Brad Warren: Well actually, there could be a strategy because you have control over:
1) your stake in each round
2) the number of games you choose to play (you can either play a minimum of 1 game of $100 or more than 10,000 games of gambling a penny each time)
I'm thinking that you are forced to play this game anyways. Playing 1 game with a stake of $100 is too risky, you could lose it all with 50% chance. Playing 10,000 games of a stake of $0.01 each time is more favorable to the convergence of the law of large numbers, this way if you are forced to play this game, this is how you can at least guarantee that you leave with your $100 still in pocket.
In thinking about possible "strategies to actually GAIN money"... I do not know what is the best one but I have this idea:
Since the stakes are $0.01 each time, we could do a simple strategy. I used to do this when I was a freshman in university with online casinos. When applied to non-1-1 payouts it worked and I won like $850 from $25 but then I stopped bc I was paranoid they were noticing my strategy and would arbitrarily interfere. For a 1-1 payout it will be a doubling strategy: Always bet on H. If you win, always start over at Step 1. If not, go on to the next step.
Step 1: Bet $0.01.
Step 2: You just lost $0.01. Bet $0.02. If you win, you'll get $0.04 back, subtract the $0.02 you put in, subtract the $0.01 cumulative loss, and you have made a profit of $0.01.
Step 3: You have now lost $0.03. Bet $0.04. If you win, you will get $0.08, subtract the $0.04 you put in, subtract the $0.03 cumulative loss, and you have made a profit of $0.01.
Step 4: You have now lost $0.07. Bet $0.08. If you win, you will get $0.16, subtract the $0.08 you put in, subtract the $0.07 cumulative loss, and you have made a profit of $0.01.
... etc
The chances of this failing correspond to you having lost so much that you are not able to recover by doubling your next bet. For this to be the case, you would have to see 14 consecutive tosses which are tails, which has a probability of (1/2)^14 = 0.00006.
As you win each time, your pot for the gambling becomes bigger. Then you can (slowly) get stronger so that you can withstand 15 consecutive losses, 16 consecutive losses, ... , and the probability of your losing it all will become infinitely small.