Martingale vs. One Big Bet - what is better?

[mr.ore]

What is better, Martingale or betting your whole bankroll? Question I will try to solve because right now I'm bored  

Suppose we have a fixed size bankroll of 1+2+4+8+16+32+64=127 units, it can survive six loses in a row when playing Martingale, series of 7 is a killer.    We wan to win 254 units, ie. more 127 thus doubling our bankroll.   

Two strategies:

a) bet all 127 units on EC:

Pwin(ALL) = 18/37 = 0. 486486

b) Martingale: Play until we double our bankroll OR lost series. What is a probability that we win one unit? It is probability that we don't lose 7 times in a row.   

Pwin(1 unit) = 1-(19/37)^7 = 0. 990548

Now, let's compute, how many units (u) we can win with same probability as winning 127 units by betting all:

Pwin(1 unit)^u = Pwin(ALL)

0. 990548^u = 0. 486486

Let's use logarithm to both sides of an equation:

log(0. 990548^u) = log(0. 486486)

Now we can use formula log(x^y) = y*log(x)

u*log(0. 990548) = log(0. 486486)

and now just put all but you on right side:

u = log(0. 486486)/log(0. 990548)
u = 75. 87
u ~= 76

So with probability 18/37 we can win only 76 units playing Martingale, and it is 127+76=203 units.   

So it is definitely better to risk all in one bet.   

Someone could say:  but what if we restart martingale after the lost series or bet all we have if not enough for eight step in progression? We have some units left, then we can play until we lose all or win!

Well, your chance would increase, but never to a better probability than betting it all at start. It is because on average you are down by 1/37*"sum of units already bet", and all what Martingale does is increase this sum as fast as possible.   

Now, why people play Martingale?

1) It's "example" system for roulette, first everyone gets to know of, and it is luring the fool.   

2) It's addictive (I know. . . ) and it provides with a lot of quick wins and adrenaline and dopamine rushes. It's nice to watch how bankroll slowly increases. It's a high, obviously.   

3) They believe they bet selection CAN avoid losing series of seven. If they COULD(cough. . . ), then Martingale would be the best strategy, because:

             MARTINGALE IS MATHEMATICALLY OPTIMAL STRATEGY TO WIN ONE UNIT,
                       ASSUMING YOU WILL NEVER GAMBLE AGAIN IN YOUR LIVE

If you COULD avoid losing series for sure, you have found a goldmine. . .   

Roulette is a special game, because you can control variance and probability, and if you by regular play decrease one(like probability), you have not been totally ripped - you have bought something: low variance. By combining different options you can make this game perform in many different ways. If you like a lot of small wins, you use progression, if you like to wait you use negative progression and it is then more like video poker slot (well, the house edge isn't). So with Martingale, you have bought yourself very low variance, almost all the time you are going up following a line, which is something like bankroll=0. 4*spin or so, not sure now exactly but I measured it once. It is not for free, therefore lower overall probability.   

Martingale is often combined with Gambler's fallacy. Why? After seven reds the probability of black is still 18/37, so why are people so "foolish"? I think there could be one explanation, there is actually some logic in Gambler's fallacy(just opinion):

If negative event, that cause system's failure have low probability(p), then the probability that the system won't fail is high(1-p), of course. But it means, that there would be LONG RUNS where the negative event just does not appear, and it is because of variance. If you want to "catch" that time, where for example a series of of 7 does not appear in 1000 spins, then you HAVE to play by Gambler's fallacy.   

But one never knows what would happen, and one bad series will take all wins and a part of starting bankroll.   

But Martingale is not that bad: use it for your LAST gamble in life, and most probably you won't regret it.   

[VLS]

Thanks Mr. Ore.

But Martingale is not that bad: use it for your LAST gamble in life, and most probably you won't regret it.

Most probably not.

Someone once mentioned a strategy: hunting for newbies in the casino, bankrolling them and having them play martingale or progression on sleeper dozen.

-It always works for newbies, Y'know.

[albalaha]

Through martingale, it is more than unlikely that u keep winning 127 units. I think chances of winning this much is practically not more than 10%-20% in any normal session through martingale.
    On the other hand if u stake 127 units in one spin on any EC bet, there is slightly less than 50% chance to win ur desired units in one go. I hate martingale so much that if forced to chose amongst martingale or double-or-nothing bet, I will prefer later.