[AaronBrown]

I have not seen this problem before, but it doesn't add up the way you describe it. I'm assuming you mean there is $10 in the pot preflop if BB calls, and SB and BB each have $50 left after the preflop betting.

If BB calls preflop and goes all in postflop, SB has to put up $50 to win $60. He'll do this if he thinks his chance of winning is $50/($50+$60) = 5/11 or better. He'll always better than 5/11 chance against a random hand. So if BB goes all-in all the time on any flop, SB calls him all the time. The worst flop for black AA is triple red suited connectors between 6 and 9, AA is 60% to win against a random hand. Even with a perfect flop, BB loses money with this strategy.

To make this work, BB must consider not only the chance of AA against a random hand, but the chance of AA against the hand BB actually holds. Let's say he goes all in on any hand for which he has a 30% chance of winning, and his average chance of winning on these hands is 46% (I just made these numbers up). If he does this, then SB will fold whenever he goes all in. So he makes $7 when this happens. If he folds with less than a 30% chance of winning, he loses $3 on those hands. If there is 1 of the first for two of the second (again, I made those numbers up), he has positive expected value in calling.

While it's possible to compute the numbrs in the last paragraph, you'd need a computer to go through all the calculations. It's not a simple game theory example. I think there must be something you left out.