I am new to Martingales. Why the betting strategy where if I loose then I double the amount (so, with the first win I get whatever I lost plus the amount of initial bet) is called the "martingale strategy" ? How does this match with the definition that the conditional expectation at time $n+1$ is same as the the observation at time $n$.
BTW, what is a good book for self-studying the theory of martingales ?
On
Martingale betting system dates from the 18th century, originating in France and is very simple to put into practice.
Here's how:
Place a first bet of a minimum odd of 2.00.
If you win you double your money. If you lose, double your initial bet.
You stop doubling when you win, in which case you make a profit equal to the initial bet. Thus, a series of six bets, you lose the first five and manage to make a profit on the initial bet if you catch the sixth.
I had always heard that the term originated as a sort of harness for a horse, and by analogy a system for the gambler that kept him on track, but the following suggests that this is not definitive. http://www.emis.de/journals/JEHPS/juin2009/Mansuy.pdf
When you then calculate your winnings under such a martingale strategy, you find $E(S_{n+1})\leq S_n$ where $S_n$ is your fortune after n plays. The little I know about martingales I've learned from Breiman's Probability, Shirayev's Probability, and Campbell, Lo and MacKinley, The Econometrics of Financial Markets.