
Originally Posted by
FightForCalifornia
you are just as likely to reach a given profit as you are betting evenly, and you are just as likely to go broke.
Technically speaking, this is untrue. If you were playing a juice-free game, then it should be apparent that the probability of your stake increasing by 50% will be twice the probability of losing your entire stake (50% * 2p = 100% * p). In general, the probability of your stake increasing by a factor of X will be 1/X times the probability of losing your entire stake.
If you're playing a binary1 game with a house edge, and if your goal were to make $X before losing your entire stake, then it can be proven that the Martingale with a single unit of $X would yield the greatest possible probability of achieving that goal (unless $X were less than your stake size, in which case you would bet as much as you had) .
Now this should in no way imply that the Martingale is a sensible strategy for a typical bettor. How many bettors do you know whose sole goal it is to make $1 before losing $1,023? And in the case of a player attempting to more than double his bankroll, that player would essentially be playing a reverse Martingale, increasing his bet after each win (and going home after his a first loss2). How many bettors do you know like that?
Following is an example of one of the few situations where nearly everyone, regardless of preferences (within reason) would be best served by using the Martingale:
You're just ingested a rare poison and are told at the emergency room that you have an hour to live. There is an antidote but it costs $10,000,000. You only have $9,999,990 and have no way of contacting anyone else for additional funds. There is a casino in the lobby of the hospital that only offers American roulette.
This would be an ideal time to use the Martingale strategy as it would provide you with a large chance (the highest possible chance given that starting position, actually) of making a small amount and a small chance of losing a large amount (of course, you'd be dead ... so it wouldn't really matter)3. Nevertheless, from a pure dollars perspective, your expectation betting in this manner is still negative.
So while that's not a particularly realistic example (one might also come up with a story involving a homicidal loan shark not willing to accept less than full payment), I hope it nevertheless illustrates a point -- if your goal from any stretch of betting were not the maximization of pure dollar profit, then betting strategy selection certainly could make a difference.
Notes:
1By "binary" I'm referring to the property of a betting game whereby it only has two moneyed outcomes; a loss of the entire bet, or a win of some constant positive proportion of the bet. Hence, baccarat and fixed odds sports betting (even with pushes -- pushes aren't moneyed outcomes) constitute binary bets. Blackjack and Asian handicap betting with a hook of ¼ do not.
2Unless his first loss occurs when the player's bankroll is greater than half (actually, greater than the reciprocal odds if the bet's odds decimal odds) of the target stake.
3Of course one might point out that if you were to win the extra $10 and receive the antidote, you probably wouldn't survive long without food or shelter. As such, it could make sense to play a Martingale with a unit greater than $10. Try as might, however, I can't contrive a single rationale for betting a Martingale with a unit of less than $10.