Given that I played a game with unknown probabilities and odds (actually they are some trades in reality) and I recorded the results into a spreadsheet. How can I calculate the best allocation in each trade if they must be all the same?
That is,
let $A_i$ be my account size after $i$-th bets,
let $r_i$ be my recorded return rates (positive or negative)
let $x$ be my allocation
Then, We have $A_i = A_{i-1} + r_i x A_{i-1} = A_{i-1} (1+r_i x) $
I want to maximize $f(x)=\frac{A_n}{A_0}=\prod_{i=0}^{n-1}{(1+r_i x)}$
I thought it should be a classic problem. What is the name of it? And how can we solve it especially in a spreadsheet? Thank you!
I am the author of this post. It tunrs out I could solve it with spreadsheet after some reformation of the problem
To maximize $f(x)$, I want to find the root of
$g(x)=\frac{d}{dx} ln f(x)=\sum \frac{r_i}{1+r_i x}$
Spreadsheet has a solver to minimize the function $|g(x)|$
But have to be aware that there are some local max that are not really global.
It can be also done without solver using Newton's method, in which we have very fast convergence.