You are talking theory.. (everyone knows that part if you have been in the gambling threads for a good time)
I am talking psychology. Many want quick profits, like to take risks. My theory is they will go on playing till they lose.
Suppose I start a site with -0.1% edge, you make a profit, you will most likely play again. You lose, you think EV is positive and still play again. IT goes on till you lose and have nothing to continue bet. (assuming bankroll of player < house and there is a max bet limit, as I said)
That's the key assumption you're making here, right? For example, if bankroll of player > house, then you will also agree that the house will lose, even if the player wants to keep playing, right?
But if the aggregate bankroll of all players is significantly greater than that of the house, doesn't it imply that the house may lose even if the house edge is positive (say, 0.5%)? What are the chances for this to happen, depending on the total players bankroll vs bankroll of the house?