Right. The coin has a fixed value for P(heads), set when your friend tampered with it. You just don't know what it is.
OK, so you are averaging over all possible flip odds, assuming a uniform distribution of them. That's what "Assume that all heads:tails ratios are equally likely for the coin." means.
Your obvious best playing strategy is to look at the history of flips and trust the apparent bias. Assuming that N is much larger than the time it takes for the apparent bias to settle to the real one (perfect modeling), your odds of winning are max(p,1-p). Let's assume p > 0.5. Your expected payout per flip is 1p-3(1-p)=4p-3. Averaged over 0.5<p<1, this gives 0. In other words, even if the coin bias is given to you, you do not come out ahead when averaged over all biases, regardless of N.
Hmm, what else am I missing?
I came up with this puzzle after reading Vaniver's excellent post on the Value of Information. I enjoyed working it out over Thanksgiving and thought I'd share it with the rest of you.
Your friend holds up a curiously warped coin. "Let's play a game," he says. "I've tampered with this quarter. It could come up all heads, all tails, or any value in between. I want you to predict a coin flip; if you get it right, I'll pay you $1, and if you're wrong, you pay me $3."
"Absolutely, on one condition," you reply. "We repeat this bet until I decide to stop or we finish N games."
What is the minimum value of N that lets you come out ahead on average?
Each game, you may choose heads or tails, or to end the sequence of bets with that coin. Assume that all heads:tails ratios are equally likely for the coin.
edit: since a couple people have gotten it, I'll link my solution: http://pastebin.com/XsEizNFL