All arguments depend upon how that question is answered, but how is it to be done? (Maybe subjective probabilities are relevant, but the St Petersburg paradox makes me unsure; the following sketch of that paradox is based on a post of mine in May:) Imagine the Supreme Being offering you the following deal, on Her fair tossing of a fair coin (you can tell somehow that it is the Supreme Being talking to you, and that She is no deceiver, and so you rightly believe all that She tells you, e.g. that the tossing will be fair)...
......The deal is that in exchange for you playing the following game (as detailed below) She will give you the entire wealth of the Universe. In effect, you would become Her appointed (and hence the absolute) Ruler of the Universe (in exchange for you owing Her an amount determined by the game below). To simplify matters, assume that She has shown you that, whether or not you take Her up on this deal, you will live forever in some form or another (e.g. as an immortal soul), and that the wealth of the Universe includes alien medical technology that can prolong your natural life within it indefinitely; and also teleportation devices, so that you could actually spend all that wealth. Conversely She could, if necessary, make you pay Her arbitrary amounts over and above your new wealth, were you to end up owing Her money (were you that unlucky at the following game), by getting you to work for Her, at a very reasonable rate of pay, in some relatively pleasant part of Purgatory.
......The game is as follows: She will repeatedly toss a fair coin, until it lands heads up, and you will pay Her back a number of cents equal to 2 to the power of (1 + the number of tails before the first head), but only if that number of tails is less than twenty times the wealth of the Universe in cents. So, if She gets a head first time, you will only owe Her 2 cents; and if She gets a tail and then a head, you will owe Her 4 cents; and if She gets two tails and then a head, you will owe Her 8 cents; and so forth, unless She throws as many tails as twenty times the wealth of the Universe in cents, in which case you will owe Her nothing. Since the chance of Her getting a head on the first toss is 1/2, and the chance of Her getting Her first head on the second toss is 1/4 (there being four equally likely possibilities for two tosses, i.e. HH, HT, TT and this one, TH), and the chance of Her getting Her first head on the third toss is 1/8, and so forth, hence Her expectation is (2/2 + 4/4 + 8/8 + … + N/N, for some N, as given by the above) cents minus the wealth of the Universe = twenty times the wealth of the Universe minus the wealth of the Universe = nineteen times the wealth of the Universe. That is, She would expect to get, were She to play this game a lot (thereby reducing the effects of chance), an enormous profit.
......Nonetheless this deal is only being offered because She suspects that you might wish to take Her up on it; so, would you? Well, what is your chance of losing much? It is clearly very small because for you to have to return as much as 20 dollars, from the vast wealth of the Universe that you would have already been given, She would have to throw at least 10 tails in a row (and if She threw less than 46 tails before the first head, which seems almost certain to occur, you would not even have to return a paltry trillion dollars of your vast wealth), and so your chance of having to work for many years in the afterlife is clearly tiny; so, would it be rational to reject such an offer, just because of something not too bad that almost certainly won’t happen anyway? Hardly; I mean, what would actually happen if the above deal were offered, and you took Her up on it? Just the actual outcome, which would surely (is there any reasonable doubt about this?) be you owning most of the Universe.
......How could turning that down be rational? (Of course, were you to take Her up on this, it being irrational not to, your future incarnations in similar Universes would surely do the same, for similar reasons, thereby ensuring that you would almost certainly spend an awful lot of your time working for nothing, because of your own well-informed and free choices :)
Deconstructing the Fisher-Neyman conflict wearing fiducial glasses (continued) - This continues my previous post: “Can’t take the fiducial out of Fisher…” in recognition of Fisher’s birthday, February 17. I supply a few more intriguing ...
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