A Curious Conclusion From The Martingale Betting Strategy

Here’s the deal. You want to double your money when playing a coin flipping game with your friend. The coin is fair, it has a 50% chance of being heads or tails and if you call the coin toss correctly you double your stake.

For example, if you bet £1 on heads and the result of the coin flip is heads, you win £2.

If you have £N and want to make £2N from this game, what strategy would you employ? Perhaps you would consider the Martingale strategy. This strategy requires the gambler to double their bet after every loss, so that the first win would recover all previous losses and win a profit equal to the original stake.

For example, if you lose the first bet of £1, then the next bet is £2. If this bet wins, you receive £4, having bet £1 on the first go and £2 on the second, leaving you with a net return of £1 (£4 -£2 -£1). This process of betting double the previous amount forms a geometric sequence:

This allows one to calculate total loss if a streak of consecutive losses reduces the player’s bank balance to zero. More compactly, to determine how much a gambler has staked and lost, one needs to compute the following sum

Conveniently, the sum of a geometric sequence is given by,

Here, a is the first term in the sequence, r is the common ratio and k is the number of terms.

Lets explore a simple example and then generalise. Imagine your first bet is £1, you then lose 5 times in a row and want to know how much you have lost by employing the Martingale strategy.

Manually calculating the sum we find the total loss is £31. Using the formula above we need to identify the variables a, r and k. a is the the first term in the sequence, which is 1. The common ratio, r, is equal to the number we multiply by to move from one term to the next, which is always double, so r is equal to 2. Lastly, k represents the number of terms in the sequence which is 5.

This is incredibly useful. Imagine manually calculating the loss if k=500!

Now lets generalise to find the total amount lost after k consecutive coin flip losses. Regardless of the size of our first bet, the common ratio will always be 2 because the next bet is always double the previous bet. To keep things simple we will assume our first bet in the coin game is £1, which yields the following general formula to calculate total loss,

To recover this loss, your next bet would need to be 2^k. So in the example above, your sixth bet would be 2⁵= £32.

Therefore, you become bankrupt once your next bet exceeds balance £N. You are no longer able to bet such that winning that bet would return everything you have previously lost and then provide you with a profit equal to your initial stake. Mathematically, you are bankrupt when,

This tells us that the number of consecutive losses, k, that can be incurred for a given bank balance, N, is given by,

It is fascinating to see how k varies with increasing N.

Moving from an initial balance of 1000 to 1,000,000, that is, multiplying your starting balance by a factor of 1000, only doubles the number of consecutive losses you can incur.

Our perception of randomness is poor and many studies show humans typically underestimate the probability of seeing, for example, 7 tails occurring in a row randomly. In one study, students were asked to write down the results of 200 coin flips. The professor in charge of the study was able to determine which results were genuine and those that were fabricated with very high accuracy.

This was because the students that faked the results rarely recorded large runs of either heads or tails. This is because we generally perceive the probability of, say 15 heads in a row, as unlikely, however, it is far more likely than our intuition would lead us to believe.

Therefore, it is important to quantify risk and establish probabilities. Before looking at doubling our money, what is the probability of winning £1? First, consider losing the game. This happens when you lose k times in a row and and run out of money. As the coin flips are independent, the probability of this is given by

Where we substituted 2^k = N from the relationship established above (to make things simpler the inequality in 2^k>N was replaced with an equality).

From this, the probability of winning after k coin flips is given by,

One trivial observation here is that with an infinite bank balance there is a certainty of winning. However, with an infinite balance, why would you play this game!

With these probabilities in hand, it is now possible to calculate the probability of doubling your money. Each time you win a coin flip you make £1, regardless of whether it was the first flip or the nth flip. Therefore, to double your money you must win £1, N times. Given the flips are independent events, the probability of winning £2N is thus,

This is simply the probability of winning once, multiplied by itself N times. The most fascinating result of employing the Martingale strategy in this game is that this probability has a limit! As your balance tends to infinity, the probability of doubling your money is equal to 1/e. How amazing that the constant e appears in this probability!

From this, we can see that as you increase your bank balance from 0 to infinity, the probability of doubling your money approaches 37%.

This brings us to our conclusion. When playing the coin flipping game, if your objective is to double your money, it is better to stake everything on the first coin flip with a 50% chance of success as opposed to employing the Martingale strategy!