People have come up with many different betting systems over the years. One of these, the Martingale betting system, is interesting because it illustrates how a high level understanding of a concept can be insufficient.

The Martingale betting system as follows:

• I will start by betting some small amount on a wager that is as close to 50/50 as I can get. So assume I’m betting $1 on a coin flip.
• Anytime I lose, I’ll double the bet
• Anytime I win, I will start over at $1
• Since I’m always increasing the stakes, anytime that I win it means that I will have cleared all my loses, and won an additional dollar. For instance, If I lose $1, $2, $4, then win $8 I will have lost 7 dollars, but then won 8 dollars for a net gain of 1 dollar

What the bettor appears to have is a system that guarantees them small winnings. They will steadily be winning the 1 dollar net bets.

Clearly this is not the case, from Gambler’s Ruin we know that you cannot win any negative expected value game long term. So the Martingale system is not a betting system that you should use. But why is it bad?

There is a surface issue, and an underlying math issue.

The First Problem With The Martingale – Bet Doubling System

On the surface, one problem is that there isn’t really a game out there that is 50/50 odds you can bet on. Betting red/black in roulette is about as close as you come, but even there your odds of winning are 16/33 or 16/34 depending on the 0’s

The Main Problem With The Martingale – Bet Doubling System

The underlying math issue is more interesting. It boils down to the fact that you cannot keep doubling your bet forever. At some point you hit either the limit of your bankroll, or the house limit. And in any finite money scenario, all you have done is set up a situation where you frequently win a little bit of money, but infrequently lose a lot of money.

So how does that work ? Let’s give an example with different amounts of bankroll.

Let’s say that you have 7 dollars total. This means that you can keep betting up to the 4 dollar limit, and if you lose the 4 dollar bet you will have lost everything. That gives you a total of 3 bets, at 1, 2, 4, before you lost.   How likely is this to happen ?

This tree illustrates the probabilities

If you play this game 8 times, then 7 times you will win 1 dollar, and one time you will lose 7 dollars. The net expected value from this is zero, which is exactly what you would expect since you are assuming a 50/50 game.

If you have 63 dollars, you can run through it and find that you have a 1 in 64 chance of losing the 63 dollars, and a 63 in 64 chance of winning 1 on any give series of bets.

Your Chance Of Going Broke Is Directly Proportional To Your Bankroll

Assuming 50/50 odds, and that you have a power of 2 bankroll, your chance of going broke is exactly equivalent to your bankroll. Do you have a 1023 dollar bankroll? Then you will go broke 1 in 1024 times. Do you have a 4095 dollar bankroll? You will go broke 1 in 4096 series of bets.

What this strategy does is shift the probability curve, but the total probability remains the same. If you have a 63 dollar bankroll, and you bet 1 dollar 64 times, keeping your bet to 1 dollar regardless of if you won or lost, your probability curve would look like this

Which can be calculated from the binomial distribution.   Doing that betting strategy, the “Normal” strategy of betting the same amount every time would give you practically zero chance of losing all your money after 64 bets.

If instead you went with the bet doubling strategy, and placed 64 strings of bets this is what the probability curve would look like.

I generated these odds plot with a this python program & this Excel file simulating 1 million gambling outings.   Assumptions made were

• You would place 64 strings of bets. With a string ending either when you won your bet, or couldn’t double the bet anymore
• If you lost all your money then you were done
• If you lost and couldn’t double your bet, but you still had some money you would start again at a 1 dollar bet

This chart is much different than the binomial based chart. Most of the probability is pushed to the edges of the chart, so you are most likely to either walk away having won 64 dollars, or having lost your 63 dollar bankroll.   There are other options, such as winning 30 dollars, but then losing your 64 dollar bet and ending up down ~ 34 dollars

The total expected value of that chart, which is the cumulative probability multiplied by net winnings, is zero.

So while Martingale strategy of doubling the bet strategy isn’t a surefire winning technique, it does change the outcome so that instead of being likely to win or lose a little bit, you are most likely to win or lose a lot.