This could get complicated. I wanted to create a two-envelope paradox where the expected utility of receiving either envelope was not infinite. It’s impossible to create a paradox when the utility is finite, but it turns out it is possible to devise one where the utility is undefined. What’s the difference between an infinite utility and an undefined utility? Well, if X is infinitely valuable, it is irrational to prefer any good with finite utility to it, whereas if its utility is undefined, such a preference would be rationally acceptable. For a simple example, consider a situation like the following.
Eris tosses a fair coin repeatedly until it falls heads. She counts how many throws that took, call that n, and then places something worth (-2)n utils in an envelope. How much is the envelope worth?
If you try and work this out, it comes to -1+1-1+1-1+1-…, which is obviously undefined. I think (and I could be wrong about this) that for any good with finite utility, it is rationally permissible to be indifferent between Eris’s envelope and that good.
That isn’t the two envelope paradox I have in mind though. It works something like this. Eris takes three fair coins, A, B and C. She tosses A repeatedly until it falls heads. Let n be the number of tosses this takes. She then tosses B and C. The amount of utility put into the two envelopes is determined as follows:
| Larger | Smaller | |
| Heads | 3n+1 | 3n |
| Tails | 5-g(n) | 5-g(n+1) |
where Heads means B lands heads, Tails means B lands tails, and g is the function recursively defined as follows:
g(1) = 2
g(n+1) = 2 * g(n) – n/4
If coin C lands heads she puts the larger amount into the blue envelope, and the smaller amount into the red envelope. If it lands tails she puts the larger amount into the red envelope, and the smaller amount into the blue envelope. She then gives you the blue envelope.
Question: How much is the blue envelope worth?
Answer (I hope): It’s undefined.
Question: Should you pay to swap envelopes?
Answer: Er, no.
Question: If you see how much is in the blue envelope, will you want to swap?
Answer (I hope): Yes – whatever you see the expected utility of swapping is at least 1/12.
Question: Was this whole thing just an attempt to get fewer readers?
Answer: No – some people find this stuff genuinely interesting. Well, at least I find it genuinely interesting and I can project.