No, now wait, Dave!

Stone sharpens scissors! That means that we now have an uneven game where scissors and stone are working together to shred paper!

So now, scissors and stone team up, shred the paper, but as the paper gets divided into more and more pieces it is capable of covering more and more stones -- thereby keeping them from sharpening the scissors. Thus, the scissors get dull and can no longer cut the paper...

...then the paper, uncut, remains a constant number and thereby controllable, so it starts losing the game because, when the wind comes up, it hasn't shown the foresight to form alliances with stone to act as a paperweight!

So now, paper, severely undermanned has no hope.

...if only paper had thought ahead.

So, both paper and scissors have a vested interest in forming alliances with stone, but stone is a cold heartless bastard! ...and he sees little value in forming alliances. He's basically satisfied to sit in the corner and remain stoned.