[Uniqueuponhim]

Your post does make a lot of sense, and answers a lot of questions, but I find myself in the dark about some things.

There is one assertion in particular that you've made which I'm unsure is correct and that is that the law of large numbers does not apply in this situation. The following is your definition of the LLN, which I completely agree with:

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What people so easily forget (or perhaps never learned in the first place) is that the value of EV comes from the Law of Large Numbers (LLN), which states that if you repeat the same wager many times independently, then your winrate will be close to the EV of the wager. By itself, EV is just an abstract number. It takes the LLN to make a +EV game desirable.

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So in this situation, what we do is look at each individual situation and say "if I performed this exact bet a large number of times, how much would I end up with on average?" and you get the answer of 150% of what was bet. Therefore, on average, you make 50% profit every time you make the wager and each individual bet has an EV of +50%. There's no need to compare that bet to past or future ones in the hypothetical situation, just like in a poker game you don't compare your current hand with past or future ones to get the EV of different actions. Rather, you *simulate* the exact hand a large number of times and look at what your average profit or loss is for each decision. That doesn't mean that that exact hand actually has to happen a large number of times, you just pretend it does and look at the average result. Even looking at individual streets within a hand where the amount in the pot (and therefore the amount wagered) depends on past streets, you still just take that exact situation and iterate it a large number of times to get your average expectation value for various decisions, and both EV and LLN still apply. So how is this hypothetical situation really any different? The amount wagered in each bet may depend on the outcomes of previous bets, but each one can still be looked at individually and iterated a large number of times to get your average EV.

However, all of this does bring me to the idea of bankroll management, where you play the highest stakes that you can while retaining a negligible Risk of Ruin. As you mentioned, playing with your entire bankroll may have the highest EV in the short term, but in the long term you'll end up broke. Calculating the stakes of the game you should play seems to be a problem very similar to the one I mentioned above: How do you balance EV with RoR? I would imagine it would have something to do with the utility of an extra dollar decreases with the number of dollars you already had. But how do you actually perform the calculations?

I'm also curious about another problem which is very similar to the first, but also different on many levels:
Say that instead of wagering money on each game, you play the following game against a friend:

Neither of you can know what your opponent is doing or how many points they have until you both decide you're finished and compare points to see who won.
You each start with one point, and each round you can decide to either wager all of your points in the same manner as the first example or finish playing. If you lose a wager you're forced to finish playing. What is the optimal strategy to win this game?