In the roulette game the martingale strategy is to bet on even money bets and if you loose you double the bet for next round until winning. The “logics” behind the strategy is that the player should eventually win - no matter how long the loosing streak is. And when he does he has an advantage over the casino of 1 original bet.

*Note: This article is more on mathematical and technical aspects of martingale strategy. For general overview and casino management aspect you should start with this article.*

Let's look at some numbers to analyze the martingale strategy. First we must define some formal rules for a martingale strategy player so we can analyze the behavior:

- the player chooses one even money bet and always bets on it (red, black, even, odd, low, high)
- after every win and at the beginning of the session he starts playing with minimum bet
- if the player looses he doubles his bet
- if he looses the game and can’t double the bet he restarts his session (with minimum bet)

On top of these rules we must define also the ratio of minimum vs. maximum bet. In order to make a simple and relatively short example we could say a 1€ minimum and 100€ maximum for a ratio of 1:100.

### Analysis of single series of games

With all of these rules in place we can look at and analyze a series of games - a series from initial bet (minimum bet) to when the player again resets his bet - either winning the last game or loosing the last game without an option to double the bet.

All the possible outcomes of this series for an example of single zero roulette are written in the table below :

Lose spin | Win spin | Prob. formula | Prob. | Tot. bet | Tot. won | Play. result |
---|---|---|---|---|---|---|

1 | $18 \over 37$ | 48.65% | 1 | 2 | 1 | |

1 | 2 | $ ({19 \over 37})^1{18 \over 37} $ | 24.98% | 3 | 4 | 1 |

1-2 | 3 | $ ({19 \over 37})^2{18 \over 37} $ | 12.83% | 7 | 8 | 1 |

1-3 | 4 | $ ({19 \over 37})^3{18 \over 37} $ | 6.59% | 15 | 16 | 1 |

1-4 | 5 | $ ({19 \over 37})^4{18 \over 37} $ | 3.38% | 31 | 32 | 1 |

1-5 | 6 | $ ({19 \over 37})^5{18 \over 37} $ | 1.74% | 63 | 64 | 1 |

1-6 | 7 | $ ({19 \over 37})^6{18 \over 37} $ | 0.89% | 127 | 128 | 1 |

1-7 | $ ({19 \over 37})^7 $ | 0.94% | 127 | 0 | -127 |

The probability of player winning the first spin (with bet 1) is $18 \over 37 $ = 48.65%. Should the player loose the first spin (probability $19 \over \ 37$ = 51.35%) the player doubles the bet and tries with second spin.

The probability of the series ending with his win in second spin is probability of his loosing the first spin times probability of winning the second spin, therefore: ${19 \over 37} {18 \over 37}$ = 24.98%. Together he bet 1 in the first spin and 2 in the second spin for a total bet of 3 and he won even money on the bet of 2, therefore 4. The result of the series is therefore 1 (won - bet = 4 - 3).

The table goes on for possibilities that he lost also the second spin but won the third spin and so on to the last case where he bet 64 in the seventh spin and lost. Since the player can't double the bet anymore (double would be 128 which is more than table maximum 100) the series ends and the player looses total bet of 127 (1 in first spin + 2 in second spin + ... + 64 in seventh spin). Note that there is an approximate probability of 1 percent of player loosing 7 spins in a row and the player looses 127 €.

From this table we can calculate a couple of additional statistics for the whole series:

- the average expected result of the player is -0.2051 (sum of the player results multiplied with probability for all the possible outcomes)
- the average bet for the whole series is 7.59 € (sum of all the total bets multiplied with probability of outcome)
- the normalised expected result (relative to the bet of 1) of the player for the series is -2.70% (average expected result divided by average bet) which is exactly the house edge of a single zero roulette
- average number of spins per series is 2.04 (sum of number of spins times the probability of outcome)

### Volatility

With this we can already see that martingale strategy did nothing for the average expected result for the player - the RTP of the roulette is the same as with normal betting. What did change is the volatility so let's calculate the volatility for the series:

Prob. | Total bet | Player result | expected average result | play. result dist. to expected result squared |
---|---|---|---|---|

48.65% | 1 | 1 | -0.0270 | 1.0548 |

24.98% | 3 | 1 | -0.0811 | 1.1687 |

12.83% | 7 | 1 | -0.1892 | 1.4142 |

6.59% | 15 | 1 | -0.4054 | 1.9752 |

3.38% | 31 | 1 | -0.8378 | 3.3776 |

1.74% | 63 | 1 | -1.7027 | 7.3046 |

0.89% | 127 | 1 | -3.4324 | 19.6465 |

0.94% | 127 | -127 | -3.4324 | 15268.9438 |

The expected average result is the total bet times the expected result for the bet of 1 (house edge actually). For the calculation of standard deviation we need to calculate the distance between the expected average result and the actual result of the spin.

From this table we can calculate:

- variance is 145.304 (sum of distances multiplied with probability)
- standard deviation is 12.05 (square root of variance) for the series
- calculated standard deviation for a single spin is 17.20 (SD times square root of average number of spins per series)

### Comparison to normal play

So if we are using martingale strategy we are expecting for each spin of roulette (average bet 7.59 €) the net result -0.2051€ with SD of 17.20. Next table shows comparison to normal roulette play (play with a single bet amount all the time) of non-normalised standard deviation. This is the variability of the income for the casino.

Strategy and bet | Average Bet | Expected result | Standard deviation |
---|---|---|---|

normal - even money bet | 1 | -0.0270 | 1.00 |

normal - straight number bet | 1 | -0.0270 | 5.84 |

normal - even money bet | 7.59 | -0.2051 | 7.59 |

martingale - even money bet | 7.59 | -0.2051 | 17.20 |

normal - straight number bet | 7.59 | -0.2051 | 44.31 |

normal - even money bet | 64 | -1.7297 | 63.98 |

The martingale strategy increases the volatility quite a bit. This is possible because there are multiple bet amounts used with martingale strategy (the volatility changes if the bet amount is varied) and the bet amount is not independent of the game result.

**However, the volatility (as related to casino income variability) is quite higher for a normal player betting the highest bet of 64 € each spin than for the martingale strategy player ranging with bet form 1 to 64 €! And similarly, the volatility is quite higher for a player betting on a straight number than a player betting on martingale strategy.**

UPDATE: A straight number bet was added to the examples of volatility.

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