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Sep 14, 2020

What can the Martingale strategy teach us about betting?

Applying the Martingale strategy to even money bets

Risk and reward using the Martingale strategy

Modelling Martingale strategy outcomes

Understanding the effects of a house edge

What can the Martingale strategy teach us about betting?

The Martingale strategy is one of the most well-known methods in sports betting. While the pitfalls are clear for all to see, there could be some potential uses thanks to the math involved. Read on to find out what bettors can learn from the Martingale strategy.

First things first, this article comes with a warning. Anytime you find anyone talking about Martingale in a positive light within the context of betting, the correct response is to tell them that they are wrong and walk away.

Martingale is a well-known progressive betting plan which aggressively increases amounts after losses in an attempt to recover them. This kind of loss recovery is to never be seriously recommended. I have written many times before about why this is so, including demonstrating the mathematical proof as to why it is flawed on Betting Resources.

As for any money management plan where your assumed advantage over the odds (positive or negative) is constant across all bets, it is impossible to change that advantage by messing about with stake sizes. All you can do is change the distribution of risks and rewards. In the case of Martingale, you are attempting to buy a lot of reward but at the cost of a small probability of a very, very big risk: total ruin.

With this is mind, I wanted to have a little bit of fun with Martingale to show how one might use it in a managed way without risking the sort of ruin that it can lead to. In doing so, I hope this article demonstrates nicely how betting in general is always a balance of risk and reward. The more reward you want, the more risk you have to take to achieve it.

How to get the most out of a holiday in Vegas

It’s always been an aspiration of mine to go to a casino in Las Vegas. Casinos, as we should all know, offer only negative expectation games. While card counting might give you an edge over the Blackjack dealer, it will also likely have you quickly ejected from the establishment.

For example, all Roulette proposition bets are losing propositions, based as they are on mathematical principles where the house edge is given by the zero. Ignoring the stories of biased wheels where teams of gamblers observed tens of thousands of spins, we can assume that you will not be holding a profitable edge.

Given this reality, I have wondered what the best way would be to maximise the longevity and fun of play whilst on holiday there. To consider this, I will concern myself only with even money propositions at the Roulette wheel, including odd or even and red or black.

How long will you last with flat staking?

The simplest form of staking is flat staking, where the same stake is used on every bet. Let’s stick a dollar on every wheel spin. What can we expect to happen if we bet for 1,000 wheel spins?

Our expected returns will conform to a binomial distribution shown below. The blue distribution shows the range of possible outcomes assuming fair odds, while the red distribution also considers the 2.7% house edge for a single zero Roulette wheel:

In-Article-1-How-to-solve-a-problem-like-Martingale.jpg

Even with a fair wheel, our chances of seeing any appreciable accumulation of our bankroll are minimal. We have just a 5.7% probability of making $50. Factor in the house edge and our prospects are even worse (0.74%). Of course, this is balanced by a similarly small risk of losing $50. Even with the house edge against us, there’s only a 23% chance of losing $50 and just 1% of losing $100.

Flat staking might be safe, but it’s not going to facilitate any significant rewards. Surely there must be a more entertaining way to spend a holiday in Vegas?

Using Martingale and inevitable losses

The simplest version of the Martingale strategy doubles stakes after each loss on an even-money proposition bet until a win. The stake is then reset to the original amount and the progression begins again. The dangers of Martingale should be obvious to anyone but the most overconfident of gamblers, losing runs are an inevitable consequence of repeated play. The more you play, the more likely it is that you will have a longer losing sequence.

As a rough estimate, the expected longest losing sequence you are likely to see in a sequence of n bets is given by the logarithm of n to the base of the odds divided by odds minus one. For even-money propositions, you are likely you see a losing sequence of three in eight bets, four in 16 bets, five in 32 bets and so on. In a series of 1,000 bets, your expected longest losing sequence is around nine to ten bets. Pinnacle have already published a detailed article on the subject of losing runs.

Modelling Martingale outcomes

Given infinite funds and an infinitely generous casino who will allow you to bet any amount, your expected profit from 1,000 wheel spins will be $500 for a fair wheel, or $486 with the single zero house edge. Of course, neither of these is a possibility. Most significantly, there will come a point when a losing run will either wipe out your funds or put a significant enough dent into them that you will lose your nerve.

To manage these limitations, the sensible approach is to identify goals and set rules and limits to your play while modelling the probabilities of different outcomes, just as we did for flat stakes. Let’s consider the following scenario:

  1. Using $1 for the initial stake in any Martingale progression, we will aim to win $500 after 1,000 wheel spins.
  2. We will limit the risk of going bust at any point during the 1,000 plays to 50%.
  3. What is the maximum bankroll loss we can accept before opting to cease playing given points 1 and 2?

