Coin Flipping Puzzle

Suppose your friend offers you the opportunity to play this game for free: Your friend will flip a biased coin 1000 times, which has a 60% chance of coming up heads each time, and a 40% chance of coming up tails. Every time it comes up heads, you are paid $100, and every time it comes up tails, you pay $100. You should definitely play this game, unless you are ridiculously risk-averse, because the chances of getting more than 500 heads are almost 100%, so you are virtually guaranteed to make money.

Suppose the game is changed slightly: Instead of agreeing upfront that the coin will be flipped 1000 times, you are given the option after each coin flip whether to continue flipping, to a maximum of 1000 flips, or to settle all payments and stop flipping (so this game is the same as the first one, with an additional option to terminate early). In order to decide whether to play this game, you think it through in your head, and you realize that, after 999 flips, you would probably end up terminating (i.e. not flipping the 1000th time), because at that time, you’d have a 40% chance of losing $100, and you are too risk-averse for that. But then you realize that, after 998 flips, you would effectively have only 1 flip left (because you’d know you would terminate after the 999th), so you’d apply the same logic and terminate at that time. You realize that this same logic can be applied repeatedly, backwards, all the way to the 1st flip, so you decide not to play the game at all.

But that doesn’t make any sense – the 2nd game is the same as the 1st, except for an option to terminate early (which makes the 2nd game superior), so it can’t make sense for you to agree to play the 1st game but not the 2nd.

So where is the logic wrong?

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Note: I thought of this game in response to hearing the financial advice which I complain about in my last post.