There is competition, but it is used in a good way. It is positive to want to go first, provided the intention is to pave the way for others, make their path more easy, help them, or show the way. Competition is negative when we wish to defeat others, to bring them down in order to lift ourselves up.
I think, that in the twenty-second century, there will be more female reincarnations at female institutions. Then there'll be competition between male lama institutions and female lama institutions. It'll be a positive sort of competition.