[Borodog]

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Ok. For all of the guys who say, "Until one of us is broke," would your answer changed if Daniel bought in for say, $2,000,000 instead of $125,000?

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500 to 2,000,000, so 1 to 4000 pot odds

You need to only win 11 times to win the total pot

AA is favorite over xx 85% of the time

.85^11= 1 in 6 !!!!!!!!!!!!!!

You will lose 500 dollars 85% of the time, 15% of the time you win 2 million... and you would refuse this offer?

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No, I would not refuse that offer.

Unfortunately, that wasn't the offer.

It'd be trivial to say, "I'll give you 4000:1 on a wager of whether pocket aces will hold up 11 times in a row versus 11 randomly dealt hands."

This is not the same question as asking, "Now that you've won 10 times in a row and have over a million dollars in front of you, five times more than your entire net worth as of last night, would you like to bet your one million dollars, most of your current life savings, on whether pocket aces will hold up in the next hand?"

Personally, I would not wager the majority of my net worth simply to double my money, even if I were an 85% favorite.

So I guess my answer to the guestion is that I would stop when my stack exceeds my net worth prior to the start of the game.

The question is basically about risk tolerance and utility, rather than expected value. The highest expected value play of course is to play until someone is broke. The expected value is not as high as you may expect though; it's only about $35k for the first case and [$300k] for the second. But the highest EV case is also the most likely case for you to end up out $500 instead of up a few thousand or tens of thousands.

As several authors including Sklansky and Brunson have noted, forgoing a +EV gamble with a significant risk of going broke, so that you are able to take advantage of future +EV gambles is not a bad thing.

If you were to keep playing until somebody is broke, it will usually be you.