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There are different ways of looking at this problem. Do we just pick an arbitrary stopping point because it 'feels' right? Or do we try to apply mathematical principles and real world scenarios?

One way one looking at this problem is to say that I only have 40% chance to get to 32K so I will stop at at let's say 8K - like pococurante did. So getting to 8K becomes a strategy to tackle this problem. That's how I used to approach this problem - with those type of 'strategies'.

But then I thought of it this way. If someone offered me 8K with a 1 to 5 odds of doubling it - and it was a one time deal - would I take it? And I can say that without reservation I would because in real life I would never get those type of odds AND because I am freerolling. Mathematically it's incorrect not to try and double up with such great odds when it is a one time deal and when I am freerolling. I then came to the conclusion that I would try and double up any arbitrary amount between $1 and $1billion if it is a freeroll with great odds and it was a one time deal.

So my question then became - why am I stopping at some arbitrary small number (like $8k) just because I am doing it multiple times in a row? Isn't each decision point independent? Is my strategy for this problem flawed? I have great odds with a exponential reward. I started looking at it this way:

If I stop at $8K I am then passing up a 28% chance of turning $8K into a million.

If I stop at $8K I am then passing up a 4.5% chance of turning $8K into a billion.

One billion is life changing to me, my family, all my friend, all my relatives, and all my favourite charities. $8K is is nothing in comparison. When in real life am I EVER gonna get such great investment odds? There is no stock, no bond, no real-estate deal that have potential for such great return for present dollars. I will never enter a $8k buy-in poker tourney in which I am 28% favourite to win a million. So why am I stopping? It's a freeroll!

Another way to think about this problem is that if you had two groups of 20 people playing this game and competing to get the most amount of money as a group. Each group then picked two strategies. One group would play each time until they got to $8K (or bust) and stop. The other played until they got to $1Mil (or bust) and then stop. The group with the $1mil strategy would crush the $8K strategy group majority of the time. So all repeating simulations point towards going deep in the game as the correct strategy.

And yet having said all that... by going for it big I am almost certain to walk away with nothing! So how can I justify that to myself or anyone else in my life who that would impact financially (if for example I busted out at the 500K or so level)?

The dilemna of this problem is that in order in to maximize your payout you have to play this game deep. And yet by playing this game deep you are almost certain to lose and get nothing.