Martingale: The gambler doubles his bet after every loss...

There are a few complicated Martingale math papers floating around and even those are mostly about roulette but I thought of a way to make a really simple math based presentation.

First of all I am assuming the goal is to make as much profit as possible from a given risk.

Lets take a very small run and calculate all possibilities. We will start with 4 units to risk over 2 trades with 4 possible total outcomes. The system has a 50% chance of winning and wins 1.1x risk when it wins. Lets define Risk of Ruin as losing 75% of the 4 unit start account. Of course we don't normally only make 2 trades and risk such a high proportion of our risk capitol but I will show later how the conclusions hold true when we add trades or increase our edge or reduce risk or reduce the martingale multiplier.

Standard risk 1 unit on each trade non martingale:
sum “L” “L”
-2 -1 -1
“W” “L”
0.1 1.1 -1
“L” “W”
0.1 -1 1.1
“W” “W”
2.2 1.1 1.1
EV: 0.4


OK now lets double the bet after each loser:
sum “L” “L”
-3 -1 -2
“W” “L”
0.1 1.1 -1
“L” “W”
1.2 -1 2.2
“W” “W”
2.2 1.1 1.1
EV: 0.5
Hey the Martingale really is more profitable! Well, of course it is, we have taken on more risk so it's not a fair comparison. notice in the martingale we have a 25% chance of RoR.

We can run the system NON martingale and risk 1.5 units on each trade and wind up with an identical RoR of 25%:
sum “L” “L”
-3 -1.5 -1.5
“W” “L”
0.15 1.65 -1.5
“L” “W”
0.15 -1.5 1.65
“W” “W”
3.3 1.65 1.65
EV: 0.6
Winner!

What if we make the martingale multiplier less than double? We can use a 1.5x martingale multiplier and a start lot of 1.2 to again come up with a 25% RoR strategy:
sum “L” “L”
-3 -1.2 -1.8
“W” “L”
0.12 1.32 -1.2
“L” “W”
0.78 -1.2 1.98
“W” “W”
2.64 1.32 1.32
EV: 0.54

Still no dice for Marty. The highest EV for a 25% RoR strategy is 0.6 achieved by the non martingale. OK well, we have a winner. Lets see if it holds true when we increase the run size to 3 trades and reduce the risk. Lets try some 12.5% RoR strategies:
Martingale 2.0 multiplier:
sum “L” “L” “L”
-3.01 -0.43 -0.86 -1.72
“W” “L” “L”
-0.82 0.47 -0.43 -0.86
“L” “W” “L”
0.09 -0.43 0.95 -0.43
“W” “W” “L”
0.52 0.47 0.47 -0.43
“L” “L” “W”
0.60 -0.43 -0.86 1.89
“W” “L” “W”
0.99 0.47 -0.43 0.95
“L” “W” “W”
0.99 -0.43 0.95 0.47
“W” “W” “W”
1.42 0.47 0.47 0.47
EV of Martingale (2.0x): 0.77

Martingale 1.5 multiplier:
sum “L” “L” “L”
-2.99 -0.63 -0.95 -1.42
“W” “L” “L”
-0.88 0.69 -0.63 -0.95
“L” “W” “L”
-0.22 -0.63 1.04 -0.63
“W” “W” “L”
0.76 0.69 0.69 -0.63
“L” “L” “W”
-0.02 -0.63 -0.95 1.56
“W” “L” “W”
1.10 0.69 -0.63 1.04
“L” “W” “W”
1.10 -0.63 1.04 0.69
“W” “W” “W”
2.08 0.69 0.69 0.69
EV of Martingale (1.5x): 0.774

Standard non Martingale:
sum “L” “L” “L”
-3.00 -1.00 -1.00 -1.00
“W” “L” “L”
-0.90 1.10 -1.00 -1.00
“L” “W” “L”
-0.90 -1.00 1.10 -1.00
“W” “W” “L”
1.20 1.10 1.10 -1.00
“L” “L” “W”
-0.90 -1.00 -1.00 1.10
“W” “L” “W”
1.20 1.10 -1.00 1.10
“L” “W” “W”
1.20 -1.00 1.10 1.10
“W” “W” “W”
3.30 1.10 1.10 1.10
EV of non martingale: 1.20

