[Pokey]

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Apparently, you didn't read (or understand) my previous point. I'll elaborate.

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The *key* when calculating EV is that you're actually taking a limint as n->infinity. In this case, your EV will be -1/4*(average bet size). Simply summing the numbers in a way that supports your hypothesis doesn't work since you are summing a series that doesn't converge.

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Sorry; I wasn't trying to address your specific point, but now I will -- you're making a different assumption from the one I started with. You're assuming we'll actually *play* an infinite number of games. I've assumed, as has the original poster, that the gambler facing the -EV gets to choose when to stop. That's all that's necessary to make the original position correct and my statements hold true. So long as one of the two players has a finite amount of money and the other has an infinite amount of money, this contest is over before it begins.

At any given time, I'm likely in debt as I play along. I'll go for long swings where I'm ridiculously far in debt. However, the expected time until my next profit is always finite, and I will eventually be positive again. Then, all I have to do is quit.

You assume we keep playing forever, at which point you're absolutely right -- the series never converges. However, that's the one feature of the game that allows us to always say that we can make a profit if we get to choose when we stop.

In short, we're both right. Of course, that's the kind of crazy thing that often happens when you have infinities running around.