Smash Vegas

There isn’t a gambling forum on the internet without hundreds of posts about “unbeatable” betting systems based on the martingale.  They are usually either fraudulent systems for sale or naive newcomers who have just discovered a foolproof betting system - and they can show it’s mathematically sound!

In fact the martingale has been around for hundreds of years.  The system is simple - by increasing the stake with each loss, when the gambler eventually makes a win it will cover all the previous losses and make a profit.

For example if we are flipping a fair coin with 1:1 payout if we guess correctly, our starting bet would be $1.  After each loss we double our stake and when we win the stake is reset to $1.  So if we lose once before winning we gain $2 - $1 = $1.  If we lose twice before winning we gain $4 - $2 - $1 = $1.  Losing three times before winning nets us $8 - $4 - $2 - $1 = $1.  And so on.

As you can see the win always nets us $1 - so mathematically it is unbeatable!

Of course there is a problem which I hope you can see.  The size of the bets grows exponentially and so the stakes quickly get very large indeed.  This means that sooner or later we will either run out of money to make the next bet, or hit the limit of what the casino will let us wager.

This happens sooner than you’d think.  In the example above after 10 losses in a row we would be betting over $1000 to win $1.

The martingale is only an unbeatable strategy if:

1. You have infinite wealth
2. There is no limit to the size of bet you may make

It tricks a lot of people because if you try it for a short period of time you are likely to finish ahead.  Unfortunately if you keep playing you will eventually hit a big loss which will wipe you out.

What the martingale really does is trade a lot of small wins now for a big loss later.

No betting system can change the expected value of a game.