. Everybody is the same in the following sense: for any single individual either A or B is the optimal action but which one it is depends on an unknown state of the world.

But in another sense people are heterogeneous. It is common knowledge that in the state of the world where A is best for people of type L then B is best for people of type R. And in the other state its the other way around. Each person observes a private signal that contains information about the state of the world.

Acting in isolation everybody would do exactly the same thing: pick the action that is best according to their belief (based on the private signal) about the state of the world. But now embed this in a model of social learning. People make their choices in sequence and each observes the choices made by people who went before.

Standard herding logic tells us that L's and R's will polarize and choose the opposite action even if they get it completely wrong (with L's choosing the action that is best for R's and R's choosing the action that is best for L's)

(A reminder of how that works. Say that an L moves first. He chooses the action that looks the best to him say A. Now suppose the next guy is an R and by chance action B looks best to him. The third guy is going to look at the previous two and infer from their choices that there is strong information that the true state is such that A is good for L's and B is good for R's. This information can be so strong that it swamps his one little private signal and he follows the herd: choosing A if he is L or B if he is R. This perpetuates itself with all subsequent decision makers.)

In effect the L's choose A just because the R's are choosing B and vice versa.