A few weeks ago, my family and I were wandering about in an unfamiliar part of the city where we live. We were getting hungry, so we started looking for a place to eat. We happened upon a block that had three restaurants in a row.
All three restaurants served types of food that we enjoy. Although it was early in the evening, one was already quite crowded. Another had a couple at one table near the window. The third appeared to have no customers.
In such a situation, many people might think that there must be some reason why no one is at the third restaurant. Maybe there’s something wrong with it. The restaurant with just one couple might also appear questionable for the same reason.
So, in the absence of any additional information, the natural thing to do would be to join the crowd in the first restaurant. It must be a good, well-known restaurant. Higher quality brings more customers. Right?
Suppose that the likelihood of someone choosing a restaurant is proportional to the number of people already in the restaurant. Given that all the restaurants are initially empty and that the first customer chooses randomly, what happens to the number of people that end up in the different restaurants?
Statistician Susan Holmes of Stanford University has created a Java applet that allows you to simulate such a situation (see “Restaurants at Niagara Falls” at http://www-stat.stanford.edu/~susan/surprise/Polya.html).
The simulation features red, blue, and yellow blocks to represent three restaurants. It starts with somebody in the red restaurant, and the probability of going into a restaurant is proportional to the number of people already in it, plus one.
“The plus one is to avoid the case where a restaurant is empty and so would never get any customers!” Holmes explains.
Most of the time, the red restaurant fills up first. Its head start is hard to overcome. And the final distribution of people among the three restaurants is usually quite skewed, strongly favoring one restaurant over the other two.
So, just because one restaurant is crowded and its neighbors have few customers doesn’t necessarily mean that the packed bistro is the place to be. By a fluke, it may have attracted the first few customers in the area, and the rest just followed the crowd.
What did we do? We happened to glimpse a restaurant in the next block that we had heard about, and we went there. Our meals were delicious.