Triple or nothing paradox

Stabilizer

You are at a casino. You have $1. A table offers you a game: you have to bet all your money; a fair coin will be tossed; if it lands heads, you triple your money; if it lands tails, you lose everything.

In the first round, it is rational to take the bet since the expected value of winning is $1.50, which is greater than what you started out with.

If you win the first round, you'll have $3. In the next round, it is rational to take the bet again, since the expected value is $4.50 which is larger than $3.

If you win the second round, you'll have $9. In the next round, it is rational to take the bet again, since the expected value is $13.50 which is larger than $9.

You get the idea. At every round, if you won the previous round, it is rational to take the next bet.

But if you follow this strategy, it is guaranteed that you will eventually lose everything. You will go home with nothing. And that seems irrational.

Intuitively, it feels that the rational thing to do is to quit while you are ahead, but how do you get that prediction out of the maximization of expected utility? Or does the above analysis only feel irrational because humans are loss-averse? Or is loss-aversion somehow optimal here?

Anyway, please dissolve my confusion.

In the first round, it is rational to take the bet since the expected value of winning is $1.50, which is greater than what you started out with.

If you win the first round, you'll have $3. In the next round, it is rational to take the bet again, since the expected value is $4.50 which is larger than $3.

If you win the second round, you'll have $9. In the next round, it is rational to take the bet again, since the expected value is $13.50 which is larger than $9.

You get the idea. At every round, if you won the previous round, it is rational to take the next bet.

But if you follow this strategy, it is guaranteed that you will eventually lose everything. You will go home with nothing. And that seems irrational.

Anyway, please dissolve my confusion.

I don't like any of the proposed solutions to that when I glanced through the SEP article on it. They're all insightful but are sidestepping the hypothetical. Here's my take:

Compute the expected utility not of a choice BET/NO_BET but of a decision rule that tells you whether to bet. In this case, the OP proposed the rule "Always BET" which has expected utility of 0 and is bested by the rule "BET only once" which is in turn bested by the rule "BET twice if possible" and so on. The 'paradox' then is that there is a sequence of rules whose expected earnings are diverging to infinity. But then this is similar to the puzzle "Name a number; you get that much wealth." Which number do you name?

(Actually I think the proposed rule is not "Always BET" but "Always make the choice for which maximizes expected utility conditional to choosing NO_BET on the next choice". The fact that this strategy is flawed seems reasonable: you're computing the expectation assuming you choose NO_BET next but don't actually choose NO_BET next. Don't count your eggs before they hatch.)