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Imagine a gambler having x dollars to start with. Each round the gambler has a chance of winning equal to p that is itself randomly chosen from [0, 1]. If he wins he gets twice the money he put in (meaning he doubles the money every time he puts all the money in and wins). And gets nothing if he loses.

Gambler can choose the amount of money he puts in the game ranging from 0 to x knowing p (probability that he can win this round).

Now, what is the optimal strategy to maximize outcome in 100 rounds?

If you put all the money in a round that has a win probability higher than a half and loose then outcome goes to 0 and at the same time it has the most outcome?

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  • $\begingroup$ We just had essentially this question from another account. It did not have the variable probability of success. Was that you? $\endgroup$
    – Ross Millikan
    Commented Sep 18, 2020 at 21:00
  • $\begingroup$ Yes I forgot to note that out :) $\endgroup$
    – Mahdi Khodabandeh
    Commented Sep 18, 2020 at 21:01

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If you are trying to maximize expected value, you should bet all your money on each round where $p \gt \frac 12$. This is discussed in this question. Many people are hypnotized by the fact that one loss means you finish with $0$, but it is compensated for by the enormous gain when you win them all. If you want to maximize something besides expected value you need to define that.

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  • $\begingroup$ The problem here is how lucky he has to be to win all the matches and end up with any amount of money. In fact I simulated this in python and found a better strategy that I'm not still sure if its the best. your strategy was almost guaranteed to end up with 0\$. My strategy: if p is greater than a half use p times your money and don't bet otherwise (bet 0$). This one ended up with 35812702610\$ on average (simulated 10000 games and 100 rounds in each game). @ross $\endgroup$
    – Mahdi Khodabandeh
    Commented Sep 19, 2020 at 8:39
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    $\begingroup$ This is me from future... Turns out that you are right. I just didn't simulate enough games for all-in to shine up. seems like that does much better than any other strategy and I'm totally convinced now that's best strategy. $\endgroup$
    – Mahdi Khodabandeh
    Commented Sep 19, 2020 at 9:23

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