[BruceZ]

I said one thing that isn't right. The average length of this game isn't infinite. The average length of the game is 2 flips. The average length of the game would be infinite if one opponent started with an infinite number of coins (so he could never go broke). That is the classical result I was thinking of, and I didn't bother to do the math for this case. I was under the influence of powerful opiates (legal ones). For this case, the probability is 1/2 that it will end in 1 flip, 1/4 that it will end in 2 flips (guy with 2 coins loses twice), 1/8 that it will end in 3 flips (LWW), etc. So the average length of the game is 1(1/2) + 2(1/2)^2 + 3(1/2)^3 + ... = 2.

It's still true that there is a possibility of the game not ending for any finite number of flips, which was the main point.

If they agree in advance to quit after a certain number of flips, it will still be 2:1 if they quit after an even number of flips, but now there will be the possibility of a tie, so it's really 2:1:t where t is some fractional number of ties. If they quit after an odd number of flips, then the guy that starts with more coins will be better than 2:1 since they are quitting when the other guy is ahead, and he would have won most of these if the game had continued. If they quit after an odd number, then it is back where it started, so it would still be 2:1 if they continued.