Everybody wants EV. Play a +EV game, and earn "Sklansky bucks." Got a decision to make? Turn off your brain and pick the one with the highest EV. EV, EV, EV. It is all about EV.
Really, EV is just an abstract mathematical concept. You cannot take Sklansky bucks to the bank. If your decision has an EV of 10 and mine an EV of 9, what do you have that I do not? An extra unit of EV. What are you going to do with it?
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.
In the situation you described, the LLN does not apply. You are not repeating the same wager, since the wagered amount keeps changing. And the wagers are not independent, since the wagered amount depends on previous results. So you cannot use the LLN to translate the +EV of each individual wager into a long term positive result. Clearly, you already realized this, but I wanted to point out that it is not a paradox. It only looks like a paradox if you forget that EV requires the LLN to be valuable.
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what is the correct (ie highest EV) action with respect to the man's bets?
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You are assuming that "correct" = "highest EV". You are falling back on a general principle (always maximize EV) without thinking critically about whether or not it applies. You are not alone in this. Many people have it ingrained in their heads that the optimal choice in any situation is the one with the highest EV. They live by this motto without ever stopping to ask themselves why it is true. Stop and think. Highest EV is correct when that EV can be translated into long term profit. EV turns into long term profit by the LLN. The LLN depends on repeating the same wager many times independently. Are you not repeating the same wager many times independently? Then you cannot use the LLN. So you cannot translate your EV into long term profit in the usual way. So maybe the highest EV choice is not in your best interest.
At any step in this game, the highest EV choice is obviously to continue betting. As you pointed out, this is clearly not the best choice, by any reasonable definition of "best." There is no 100% objective answer to when you should quit. After every win, you will need to make a subjective evaluation to decide whether or not to continue.
You may be interested in reading about the concept of utility functions. The Kelly criterion is also connected to your question. In addition, you can read
this post for some comments of mine which are related. Here is the relevant part:
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Now, you also brought up EV and decisions. Probability theory (and science in general) cannot tell you what decisions you should make. It can only tell you the consequences of the various decisions you are considering. It is up to you to decide which consequences you prefer. Physics, for example, cannot tell us whether we should split an atom. It can only tell us what will happen if we do. The same is true of probability theory.
As far as EV is concerned, the relevant theorem is the Law of Large Numbers (LLN). The LLN, as you probably know, says that if you perform a sequence of independent and identically distributed wagers, then the overall average rate of change of your bankroll will converge to the EV of a single wager in that sequence. People often ignore the details of what the LLN says, and simply act as though it says you should always maximize your EV. But it is a mistake to ignore the details. In particular, it is mistake to ignore the fact that the LLN contains hypotheses which must be satisfied before it can be applied. In particular, the LLN requires not only that you have a sequence of wagers, but also that they are independent and identically distributed.
For example, suppose you are offered a 3:2 payout on the flip of a fair coin. What percentage of your bankroll should you wager on this bet? If you unthinkingly try to maximize your EV, then you will bet your entire bankroll. However, if you did this many times, you would eventually go broke. The LLN does not apply to a sequence of wagers in which you always bet a fixed proportion of your bankroll, because these wagers are not independent and identically distributed. The Kelly criterion will recommend a fraction smaller than 100% of your bankroll. If you accept Kelly's recommendation, then on a single wager, your raw EV will be smaller than it would be if you bet your whole roll. So Kelly, strictly speaking, is not recommending that you maximize your EV. If you study the Kelly criterion, you will find theorems that explicitly describe the long term consequences of following the Kelly system. The theorems do not say, "follow the Kelly system." They say, "if you follow the Kelly system, then here is what will happen." In some circumstances, the LLN is valid. In others, the Kelly theorems are valid. And in still others, neither might be valid and we may need to turn to something else in order to discover the consequences of our considered actions.
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