Your balance isn't really outside information. In a game like Texas Holdem Poker, your balance is a key factor in how much you bet.
But if we change the game so that "if you flip more than two heads in a row, your chances of flipping a third head are only 10%; if you flip two tails in a row, your chances of flipping a third tail are 90%", the result is the same, the odds turn more negative over time than Game A and you should switch to A after two consecutive flips.
It's not your balance that matters there, it's the amount of money you choose to bring into the game at the start. If I could boost my odds by bringing in only $498 dollars of my five hundred then I would do so every time.
Your actual balance is ridiculous to include in a game's calculations.
In texas hold'em your actual balance is a very significant factor in your optimal behaviour in a particular hand. Short-stack tactics are different to big stack tactics.
Okay, I didn't make myself clear enough. Let's say you have a thousand dollar bill in your hand. You decide to convert $600 into chips because that's a multiple of three, and then you put $400 into your room safe. Your real balance is 1000, but your balance for the game is 600. Your stack tactics are based on the 600. Your chance of winning is based on the 600 you chose to bring. Anything that counted the 400 in your room vault would be ridiculous. Yet the game in the blog would.
But if we change the game so that "if you flip more than two heads in a row, your chances of flipping a third head are only 10%; if you flip two tails in a row, your chances of flipping a third tail are 90%", the result is the same, the odds turn more negative over time than Game A and you should switch to A after two consecutive flips.