This is a common misconception about why casinos win. That 'casinos have infinite bankroll so they cannot lose' isn't actually an argument because casinos have limited bankroll.
If they would allow me to play a +EV game I can crush them even if they had infinite bankroll by varying my betsize. The 'random walk' argument only works for fixed bet size.
I actually made a simulation of this because it seems obvious - you can actually double your bet each time you lose, and this basically guarantees that you eventually win each time!
Turns out, you have to go surprisingly high for this to be true/effective on the long run. And the casino will cut you off way below the limit (i.e. there's of course a maximum bet). Plus, with this strategy, each win is a small one... but when you lose (hitting the max bet), the loses are significant.
> I actually made a simulation of this because it seems obvious - you can actually double your bet each time you lose, and this basically guarantees that you eventually win each time!
The expected value of that strategy is still negative.
You get a high chance of winning 1 dollar, with a low chance of losing everything. Only the chance of losing everything isn't low enough.
Yeah, the doubling of your bet tactic when you lose is an old Blackjack strategy. But with the introduction of a max bet you have to look at the scale difference between the min/max bets, usually it means for example that you only need to lose 5-7 times in a row to lose everything.
If they would allow me to play a +EV game I can crush them even if they had infinite bankroll by varying my betsize. The 'random walk' argument only works for fixed bet size.