[StellarWind]

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To me it seems that if you lost the first bet, the 2nd bet would have the same EV as the first one, just that if you keep losing you might never get back to even.

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If you lose the first bet, the next bet will not be +EV. The point is that winning the second bet will lose the rakeback you already earned.

Individual bets are only +EV when your current cumulative result is breakeven. If you are ahead you cannot get rakeback on your losing bet. If you are behind you can get rakeback by losing but you can also lose rakeback by winning.

All of this assumes the oversimplified model of blackjack in which each hand is an even-money bet on the flip of a coin that is slightly biased against you. It also assumes you bet as much as you can (how much is that?) because that is the optimal strategy. You maximize profit by maximizing variance.

This analysis completely neglects a major issue. The player's normal disadvantage is very small. The player's advantage when he plays a rakeback-protected hand is large. It is +EV to play a second hand after you lose the first one. In exchange for a slight immediate disadvantage you gain a (nearly) 50% chance of returning to breakeven and getting another chance to play a high-EV protected hand.

For the same reason it is +EV to play a second hand after winning the first one. You incur a slight immediate disadvantage but you get to play another rakeback-protected hand if you lose.

Now mathematical induction kicks in. We just determined that it is +EV to play another hand when you are up a bet (+1) or down a bet (-1). It's not as +EV as playing a breakeven hand (0), but it's still more than the EV cost of normal blackjack. That makes it +EV to play +2 or -2 hands because you might get a chance to play a +1 or -1 hand.

In fact you can keep playing until you are a fair distance from breakeven before the slight house edge outweighs the possibility that you might eventually work your way back to zero. You keep playing until you have won so much or lost so much that it is no longer +EV to continue. This strategy results in a substantial win or a substantial (rakeback-cushioned) loss.

I suspect experienced BJ players intuitively recognize this. BJ is appealing to fish precisely because in the short run your chance of winning a lot of money is almost as high as your chance of losing a lot. The long run is a long way off and a 20% rebate on a losing session will allow you to play quite a lot of hands before the EV of the entire session becomes negative.