For a perfectly balanced game, see tic-tac-toe. Two people who understand the game will always end with a tie. Perfectly balanced, and so no reason to play once you understand it.
I have to wonder what your definition of balance is. As Moonsong & others pointed out, Chess is
not quite perfectly balanced, because white has an advantage in going first, in a game with, potentially, a great many moves between that first one and victory and much potential for depth of play that can shift momentum to black. Tic-tac-toe, OTOH, is very short game - 9 turns maximum, 5 of them going to X - so the advantage of X, going first, is much greater. That, alone, renders it imbalanced (really, unfair, more than imbalanced, which can be addressed by playing an even number of games until one player wins more than the other - good luck with that).
Now, the definition of balance I've seen that seems to work best for RPGs goes something like this: a game is better-balanced the more choices it presents to players that are both meaningful and viable.
Tic-tac-toe presents X with 9 initial choices, obviously. However, 6 of them are meaningless: it doesn't matter which corner or side you place your X in, the game's potential results set will be the same, regardless (there's nothing special about the top/bottom/left/right of the grid, it could be rotated with no effect on play). There's three meaningful choices for X: center, corner, or side. One of those, choosing a side square, is decidedly inferior to the others, the choice of corner square is clearly the best (again, it's a solved game, so we know that X starting in a corner & playing optimally leaves O only two paths to stalemate, both of which start with the center square). On his first move, O would seem to have 8 choices, but, if X has played corner & is playing optimally, has only one viable choice: center, the others result in certain victory for X.
If X is playing optimally, O can force a draw by playing optimally, but, if O makes one mistake, X can force a win. OTOH, if O is playing optimally, X must make two mistakes to allow O a chance at victory.
Not remotely balanced, sorry.
But, it does illustrate why imbalance is bad for games, including RPGs. It's really the same as the issues D&D has had with 'class Tiers' and 'must-have feats' and '5MWDs' and "the greatsword is strictly superior to the greataxe because 0.5 DPR" and the like, just with D&D (thankfully) having a thick insulation of complexity to keep it from being entirely solved in the sense tic-tac-toe has been.