Quote from: nicolaennio on December 08, 2013, 07:13:58 PM

Quote from: tucenaber on December 07, 2013, 11:38:53 PM

Martingale starting at 1
p = 1-0.505⁷ = 0,99162394
EV = p*1-(1-p)*127 = 0,99162394 - 0,0083760574 * 127 = -0,0721353 !!!!

The martingale is clearly much better. The expected value is much closer to zero than the single bet.

The average bet size of the martingale is 7.21B and the average return is -0,0721353, so there is no change in Kelly or more expected return or whatever problem of any sort. However that idea that with a Martingale I can better use my money (achieving a profit with a smaller average bet size) is interesting for a theoretical speculation.

You are correct. The expected value will always be 1% of wagered amount regardless of strategy so everything is fine. This is of course in accordance with our intuition. (Mathematically it's still not quite obvious to me but this is not the math forums so let's drop that)

Regarding the paradox where a martingale strategy is both better and no better at the same time, I think the answer lies in a trade off between expected value and variance. In one case the bet size is fixed and in the other case the bet size is a random variable. The latter thus has a higher variance.