## LESSWRONGLW

[anonymous]6y0

Are lotteries allowed to have infinitely many possible outcomes? (The Wikipedia page about the VNM axioms only says "many"; I might look it up on the original paper when I have time.)

There are versions of the VNM theorem that allow infinitely many possible outcomes, but they either

1) require additional continuity assumptions so strong that they force your utility function to be bounded

or

2) they apply only to some subset of the possible lotteries (i.e. there will be some lotteries for which your agent is not obliged to define a utility).

I might look it up on the original paper when I have time.

The original statement and proof given by VNM are messy and complicated. They have since been neatened up a lot. If you have access to it, tr... (read more)

0jsteinhardt6yI'm not sure, although I would expect VNM to invoke the Hahn-Banach theorem [http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem], and it seems hard to do that if you only allow finite lotteries. If you find out I'd be quite interested. I'm only somewhat confident in my original assertion (say 2:1 odds).