# Questions on optimal strategy

#### robertson

With \$100 capital, you can flip the fair coin maximum n times, head you double your bet, tail you loose your bet. What is the optimal strategy to invest your capital?

Is the Kelly Criterion applicable here?

#### gamma

is there any book where this problem is solved?

#### Ian Kaplan

I don't think that the Kelley Criteria applies here because it tells you to only bet when you have a good chance of winning and then pull back when you don't. Or put another way, it tells you how to scale your bets depending your predicted chances of winning. In this case it's 50-50.

I was tempted to say that the best should be a classic Martingale. But that is only optimal with an infinite amount of cash.

#### Dawei WANG

Well this reminds me of the martingale betting strategy which goes like this :

- You bet one dollar initially
- If you win, you quit
- If you lose, you bet 2 dollar next time

The winnings in this gam can be written as

$$W_n = \sum_{j=1}^{n}B_j X_j$$

Where $$B_j$$ is the bet where $$B_1=1$$ and for $$j>1$$,$$B_j=2^{j-1}$$ if we lost j-1 times consecutively.

Therefore our expected winning will still be zero since it's a fair game.

$$E[W_n]=1\times [1-2^{-n}]-[2^n-1]\times 2^{-n}=0$$

However, we'll eventually win which means that with probability one

$$W_{\infty}=\lim_{n->\infty}W_n=1$$

Well in your case, we can assume we bet x dollars initially. Then I think the problem will be choosing an optimal x, such that $$x \times [2^n-1] < 100$$ and n is big enough since our probability of eventually wining x will be $$1-2^{-n}$$

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