I simulated the win ratio progression, where you try to win back your loses by leaving the bet constant and increasing the amount you win, verses a regular martingale versus a single bet.

The bank started at 1000, the risk was 0.1% of the bank with each bet or 1 for the win ratio progression.  The exit was going bust or winning 10%.  300000 runs were done for each of the progressions.

The win ratio progression lets you last about twice as long and takes twice as long to win.  The Martingale is slightly more expensive than a single bet, the win ratio progression is slightly less expensive.  The differences in profitability are insignificant.

-Edit-  The previous version of this post in which I said the edge was overcome was wrong.  I ran the simulation with a 0.1% edge by mistake.  This simulation is run with a house edge = 1%.


                           single bet  single bet, edge = 0  bet size progression  win ratio progression
mean number of bets                 1                     1                   428                    734
mean number of bet to win           1                     1                   443                    667
mean win                         1100                  1100           1100.740049             1100.02207
probability of win                0.9           0.909090909              0.897828               0.903173
equivalent house edge (%)         1.0                   0.0                 1.173                 0.6489

I've not tried repeating your experiment, but I can't help but notice the minus sign in the 'effective house edge' row.

If that's true, you have found a way to beat the house.

I suspect in reality you just didn't run your simulation for long enough.

Which is it?  Finding a way to beat the house would be relevant to my interests.

I would not worry.  I went to run the simulation again and noticed a typo.  I had run it with a house edge of 0.1% instead of 1.0%.  I have rerun the simulation and fixed the post.

when calculating various characterstics of the betting strategies, you have to be careful
to check whether they are well-defined.
Specifically, expectations of some of the random variables do not exist (= are infinite),
and of course the same holds of the variance.

For example, expected number of bets to win, with 0 house edge is
\sum_{i=1}^\infty i*P(lost i-1 times)*P(won on i'th time) =
\sum_{i=1}^\infty i*1/i*1/(i+1) = \sum_{i=1}^\infty 1/(i+1) = infinity

which does not mean that on average you have to wait infinitely long to win.
It just means that the concept of mathematical expectation is not useful for
this analysis, and so emprical expectation (mean time to win) may be very misleading.

Site seems to be down. doog will be home in about 10 minutes and will take a look at it.

Deb

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