Let's talk stocks, baby. 

(You'll soon see that what I actually want to explain has zero to do with stocks. I know nothing about stocks anyway.)

Three novice day traders invest in the market. Their names are Gekko, Fox, and Mannheim. Each buys a few stocks:

#include <portfolios.html>

Now, here's a question: 

Which pair of the three portfolios are most similar to each other? Gekko and Fox, Fox and Mannheim, or Mannheim and Gekko?

Before you answer, consider the portfolio of Mr. Sato, a Japanese investor: 

Which of the other three portolios is his most similar too? 

Well, none of them. If we consider each of Sato's stocks, and try to compare the number of shares to each of the stocks in each of the other portfolios... well, we can't. Neither Fox, Gekko nor Mannheim happens to have bought any of the same stocks as Mr. Sato.

Thus we can see, intutively, that the similarity of Sato's portfolio to the others is zero. 

Looking purely at who owns which stocks, we would say that Gekko is most similar to Fox. They both have stock in HTDG, HAMB, FFRY, CHZB. But Fox is a newbie, he barely owns barely any any shares at all, whereas Mannheim has only 3 investments in common with Gekko, but he has precisely the same amounts in each of them. 

So, how can we quantify the degree of similarity of the four investors' porfolios?