I am offered a game where there is a unfair coin $p > 0.5$ that heads comes up. I start with 1 dollar and I can bet fractional amounts. Payouts are 1 to 1. What is my optimal betting strategy if I know there are two flips?

$ E ( 1 \text{ flip from betting x } ) = p(x) + (1-p)(-x) = 2px - x$. Since I have two flips, I will always bet whatever I have left after the first flip on the second one. So then is my final expression for optimal amount $x$ just a composition $ 2p(2px-x) - (2px- x)=(4p^2-4p+1)x$? However, this is a linear function and so the max just occurs at $x=1$? So then each round is to bet 1 dollar?

Kellys rule seems to imply that I bet $2p-1$ instead?

  • $\begingroup$ In your first flip if you won, you have more than 2px, because you also still have the 1-x that you did not put in. $\endgroup$ – DannyDan Nov 7 '13 at 18:41

You are correct. Generally, if a bet is winning, you want to bet as much as possible for greatest expectation.


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