This analysis extends to the stronger version presented here. At some point, the significance of the monetary difference between the two envelopes *even knowing that one has had 5% removed (or whatever)* is just swamped by the utility of discovering a sufficiently large amount in the first envelope opened. Past a certain point, even orders of magnitude difference mean nothing when it comes to a choice between different quantities of money. The 5% is just noise in the calculation, nothing more. If we assess the standard principles of decision theory w/r/t utility rather than money, there is simply no puzzle here.

]]>Of course I wouldn’t believe any person who claimed to be offering me such a game. But decision theory should be able to deal with any epistemic situation one might conceivably be in, and not just the ones that some human could offer you. It’s easier to describe these things in terms of someone offering you a bet or a gamble or whatever, but it’s really supposed to be something where some action you take is going to have consequences with this much utility, with such and such a probability, given your degrees of belief about how the world is. The payoffs wouldn’t be money, but rather complete ways everything you care about could go, measured in utility terms.

If you’re only ruling out distributions on the basis that their expectation is infinite, then you’d be fine with a distribution like this where the coin has any bias with probability <1/2 of tails but the payoffs are the same, or a distribution like this where the payoffs are of size where but the bias is the same. It seems very strange to allow the same probabilities of different payoffs, and the same payoffs with different probabilities, but not this particular combination of payoffs and probabilities.

Another idea would be to say that no game with unbounded payoffs is possible. But then you allow every version of this game that is truncated at N flips, for any arbitrarily large integer N. Again, it seems quite strange to allow all of these, but not this limit game, which has extremely tiny probability of differing from any particular one of them, when N is large enough. The fact that there really do seem to be infinitely many distinct ways the world could turn out, and unboundedly large values that situations could have, suggests that we should be able to combine these into a description of some possible epistemic state.

Another idea would be to say that not only are unbounded payoffs in a particular game disallowed, but that in fact there is a uniform upper bound for the payoffs that are possible. I suspect that this would be the best way to go, but then we do end up with the slightly odd situation that there is some N such that nothing is N times as good as gaining a dollar. It's not obvious that there couldn't be such an N, but it does seem slightly strange.

Anyway, you're definitely right that these cases are odd and not likely to come up in any sort of application of decision theory. But since the theory is often considered to be an analysis of the notion of what one rationally ought to do in a given situation, it should apply to any conceivable situation, and not just the ones that we expect to run into in ordinary circumstances. I know this won't settle all your worries, but this is the sort of thing that motivates people to consider these strange infinitary scenarios. (Well, that together with the fact that it's fun to work with infinity.)

]]>Why shouldn’t I just say that this hypothesis is impossible? The expected amount of money in the envelope is infinite. There is no way for you to credibly commit that you will put the money in the envelopes according to this rule.

]]>Are you SURE that you are that much against “strong metaphysical claims”?

More seriously my point is, before trying to answer a question (any question!) is it not worthwhile to ponder HOW the question came about?

Ready made questions seem to me as suspicious as ready made answers.

P.S. For some strange reason posting a comment from an Opera 9.64/Linux browser doesn’t work, Javascript quirks I guess.

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