The end results will always be equivelant to the house edge. This study by the wizard of odds proved that
This one is played against roulette testing three different systems. Player 1 flat bet a $1 each time. He was not using a betting system. Player 2 started a series of trials with a bet of $1 and increased his wager by $1 after every winning bet. A lost bet would constitute the end of a series and the next bet would be $1. Player 3 also started a series of bets with a bet of $1 but used a doubling strategy in that after a losing bet of $x he would bet $2x (the Martingale). A winning bet would constitute the end of a series and the next bet would be $1. To make it realistic I put a maximum bet on player 3 of $200. Below are the results of that experiment:
Player 1
Total amount wagered = $1,000,000,000
Average wager = $1.00
Total loss = $52,667,912
Expected loss = $52,631,579
Ratio of loss to money wagered = .052668
Player 2
Total amount wagered = $1,899,943,349
Average wager = $1.90
Total loss = $100,056,549
Expected loss = $99,997,018
Ratio of loss to money wagered = .052663
Player 3
Total amount wagered = $5,744,751,450
Average wager = $5.74
Total loss = $302,679,372
Expected loss = $302,355,340
Ratio of loss to money wagered = .052688
As you can see the ratio of money lost to wagered is always near the house edge. Would this mean that playing a zero edge game would end in a gain or loss of zero? I believe so. But that is with a hugeeee bankroll. In reality you would run out of money during a cold streak and bust out. It never wins.