This is sort of spawned by a thread from /r/bjj and sort of spawned by me earning a pair of submissions against my instructor this morning. I use the word "earning" very much intentionally, because I didn't 'submit' my coach. I was able to generate a situation that gave me an opportunity, and because I was doing it correctly and well I was able to get to a position to attempt a submission. Because I was doing that correctly and well I got the tap. At multiple points my coach could have just shut me down and crushed me under side control for as long as he wanted to and then subbed me. He did it multiple times during our rolls today, but it helps both of us grow if he eases of just enough to make me work hard and do everything correctly in order to EARN a submission.
The same thing applies when I'm rolling with white belts or new blues (Even senior blues at this point sometimes). I'm not worried about them tapping me. If they do everything right I'll even defend slightly less vigorously than I might otherwise, just like my instructor does for me, so that they can get to the tap if they do everything right. That doesn't at all mean that I'm GIVING them the submission. If they make a positioning mistake, or give me too much space, or anything like that then I'm going to counter and escape. But I'm not going to just refuse to give them the opportunity to succeed.
And when they get that tap they SHOULD be proud. Not because they 'beat' a purple belt, but because they did everything CORRECTLY and earned that submission. They earned that success and they should be happy about it. By the same token lower belts should understand that they aren't 'beating' people in class. I've talked about this before, but people may be approaching rolls in class in a variety of different ways, some of which may lead to a brown belt getting tapped by a blue belt a dozen times. That doesn't mean the blue belt should be a brown belt, or even that they are anywhere NEAR the same skill level. Take your successes in isolation. You do something right, you succeed, be proud of it, but don't let it fuel your ego.