Many have been asking for an intuitive explanation of martingales. The following is the historical context, which helps understand some of the motivation.
suppose you have a fair coin, and you bet 1 dollar that it comes up heads. if it does, then you get 2 dollars, and you gain 1 dollar and exit.
suppose you lose. then put $2 on the next being head. if you win, you get double, for a gain of $4-($2+$1)=$4-$3=$1.
on third toss, put $4. if it comes up heads, you get $8. the gain is $8-$7=$1.
so every time you lose, double the previous bet. you can exit with $1 gain on your first win.
this was the original, flawed, martingale strategy. what's wrong?
then we realize that there is one outcome where you lose all your stake. On that path, your loss is exponential in the number of rounds!
another way to state that is that the expectation of the money after any round is $1, your initial stake. This is where the insight of expected value being preserved comes from.
Confusingly, while this betting strategy is the original "Martingale", it's not really what's meant by the term in probability theory.
An intuitive explanation of martingale in the more technical sense is simply that "your best guess for the future value of a series is its current value". I.e. on average it won't deviate up or down from where it is now, regardless of past values.
But, crucially, that applies at any given time. It means that every time the value moves up and down, you should reset your expectation and believe that whatever value it is now is the new normal.
Some things that are not martingales are
- processes that are biased to move more up than down, or vice versa,
- processes that are mean-reverting, i.e. when they have gone up for a while they are likely to go down again, and
- more generally, processes whose future value can be predicted better by using more information about their recent past.
suppose you have a fair coin, and you bet 1 dollar that it comes up heads. if it does, then you get 2 dollars, and you gain 1 dollar and exit.
suppose you lose. then put $2 on the next being head. if you win, you get double, for a gain of $4-($2+$1)=$4-$3=$1.
on third toss, put $4. if it comes up heads, you get $8. the gain is $8-$7=$1.
so every time you lose, double the previous bet. you can exit with $1 gain on your first win.
this was the original, flawed, martingale strategy. what's wrong?
then we realize that there is one outcome where you lose all your stake. On that path, your loss is exponential in the number of rounds!
another way to state that is that the expectation of the money after any round is $1, your initial stake. This is where the insight of expected value being preserved comes from.