Non Martingale still best, what if our edge is better?
Lets try more 3 trade samples with an edge of 1.5 instead of 1.1 again all runs have 3 trades and 12.5% RoR
Martingale2.0
sum “L” “L” “L”
-3.01 -0.43 -0.86 -1.72
“W” “L” “L”
-0.65 0.65 -0.43 -0.86
“L” “W” “L”
0.43 -0.43 1.29 -0.43
“W” “W” “L”
0.86 0.65 0.65 -0.43
“L” “L” “W”
1.29 -0.43 -0.86 2.58
“W” “L” “W”
1.51 0.65 -0.43 1.29
“L” “W” “W”
1.51 -0.43 1.29 0.65
“W” “W” “W”
1.94 0.65 0.65 0.65
ev 3.87

Martingale1.5
sum “L” “L” “L”
-2.99 -0.63 -0.95 -1.42
“W” “L” “L”
-0.63 0.95 -0.63 -0.95
“L” “W” “L”
0.16 -0.63 1.42 -0.63
“W” “W” “L”
1.26 0.95 0.95 -0.63
“L” “L” “W”
0.55 -0.63 -0.95 2.13
“W” “L” “W”
1.73 0.95 -0.63 1.42
“L” “W” “W”
1.73 -0.63 1.42 0.95
“W” “W” “W”
2.84 0.95 0.95 0.95
ev 4.65

Standard
sum “L” “L” “L”
-3.00 -1.00 -1.00 -1.00
“W” “L” “L”
-0.50 1.50 -1.00 -1.00
“L” “W” “L”
-0.50 -1.00 1.50 -1.00
“W” “W” “L”
2.00 1.50 1.50 -1.00
“L” “L” “W”
-0.50 -1.00 -1.00 1.50
“W” “L” “W”
2.00 1.50 -1.00 1.50
“L” “W” “W”
2.00 -1.00 1.50 1.50
“W” “W” “W”
4.50 1.50 1.50 1.50
ev 6.00

It just doesn't matter, add more trades, reduce risk and lot size or increase the edge we are still better off staying away from martingale when it comes to maximizing profits for a given RoR. If we want to take on more risk to increase profits, it's always better to simply increase base lot size instead of going martingale.

But wait, the trend seems to be to reduce risk after loss, not increase it! lets try and risk a % of current equity on each trade instead...
Here we find that a 3 trade run with 1.5 edge risking 37% of equity on each trade gives a 12.5% RoR Run to compare to the last three runs:
sum “L” first trade bal “L” 2nd bal “L” fin bal
-3.00 -1.48 2.52 -0.93 1.59 -0.59 1.00
“W” “L” “L”
-1.53 2.22 6.22 -2.30 3.92 1.45 2.47
“L” “W” “L”
-1.53 -1.48 2.52 1.40 3.92 -1.45 2.47
“W” “W” “L”
2.09 2.22 6.22 3.45 9.67 -3.58 6.09
sum “L” first trade bal “L” 2nd bal “W” fin bal
-1.53 -1.48 2.52 -0.93 1.59 0.88 2.47
“W” “L” “W”
2.09 2.22 6.22 -2.30 3.92 2.17 6.09
“L” “W” “W”
2.09 -1.48 2.52 1.40 3.92 2.17 6.09
“W” “W” “W”
11.04 2.22 6.22 3.45 9.67 5.37 15.04
ev 9.73
Nearly triple the EV of the martingale 2.0!!!! I can assure you will find the risk percent of equity strategy the best ratio of profits to RoR no matter what. Always always always risk a % of your account and stay away from Martingale when it comes to going for the best profit to RoR ratio.

Sure if you add enough starting balance you can make the ev differences more subtle, but now the true danger of Martingale becomes apparent! It's possible to make an unprofitable system look profitable or hide its draw-down risk for many trades by using Martingale with a larger starting account vs risk. If you check out the martingale runs above you will notice the number of runs that were profitable were disproportionate to the number of losing runs. This is how the illusion is produced, the double down effect makes it much less likely to end on any given trade in a loss (even if the system is a loser). This is not a good thing, aside from the reduced ev proven above, it masks true performance and risks of the system (It takes a much much larger sample size to converge on performance if a tiny start lot/huge start account/martingale is used). Go ahead and checkout any martingale EA, turn Martingale off, make up some unprofitable settings, make a tiny lot size compared to you starting balance turn martingale back on and see you can (usually) make an unprofitable system look profitable for some time. In fact this is the concept behind that scam Roulette Bot going around.

If you want maximize profits vs RoR and you want to clearly asses the risks of the system then demand your EA provider/signal provider not use martingale. Demand from the market that no risk masking money management be allowed. Watch out for EA's with built in but hidden dangerous risk hiding MM. If you want to gamble it up, reduce your ev, increase your RoR, hide the true risks of your systems than by all means increase your lot size after each loss.