Martingale strategy for Dice

There are many betting strategies out there in the wild and probably one of the most famous ones is most definitely the Martingale. Most strategies originated in France which is of no surprise to me seeing as that so much originated from there throughout history. Basically the way how the strategy works is the player doubles every bet he makes only if the previous result was a loss. If the result is a win however, the player will not only win back double his bet, but also the total of all the previous losses.

Martingale

What happens if the dice result is only ever losses and never a win while using the Martingale strategy? Well in that case there’s logically no way to win anything back and I would advice you to check that the results were in fact correct. What goes up, must come down, and therefore it’s relatively safe to assume that your losses will eventually change back into wins. The risk is still high though because the player will always be limited to things such as their bankroll, as well as the casino limits. Another thing to take into consideration is exponentiation and numbers can get really high, really quickly. Let’s take a look at the example of placing a $1 bet:

$1 bet = loss (next bet is double)
$2 bet = loss (next bet is double)
$4 bet = loss (next bet is double)
$8 bet = loss (next bet is double)
$16 bet = loss (next bet is double)
$32 bet = loss (next bet is double)
$64 bet = loss (next bet is double)
$128 bet = loss (next bet is double)
$256 bet = loss (next bet is double)
$512 bet = loss (next bet is double)
$1024 bet = win

So we can see that the losses were (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512) which resulted in a total loss of 1023. The last bet however was a win and we not only recovered all our losses, but we also made a profit of our first bet of $1.

What would this look like within a cryptocurrency casino? Let’s take a look at the example of placing a 0.00000100 bitcoin bet:

0.00000100 bet = loss (next bet is double)
0.00000200 bet = loss (next bet is double)
0.00000400 bet = loss (next bet is double)
0.00000800 bet = loss (next bet is double)
0.00001600 bet = loss (next bet is double)
0.00003200 bet = loss (next bet is double)
0.00006400 bet = loss (next bet is double)
0.00012800 bet = loss (next bet is double)
0.00025600 bet = loss (next bet is double)
0.00051200 bet = loss (next bet is double)
0.00102400 bet = win

So we can see that the martingale losses were (0.00000100 + 0.00000200 + 0.00000400 + 0.00000800 + 0.00001600 + 0.00003200 + 0.00006400 + 0.00012800 + 0.00025600 + 0.00051200) which resulted in a total loss of 1023. The last bet however was a win and we not only recovered all our losses, but we also made a profit of our first bet of 0.00000100 bitcoin. Generally wen the losses are more than 10 – 15, the needed bankroll to continue playing becomes really scary.

This becomes less of an issue while playing with cryptocurrencies due to them being in the decimal range as well as being able to place many bets in a short duration of time. If looking at the most consecutive losses in a row for a typical dice game with 1% house edge, in 1 billion simulated rolls, we can see the range is up to 25 losses in a row. Knowing this, let’s simulate our own rolls to survive (bankroll survival) exactly that amount of losses:

0.00000001 bet
0.00000002 bet
0.00000004 bet
0.00000008 bet
0.00000016 bet
0.00000032 bet
0.00000064 bet
0.00000128 bet
0.00000256 bet
0.00000512 bet
0.00001024 bet
0.00002048 bet
0.00004096 bet
0.00008192 bet
0.00016384 bet
0.00032768 bet
0.00065536 bet
0.00131072 bet
0.00262144 bet
0.00524288 bet
0.01048576 bet
0.02097152 bet
0.04194304 bet
0.08388608 bet
0.16777216 bet
0.33554432 bet

The total bankroll therefor needed to survive 1 billion bets would be (the final winning amount * 2) – base bet which was 0.00000001 resulting in

final bet – base bet
(0.33554432 * 2) – 0.00000001
(0.67108864) – 0.00000001
0.67108864 – 0.00000001
0.67108863 bitcoin

Now lets take a look at the possible profit to make with this bankroll. If we assume that the player is playing a traditional cryptocurrency dice game with the results ranging from 0 – 99.99 as well as a house edge of 1%, we could arrive at the conclusion of winning 49.50% of our bets out of 100%. It’s not a clear cut 50% due to the house edge which does tip the balance onto the casino’s side. We get here as follows:

100 / 2
50 (percent chance of getting tails or heads, or in our case, high or low on the dice)
50 – 1%
50 – 0.5
49.50%

Calculating our chance of winning on the amount of rolls we did in the previous examples of using 1 billion:

bets – win chance
1000000000 – 49.50%
1000000000 – 495000000
505000000 losses
495000000 wins

Recall that every time we win, we not only recover our previous losses, but we also make a profit of the base bet. This means that if our average 495000000 wins will always result in the profit of the basebet, we can then do a simple multiplication to find out what our projected total profits will be for these bets as follows:

495000000 wins * 0.00000001 base bet
495000000 * 0.00000001
4.95 bitcoin estimated profits

In other words, with a bankroll of less than 1 bitcoin you could make x5 that amount quite safely. On the other hand you could also lose your bankroll if you were unlucky and got a streak of reds higher than 26 seeing as though you only planned for 26 losses in a row.

