Originally posted by Harrier88
The martingale system has obviously been debunked numerous times, but that's not quite what's going on here. Richie didn't refer to doubling up his stakes, but to the optional "double or nothing" mini-game that you can play after winning.
Ah, OK. That mini-game has zero EV. The optional stopping theorem applies to this case too - there's no value to be gained by having any rule for when to play the mini-game. The only way of making a game vs the house +EV is bonus abuse, but the downside (the risk of confiscation) is greater than the upside.
A lot of online casinos expressly prohibit decreasing the stakes when in profit while clearing a bonus. A subtler version of abuse that's not banned explicitly is playing the double-up game when in loss (to increase the probability of busting the bonus early, thus placing fewer bets per bonus on average and paying less juice on average to the casino) and not playing it when in substantial profit, but I guess that casinos' security services can detect such a tendency too and ban the user for it.