On Martingale betting system

Solution 1:

The probability that you get e.g. five heads in a row is low, but nonzero. It'll happen to you eventually.

I wrote a quick script in Matlab to simulate players following the Martingale strategy. I started 100 players with a fortune of 100 each, and had them all play the martingale redoubling strategy. Here are the results of one run:

mgale

As you can see, there are only 9 players who make it past 2,000 time steps, only 5 who make it past 4,000 and only 3 who make it past 6,000. By timestep 10,000 everyone has gone broke. "In the long run, we're all dead."

It's a common psychological bias to discount low-probability events when making decisions (even when you know about the effect!) which is why the martingale strategy seems like such a good deal on first examination.

Solution 2:

Imagine a huge number of martingale players all starting to bet at the same time. After $n$ rounds, an overwhelming proportion of players, say $1-\varepsilon_n$, is broke but the tiny proportion $\varepsilon_n$ not broke yet has some good chances of being rich.

To see this, note that statistically the total amount of money of the players does not change hence, if $N$ players start with a given fortune $F$, the total amount of money at any time is about $NF$. After $n$ rounds, about $(1-\varepsilon_n)N$ of the players are broke, their fortune is $0$, hence the players not broke yet won about $(1-\varepsilon_n)NF$ globally. There are about $\varepsilon_nN$ of these hence the mean gain of each is about $(1-\varepsilon_n)F/\varepsilon_n$. Since $\varepsilon_n\to0$, this is about $F/\varepsilon_n$, which is huge if $n$ is large.

The trouble is that, when you start playing, nothing guarantees that you will be one of the around $\varepsilon_n N$ lucky players after $n$ rounds, and that the chances that you will be one of the unlucky players whose fortune is $0$ become huge when $n$ becomes large.

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