Last fall I took a class in comedy writing for late night/variety shows and learned that Dave Letterman doesn't watch any of his competitors (Tonight Show, Kimmel, etc.). Leno keeps an eye on his competition to make sure he's not repeating jokes, but Letterman apparently doesn't care about what anybody else does. If he wants to do a certain joke on his show, it's going on the air, no matter what other hosts are doing it.
Now I also learned that most times you see a joke repeated it's a case of great minds thinking alike, not plagiarism. And I'm sure that all of the following shows came up with this same joke on the same night and had no way of knowing that the other shows were doing the same joke. But Letterman's set-up to his clip is awesome.