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The Martingale System – in a nutshell.

April 11, 2019

The basis of the martingale betting system is a strategy in which the gambler doubles the bet, such as a coin toss, after every loss, so that the first win would recover all previous losses plus a profit equal to the original stake. The martingale strategy has been applied to roulette in particular, as the probability of hitting either red or black is near to 50 per cent.

Take the case of a gambler who wagers £2 on Heads, at even money, so he profits by £2 if the coin lands Heads and loses £2 if it lands Tails. If he loses, he doubles the stake on the next bet, to £4, and wins £4 if it lands Heads, minus £2 lost on the first bet, securing a net profit over both bets of £2 (£4 – £2). If it lands Tails again, however, he is £6 down, so he doubles the stake in the next bet to £8. If it lands Heads he wins £8, minus £6 lost on the first two nets, securing a net profit over the three bets of £2 (£8 – £6). This can be generalized for any number of bets. Whenever he loses, the gambler secures a net profit over all bets of £2.

The strategy is essentially one of chasing losses, therefore. In the above example, the loss after n losing rounds is equal to 2+2²+2³+…2

So the strategy is to bet 2+2²+2³+… +2

In this way, the profit whenever the coin lands Heads is 2.

For a gambler with infinite wealth, and hence an infinite number of coin tosses to eventually generate heads, the martingale betting strategy has been interpreted as a sure win.

However, the gambler’s expected value remains zero (or less than zero) because the small probability of a very large loss exactly balances out the expected gain. In a casino, the expected value is in fact negative, due to the house edge.

The martingale strategy fails whenever there is a limit on earnings or on bets, as is the case in the real world. It is only with infinite wealth, bets and time that it could be argued that the martingale becomes a winning strategy.

Exercise

Show that the expected value of  martingale strategy in a fair game of heads/tails is zero. Show how can this be reconciled with the fact that whenever the player wins, the net overall profit to the player is positive.

 

References and Links

Martingale (betting system). Wikipedia. https://en.m.wikipedia.org/wiki/Martingale_(betting_system)

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