Reverse martingale

As the name implies, this is the opposite of martingale and the actions taken are simply reversed.

if loss then double the wager
if win then half the wager

What this means is that when a player loses, they basically cut back on costs by halfing the bet amount and upon winning they increase the bet which is really great if the next result turns out to be a win. However as with the traditional martingale there needs to be some form of limit that you’d have to apply in order to to max out. This isn’t necessarily a good strategy however it’s still worth mentioning it. Lets look at some implementations of possible reverse martingales:

0.00000001 bet – win
0.00000002 bet – win
0.00000004 bet – win
0.00000008 bet – win
0.00000016 bet – loss
0.00000008 bet – loss
0.00000004 bet – loss
0.00000002 bet – win
0.00000004 bet – loss
0.00000002 bet – win

This would result in the wins being(1 + 2 + 4 + 8 + 2 + 2), and the losses being(16 + 8 + 4 + 4) which would look as follows:

(total wins) – (total losses)
(0.00000001 + 0.00000002 + 0.00000004 + 0.00000008 + 0.00000002 + 0.00000002) – (0.00000016 + 0.00000008 + 0.00000004 + 0.00000004)
0.00000019 – 0.00000032
-0.00000013 profit loss

However if we look at one of our earlier examples with loss streak and also reverse the loss to a win and calculate it, the picture changes dramatically:

0.00000001 bet
0.00000002 bet
0.00000004 bet
0.00000008 bet
0.00000016 bet
0.00000032 bet
0.00000064 bet
0.00000128 bet
0.00000256 bet
0.00000512 bet
0.00001024 bet
0.00002048 bet
0.00004096 bet
0.00008192 bet
0.00016384 bet
0.00032768 bet
0.00065536 bet
0.00131072 bet
0.00262144 bet
0.00524288 bet
0.01048576 bet
0.02097152 bet
0.04194304 bet
0.08388608 bet
0.16777216 bet
0.33554432 bet
Total profit of 0.67108863 bitcoin

So we can clearly see there is potential for this martingale strategy as well depending on the angle the player decides to take. A somewhat odd approach also seen with this is by taking the total loss amount and then choosing, if the bet amount where to be won, and this sequence was reversed, how many wins you’d want to reset on. Here’s an example:

let’s assume the total loss is 0.00000128
let’s also assume the base bet was 0.00000001
we chose a reset number of 2
this means we’ll need to recover the loss within 2 (won) bets

Formula

Formula of the next bet in this could would be as follows:

((total loss + base bet) / 3) / 2
((0.00000128 + 0.00000001) / 3) / 2
(0.00000129 / 3) / 2
0.00000043/ 2
0.00000022 = next bet

Now that we have the value to bet next we bet it, and double the bet on the win which would complete the reverse martingale as well as achieving it within 2 steps:

0.00000022 – win (0.00000044 profit recovered)
double this amount
0.00000044 – (0.00000088 profit recovered)
0.00000088 + 0.00000044
0.00000132 total profit
total profit – total losses
0.00000132 – 0.00000128
0.00000004 = actual total profit

By constantly updating the value that we’d use in order to place the next bet, we reduce the cost needing to break even or turn a profit. This is better seen when reaching the limits of your bankroll as the next bets required would be so much smaller:

let’s assume the total loss is 0.67108863
let’s also assume the base bet was 0.00000001 – phew what bad loss streak
we chose a reset number of 5
this means we’ll need to recover the loss within 5 (won) bets + some profit as well
((total loss + base bet) / target + 1) / target
((0.67108863 + 0.00000001) / 5 + 1) / 5
(0.67108864 / 6) / 5
0.11184811 / 5
0.02236962
if encountering a number with many digits after the radix, simply round up to top 8 places after dot

0.02236962 bitcoin is therefore our next bet and if we win 5 times in a row as we specified in our target, we not only recover all previous losses, but also make a nice profit on top of that and without the need of a bet over 1bitcoin!