## The decision to double down or go home…

Photo by Chris Liverani on Unsplash

Have you ever tried a strategy where you double your bets every time you lose, hoping that the next win allows you to recoup all accumulated losses? Turns out, you were using the martingale strategy, a popular gambling strategy that started in 18th century France.

This article attempts to draw some conclusions regarding the adoption of this strategy through the use of 2 simulations:

- A martingale strategy without any restrictions
- A martingale strategy with a maximum lower limit on wealth (also a martingale with stopping time)

In these simulations, we are going to play European roulette. We bet on the colour the ball ends on, giving us a 48.6% chance of winning with a 1:1 payout. The term wealth will be used a lot and it refers to the amount of money we have at any point in time. Here is the strategy:

- We start with a bet of $1
- The next bet will continue to be $1 if we won the previous round
- We will double the bet of the next round every time we lose a round

To exemplify:

It seems like an infallible strategy, right? We cannot be losing indefinitely and will eventually have to win (and recoup all accumulated losses). To see where this strategy gets us, we will take a look at the below results.

This simulation involves running 10,000 trials (think 10,000 possibilities or 10,000 alternate universes), each playing 1000 games of roulette.

Though it is not clear in the image, there is a general upwards trend in the value of wealth as more games are being played. However, it is apparent that for increases that are considered minimal ($1 profit), we have to be ready to take huge losses temporarily; in this case, there were 3 trials where losses exceeded the $2 million mark.

Evolution of wealth across 1000 roulette games for 10,000 trials

Here, while most games end around the range of $490, it can be seen that there was an unlucky trial that ended above a $30,000 loss and several more that ended with a loss of more than $1,000.

Ending wealth of 10,000 trials

By taking the average value of wealth across all 10,000 trials, it is observed that there is an upward drift in the average wealth. Based on the simulation, this would lead us to imply that the strategy is able to generate more gains as more rounds of roulette are played.

Theoretically, the average value of wealth line should be horizontal at $0. The reason why the line has an upwards drift is because the sample in the simulation does not include all the possible outcomes.

Average wealth across all 10,000 trials at each point in time

While the simulations show some promise, there are reasons why the martingale strategy does not work out in practice:

- Limited amount of wealth — The amount of losses we can take is assumed to be limited to the amount of money we have. This means that it only takes a finite number of continuous losses before we have to exit the game at a loss.
- Maximum size of bets — Casinos may have upper limits on the size of bets; this means that there is a possibility we will need to win 2. 3 or more rounds in a row to clear our accumulated losses.

To make the simulations a little more realistic, we go a step further to consider the fact that our money is limited.

The lines are more defined in the below chart because the y-axis is much smaller. Putting it in perspective, in order to achieve the average wealth of $490 after 1,000 roulette games, there are a sizable number of trials where the net losses at certain points in time far exceed it (observe the number of lines reaching close to -$20,000 wealth).

Evolution of wealth across 1000 roulette games for 10,000 trials

Unlike in Part 1, there are now many more trials with an ending wealth around -$120,000. This represents the number of trials where we would have to walk away from the roulette table because we have no money left.

Ending wealth of 10,000 trials

The average wealth is no longer showing a positive trend. This means that a gain in wealth is much less likely when restrictions are introduced.

Average wealth across all 10,000 trials at each point in time

The theory of the martingale strategy appears to sound as it is impossible for a person to lose indefinitely and winning just once after a losing streak brings the wealth back to a positive value.

However, by considering the amount of money we currently own, eventual gains are far less likely. The expected earnings increases as more rounds of roulette are played (assuming that at no point we have to exit the table), however, this also increases the chances of encountering a losing streak that will cause us to exit.

To exemplify, there is a higher chance of encountering a losing streak of 20 rounds in 1,000,000,000 rounds of roulette than in 100 rounds.

From a risk-reward perspective, the martingale strategy does not seem a logical choice either. At many points in time, we took on significant losses temporarily in order to earn a profit of $1.

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