To answer this question, we can turn to the Monte Carlo simulation. The table below shows the results from a series of 10,000 Monte Carlo simulations. For each simulated series of 1,000 wheel spins, if the bankroll falls below a defined threshold, play stops and the strategy is considered to have failed. Otherwise, play continues for 1,000 wheel spins and the strategy is considered to have succeeded. Here, the roulette wheel was assumed to be fair with no zero:

Bankroll threshold

Average bankroll win

Average bankroll loss

Bankruptcies

Proposition odds

-50

503

-106

82%

5.69

-100

501

-188

73%

3.68

-150

500

-270

65%

2.82

-200

500

-341

59%

2.47

-250

500

-443

54%

2.15

-300

499

-496

50%

1.99

-400

498

-608

45%

1.81

-500

498

-774

40%

1.65

-750

498

-1088

31%

1.45

-1000

496

-1725

22%

1.29

A reformulated betting proposition

Unsurprisingly, we can see that the more we are prepared to lose at any one point in the series, the more likely it is that we will ultimately prove to be successful. A threshold of about minus $300 gives us about a 50% chance that we will make our $500 after 1,000 wheel spins. The other half of the time we can expect to lose an average of close to $500. Effectively, we have reformulated the betting proposition: risk $500 to win $500, giving us proposition odds of about 2.00 (or 1:1 in fractional notation).

Notice that these proposition odds reflect closely the average win to average loss ratio; effectively this ratio is a measure of your real odds. For a fair wheel with no zero, these two sets of odds should be the same.

If the threshold is instead at minus $1,000, we can accommodate longer losing runs. Hence, we experience failure less often and our proposition odds are shorter (1.29). Of course, if you didn’t want to risk losing a total of $1,725, you could simply reduce the initial progression stake size. Using $0.29 would then give you a proposition of risking $500 to win $145 or thereabouts.

Compared to the more conservative flat staking strategy, there is now much greater risk of a much greater loss. However, this buys a much greater chance of a much greater win. You may never lose $500 in 1,000 roulette plays betting $1 a play, but you’re never going to win $500 either. The probabilities of either are impossibly small.

Understanding the effects of a house edge

Introducing a 2.7% house edge by means of the zero does change things. The chart below compares the failure (or bankruptcy) rates between wheels with and without a single zero. With a zero introduced at a threshold of minus $300, you can expect to fail about 58% of the time.

In-Article-2-How-to-solve-a-problem-like-Martingale.jpg

To ensure you still have a 50% proposition you would need to increase the threshold of your maximum acceptable loss to about minus $440. Such a scenario sees you risk losing an average of about $670 to win $486. Remember, since your bet win expectation is now 48.6%, your expected profit for a successful series will be around $486 and not $500.

This ratio now implies odds of 1.73, considerably shorter than the proposition odds of 2.00 calculated by the failure probability. Essentially, this illustrates a loss of value and notice that it is considerably greater than the house margin. Your implied value is 1.73 divided by 2.00, or 0.865. This may seem a high price to pay for using Martingale.

The expected value for an even-money roulette bet is 36 in 37 or 0.973. However, consider this repeated over 1,000 bets. With flat stakes, you have about a 20% chance of making some kind of profit, compared to 80% for some kind of loss. You can observe this by comparing the areas under the orange curve in the binomial chart above to the left and right of the zero-profit line.

Your implied odds of success are then 5.00 and your implied value is 2.00 divided by 5.00, or 0.40 relative to a fair wheel without a zero.

In fact, your loss of value using this managed Martingale strategy varies depending on your minimum bankroll loss threshold and your proposition odds. The shorter the proposition odds (and the lower the failure rate), the less value you lose. For a threshold of minus $100, the proposition odds implied by the failure rate are about 5.00 with the single zero wheel compared to about 3.68 without, implying a value of 0.74.

By contrast, with a threshold of minus $1,000, the odds are about 1.4 and 1.29 respectivelt, implying a value of 0.92. If you used a threshold of $10,000, the value would be almost 0.99. This relationship seems oddly reminiscent of the favourite–longshot bias.

The distribution of risks and rewards

We can visualise how using a managed Martingale strategy like this changes the betting proposition by plotting the distribution of possible outcomes. The chart below shows the distribution of the 10,000 Monte Carlo simulation runs for the minus $300 bankroll threshold scenario using a fair roulette wheel. You can see how it is divided into distinctive zones of either success or failure. Compare this to the original distribution from flat staking (shown in the dotted orange curve).

In-Article-3-How-to-solve-a-problem-like-Martingale.jpg

What happens when you change the odds?

Whilst this discussion of a managed Martingale strategy deals with simple even-money propositions, you could apply it to any odds and any betting market, including sports. All that’s needed is a change to the size of the Martingale progression value. This is given by odds divided by odds minus one. Thus, for odds of 3.00, stakes after losses are increased by a factor of 1.5. For odds of 1.50, the factor is three.

Unsurprisingly the longer the odds you bet on, the larger the effective stake you must risk to ensure you have reformulated you bet proposition as an even-money bet. Obviously, betting on longer odds means longer losing sequences.

For example, betting at fair odds of 5.00 means you have to risk about $800 to win about $800. By contrast, betting at odds of 1.50 results in an effective proposition of risking $333 to win $333. Of course, you can always adjust the initial progression stake size to take this into account, as described earlier.

Don’t try this at home (or with a bookmaker)

This article has really been a bit of fun and the last thing I want to do is advocate the use of Martingale, or any progressive betting system for that matter. However, what it has done is provide another means of illustrating how betting is a game of risk and reward, and how money management can be used to reformulate the balance and distribution of your risks and rewards.

With regards to your holiday, you could of course choose to simply bet $500 on the first play, have only one play and walk away whether you win or lose. However, you would then have to find something else to do with your time in Vegas.

Finally, in sports betting, unlike in casinos, there exists the opportunity for positive expectation. Should you be one of the few bettors to genuinely hold it, you won’t need to bother with Martingale or any progressive betting system at all. Just let the law of large numbers work slowly for you.

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