Brad Rees

I was at a loose end a few weekends ago and thought that I would build myself a little Roulette simulation and give one of the most well known betting strategies a go, the Martingale. The principle of the Martingale is that you double your bet if you lose, and then reset back to your starting bet if you win.

The theory is that you will always win back your losses once you have a win. For example; using a starting bet of 1 and having 3 losses before a win the bets would be: 1 (loss) + 2 (loss) + 4 (loss) results in a total loss of 7. The next bet should be 8 to recover the losses. As it turns out for every win you increase your pot by one, regardless of the number of losses. It sounds perfect on the surface, but can it really work?

It is commonly cited that the casino would have a limit on the table, resulting in a hypothetical punter eventually not being able to bet the required amount to get back all the losses after a large losing streak, even if they had the funds available. With a large number of gambling sites online I thought that there is a possibility that you may be able to find a venue willing to take even a large bet, hence I excluded this constraint from the testing. As it turns out it is very hard to make the system work even with some basic real world constraints, such as credit limits.

In fact if we were to change the multiplication from a factor of 2 to a factor of 3 (1,3,9,27, 81 etc), or a factor of 10 (1, 10, 100, 1000) the theory would still work and possibly the wins would be greater. A win after 3 losses would net 14 from a cost of 13 using a factor of 3, and 889 from a cost of 111 for a factor of 10. As such I though these higher factors should also be investigated as I had not read about people trying to use them.

Firstly, lets start with the ideal case. I have used a simulation of European Roulette, where the table only has one house spot (a single zero), as opposed to American Roulette, which has two house spots (zero and double zero). This would help tip it even further in favour of the punter. Each one of the graphs below represents a typical run, however, each simulation would vary quite a bit.

1000 Games of Roulette - Martingale strategy with no external rules.

After playing 1000 games using no real world constraints we can see that in fact the punter has come out on top. The final amount won was in fact 471, which is quite close to what a statistical model would predict (one unit for each win, with an overall probability of around 0.47). The big issue is that in order to get to that level there were 3 occasions where very large losses had been incurred, putting the punter well in the red. In this specific simulation over 8000 was risked on one losing streak, while already 8000 in the red. This is only for an end payout of 471 – not a good risk in my books.

Martingale - only small bets are allowed once in debt shows a loss overall

If we make it so the punter can only make single unit bets once they are in the red then we see how the picture changes – after only a few games the punter has blown their reserve cash trying to win it all back. From there it is a constant downhill spiral. This is not quite realistic, generally a player would go in with a larger reserve.

No debt allowed and 100 starting cash.

Updating the simulation to allow for a starting kitty of 100 we see that for a while the punter is doing well, however, as soon as there is a run of 6 losses the game is over. In fact, compared to no starting cash the player is actually worse off by about 90 units for this simulation.

At this point I thought it might be worth looking into using higher multiplication factors to see if the situation could be improved. My initial suspicions were that while the wins would be greater, so would the losses. I couldn’t quite get my head around the idea that perhaps if the multiplication was larger than Euler’s number (2.71828) then perhaps something magical would happen regarding exponential growth. It didn’t, so at least I proved something to myself with this experiment.

Triple up each losing bet - no limits

Tripling up showed some interesting results – in this case we have had to bet over 28000 for a final payout of around 19000. The numbers are larger, however, so is the risk. Some simulations have very large losing streaks resulting in any real player being long since wiped out. The situation is much the same when multiplying each loss by 10.

Multiply by 10 each loss - no limits

Wow, almost one billion made after only 500 games! Unfortunately, the bet was over one billion so the chances of any normal person getting a loan to cover a bet like that are, well, small.

I did not stop simulating there, trying all sorts of combinations using different numbers and different strategies. Nothing would work in the long run while some reasonable real life constraints were placed on the simulation. This is not a surprise result, as Einstein apparently once said: “No one can possibly win at roulette unless he steals money from the table while the croupier isn’t looking”. Nonetheless, it was still an interesting exercise.