There are no big 3 globally, just the big 1 - .com Locally it's usually .com + ccTLD at best.
I think the new gTLDs will affect peak prices, even for .com, but not for a while, there is just too much money invested into development in existing extensions for existing domain / website owners to port.
Where it will make a difference will be the entry level or bootstrapped websites / businesses who want to use something brandable but don't want to spend $x,xxx on a good .com