using AAM's to shoot down other AAM's

On the R-77 Wikipage it says the following:

Gennadiy Sokolovski, general designer of the Vympel Design Bureau, said that the R-77 missile can be used against medium and long range air-to-air missiles such as the AIM-120 AMRAAM (…).

Would this work? Considering the speed and small sizes of both missiles.
Although I understand that this is the idea behind the RIM-116, to shoot down incoming anti-ship missiles, so shooting down a missile with another missile seems to be feasible.
Also because these missiles would not be programmed to avoid incoming attacks, and if they did that would push them off course, causing them to lose target lock or at least reduce their range.

This is pretty relevant because it would mean that modern combat comes down to how fast you can detect enemy missiles, and how many missiles you carry to shoot down enemy missiles.