24th July, 2013
0. Introduction
After watching more tens of thousands of US$ being thrown about on just-dice.com like foam boulders at a Star Trek away-team, I realised that those martingaling gamblers are only using the standard "Bet at 50% and double up if you lose" method. This means that there are many variations of the martingale betting method that are not being used, and that gamblers might find better for them - for example they might find an 8.8% game or an 88.8% game more auspicious. So this post explains how to martingale at other probabilities than the standard 50%. Although the house edge will be ignored in all that follows, it should be kept in mind that a house edge will always guarantee a negative expected profit.
1. The martingale betting method
Wikipedia has a nice history and explanation about the martingale betting method. The traditional method is to gamble an amount on a game with a probability of 50% to win, and to double the amount bet if the result is a loss. Once a win occurs, the next bet returns to the minimum, for example
Bet: 1 2 3 4 5 6 ...
Amount bet: 1 2 4 8 16 1 ...
Result: lose, lose, lose, lose, win, win
Cumulative profit: -1 -3 -7 -15 1 1 ...
For any "lose, lose ..., win" sequence, the profit will always be 1, assuming an infinite gambler bankroll, house bankroll, and an unlimited maximum bet.
2. A generalised martingale method
The main features of the martingale betting method are:
- A particular probability game is played until a win occurs;
- An increasing amount is bet each time, so that the profit on winning will be a constant.
Definition: The generalised martingale betting sequence
(The term "generalised martingale sequence" is my own; however someone else probably discovered it - if you know who and when let me know so I can use the correct name)
For a game with probability p and starting amount a, the generalised martingale betting sequence uses an amount per bet and generates a constant cumulative profit per completed martingale sequence as follows:
Derivation:
The basic feature of the generalised martingale betting sequence is that on the first win the sum of the amounts bet will be a constant amount less than the amount won.
So, if C denotes the constant profit per sequence,
Using the formula for a geometric series ( familiar from last post ? ), and substituting q = 1 - p:
The starting amount a you choose depends on your bankroll and the longest sequence of losses you think you might encounter. I'll address in a later post ways to define "longest" and to determine the expected maximum losses in a row / longest expected martingale sequence.
3. Examples
i. Game with 25% probability to win, 1 btc starting bet
a = 1
p = 0.25
Sequence:
bet cost win profit on win
1 1.00 1.00 4.00 3
2 1.33 2.33 5.33 3
3 1.78 4.11 7.11 3
4 2.37 6.48 9.48 3
5 3.16 9.64 12.64 3
6 4.21 13.86 16.86 3
ii. Game with 88.8% probability to win, 0.88 btc starting bet
a = 0.88
p = 0.888
Sequence:
1 0.88 0.88 0.99 0.11
2 7.86 8.74 8.85 0.11
3 70.15 78.89 79.00 0.11
4 626.37 705.26 705.37 0.11
5 5592.56 6297.82 6297.93 0.11
6 49933.56 56231.38 56231.49 0.11
iii. Game with 8.88% probability to win, 8.88 btc starting bet
a = 8.88
p = 0.0888
Sequence:
bet cost win profit on win
1 8.88 8.88 100.00 91.12
2 9.75 18.63 109.75 91.12
3 10.70 29.32 120.44 91.12
4 11.74 41.06 132.18 91.12
5 12.88 53.94 145.06 91.12
6 14.14 68.08 159.20 91.12
4. Profit per roll
Assuming infinite gambler bankroll, no betting limits and no house edge, the cost / profit ratio is a reflection of the expected number of losses per sequence - from last post you'll remember that the expected number of losses per sequence is 1/p-1, so when p = 88.8%, the expected length of the sequence (number of losses) is 0.1261261; when p = 8.88% the expected length of the sequence is 10.26126.
More generally, the profit per expected number of rolls is a, since the profit per sequence is a(1 - p) / p and the expected number of rolls per sequence is 1/p -1:
Not only does the starting amount influence to a large extent how far your bank roll will last, but it is the only factor that determines the expected amount per roll.
5. Summary
If you wish to play non-standard martingale sequences the generalised martingale sequence allows you to plan for any probability game, and (hopefully) stay within your fiscal limits.
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