# maximization of earrnings

Best strategy to maximize earnings by plying the following game?

The goal is to earn as much money as possible by the end of the game. The following rules apply:

• You start out with 1000 Litas.

• You can only play with what you have. You cannot borrow from the bank.

• Each game involves 20 flips of the magic coin.

• If the coin is a head, you earn what ever you bet.

• If the coin is a tail, you lose what ever you bet.

• P (Head) = p and P (Tail) = q = 1 - p.

• You will be told which coin you will play with before you choose your strategy.

• One of eleven (11) coins will be chosen for each game when you play where: p = (0.0; 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9; 1.0).

Without loss of generality, let $B_0$ represent the total money you have at the beginning (1000 Litas for this question specifically).

Let $B_i$ be the total money you have after step $i$ under certain strategy. The goal becomes, after step $n$ (specifically $n=20$ for this question), what strategy can get $\max{E(B_n)}$, where $E(B_n)$ is the expected value of $B_n$.

Because the event of winning or losing each bet is I.I.D., which is only determined by $p$ featured by a certain coin, the strategy should be identical for each step. Let $r$ be the portion of existing money you want to use to bet on each step, $0\leq r\leq 1$. Then $$E(B_1)=B_0(p(1+r)+q(1-r))=B_0(1-r+2pr)$$$$E(B_2)=B_0(1-r+2pr)^2$$$$...$$$$E(B_n)=B_0(1-r+2pr)^n$$

Thus we have $$\max{E(B_n)}=\begin{cases} E(B_n)|_{r=0}=B_0, & \mbox{if } 0\leq p<1/2 \\ E(B_n)|_{0\leq r \leq 1}=B_0, & \mbox{if } p=1/2\\ E(B_n)|_{r=1}=B_0(2p)^n, & \mbox{if } 1/2<p\leq 1\end{cases}$$ Go back to the original question, there are three cases accordingly:

1) When a coin has $p\in\{0.0, 0.1, 0.2, 0.3, 0.4\}$, do not bet anything and just keep the original 1000 Litas.

2) When a coin has $p=0.5$, bet whatever you want as long you can afford. You may win or lose in one flip, but the total won't change too much statistically in the long run.

3) When a coin $p\in\{0.6, 0.7, 0.8, 0.9, 1.0\}$, in order to maximize the average gain, bet all you have each run.