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I want to stake a friend of mine for poker. I know he is a winning player. That means that he will play 'n' hands and if he ends up down I cover the loses while if he wins we split. The tricky part is deciding what % I keep when he wins.

I think the best way to calculate this is through a continuous function that inputs the amount of hands played and outputs what the split percentages are, given that this person doesn't know how much he will play.

I think this is possible if I estimate a winrate and the variance of his game(for which I can make accurate predictions) while declaring what expected value I would like for myself for the risk taken.

Can anyone tell me how to approach this problem?

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Say you give the player $K$ dollars and they wind up with $X$ dollars after playing for the established amount of time. Say you take fraction $f$ of the winnings. Then your P/L is $$ W(X,K,f) = \left\{\begin{array}{ll} f(X-K)& X-K>0\\X-K & X-K<0\end{array}\right.$$

If you know the distribution $P(X)$ of the player's final amount then to get your average payoff, you integrate $$ E(W) = \int_0^\infty W(X,K,f) P(X)dX.$$ Then you have your expected payoff as a function of $f$ and $K,$ and can set $f$ to get the desired average value winnings.

You could estimate the distribution $P(X)$ based on past winnings, etc.

As one possible approach, appropriate for a small stake relative to his variance, consider if he plays until his total capital is about $mK$ with $m>1$ or he loses it all. (Say you stake him 1000 and he plays till he's up 10000. Then m =10, and whenever he wins you collectively profit 9000.)

In this scenario, you just need to estimate the probability $p_w$ of him winning. Then your expected P/L are $$ E(W) = f(m-1)Kp_w -K(1-p_w).$$

So if you have a good estimate of $p_w,$ then you have a nice simple function of $f$ you can tune to get your desired return.

Of course estimating $p_w$ (or $P(X)$ more generally) is the key here. How to do that would depend on a million things.