The point isn't that efficiency goes up with increased attempts, it is that it does not necessarily go down, as you originally claimed.
You can go back through all of those quarterbacks' seasons and see that very plainly -- their highest rating split is often at attempts 21-30 or 31-40. For example, last year, Eli's rating was higher from 31-40 than from 1-10 or 21-30. In 2011, his rating went up with more attempts ... from 76.8 on attempts 1-10 to 86.5 on 11-20 to 102.1 on 21-30 to 111.3 on 31-40. Same thing with Newton last year ... from 90.3 on 1-10 to 100.2 on 11-20 to 102.4 on 21-30 to 118.1 on 31-40.
There is nothing about having more attempts that causes a quarterback to be more inefficient. What you WILL see in many cases, however, is that a quarterback often has to throw more in games when he is already inefficient, and that inefficiency might (or might not) continue, whereas greater efficiency early in the game often reduces the need to throw more later in the game. In addition, inefficiency often means that it takes more throws to accomplish the same thing that can be accomplished in fewer throws (such as going 5-for-6 on a touchdown drive in the first quarter, compared with going 5-for-10 on an equivalent drive). Again, it's the efficiency or inefficiency that helps determine the number of attempts, not the other way around.
