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Say you have a total of c chips, and I present to you a game with a probability $p$ of winning, and payout $k:1$ (where $p > 1/(k+1)$, so this is a +EV bet). How many chips would you risk to take this bet?

For a more concrete example, say you have 100 chips, and I present to you a game with probability of winning, $p=0.66$. If you win, I will pay 1:1 odds, how many chips would you bet?


I kind of synthesized this question from poker theory. I'm curious to know if there's an optimal solution. My thinking is that this just measures your risk adverseness and there's no mathematical way of finding a bound on c. Not entirely sure.

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  • $\begingroup$ There is a vast literature on this kind of topic. The most successful practitioner was probably Edward Thorp It is well worth reading some of his stuff. $\endgroup$ – almagest Sep 24 at 21:23
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I would play all $c$ chips, but if I am allowed I would e.g., bet each chip one-by-one, assuming that the probability that I win this time is independent of the previous outcomes. Then if $c$ is large then whp I would have something close to my expected winnings.

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The mathematical analysis of this is usually done on expected value, where you want the highest average outcome over the long term. In that case, if the bet is in your favor you should take it and bet all your money. If there is a series of bets that will maximize the expected value, even though the probability you go broke is very high.

You may believe there is a declining marginal utility to money, so winning an enormous amount doesn't count as much as it would seem and going broke is terrible. If so, you have to define the utility curve, then the result can be different. You might see the Kelly criterion for a result along this line.

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  • $\begingroup$ I think I'm looking more into maximizing my bank role rather than in terms of economics. Like suppose "c" represented my entire wealth, then I don't think I would risk all of it especially with only .16 edge $\endgroup$ – Sameer Lal Sep 25 at 3:16
  • $\begingroup$ Then you need to define your utility function. That says what you are trying to maximize the expectation of. The Kelly criterion is based on the utility being the log of wealth, so each doubling is equally valuable to you. $\endgroup$ – Ross Millikan Sep 25 at 3:37
  • $\begingroup$ If you bet a fixed fraction of your bankroll each time, the final result only depends on how many wins you get, not on the sequence. That makes the analysis much easier. $\endgroup$ – Ross Millikan Sep 25 at 3:41
  • $\begingroup$ I'm trying to maximize the expectation of my profits. That makes sense what you are saying. $\endgroup$ – Sameer Lal Sep 25 at 15:42
  • $\begingroup$ If you want to maximize the expectation, bet all your chips every time. See here for an extreme example $\endgroup$ – Ross Millikan Sep 25 at 15:49

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