Since utility functions are only unique up to affine transformation, I don’t know what to make of this comment. Do you have some sort of canonical representation in mind or something?
In the context of this thread, you can consider U(status quo) = 0 and U(status quo, but with one more dollar in my wallet) = 1. (OK, that makes +10000 an unreasonable estimate of the upper bound; pretend I said +1e9 instead.)
Yes, this seems almost certainly true (and I think is even necessary if you want to satisfy the VNM axioms, otherwise you violate the continuity axiom).
Yes I’m quite aware… note that if there’s a sequence of outcomes whose values increase without bound, then you could construct a lottery that has infinite value by appropriately mixing the lotteries together, e.g. put probability 2^-k on the outcome with value 2^k. Then this lottery would be problematic from the perspective of continuity (or even having an evaluable utliity function).
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, try “Follmer H., and Schied A., Stochastic Finance: An Introduction in Discrete Time, de Gruyter, Berlin, 2004”
See also Kreps, Notes on the Theory of Choice. Note that one of these two restrictions are required in order to specifically prevent infinite expected utility. So if a lottery spits out infinite expected utility, you broke something in the VNM axioms.
For anyone who’s interested, a quick and dirty explanation is that the preference relation is primitive, and we’re trying to come up with an index (a utility function) that reproduces the preference relation. In the case of certainty, we want a function U:O->R where O is the outcome space and R is the real numbers such that U(o1) > U(o2) if and only if o1 is preferred to o2. In the case of uncertainty, U is defined on the set of probability distributions over O, i.e. U:M(O) → R. With the VNM axioms, we get U(L) = E_L[u(o)] where L is some lottery (i.e. a probability distribution over O). U is strictly prohibited from taking the value of infinity in these definition. Now you probably could extend them a little bit to allow for such infinities (at the cost of VNM utility perhaps), but you would need every lottery with infinite expected value to be tied for the best lottery according to the preference relation.
I’m not sure, although I would expect VNM to invoke the Hahn-Banach 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).
Also, it’s well possible that your utility function doesn’t evaluate to +10000 for any value of its argument, i.e. it’s bounded above.
Since utility functions are only unique up to affine transformation, I don’t know what to make of this comment. Do you have some sort of canonical representation in mind or something?
In the context of this thread, you can consider U(status quo) = 0 and U(status quo, but with one more dollar in my wallet) = 1. (OK, that makes +10000 an unreasonable estimate of the upper bound; pretend I said +1e9 instead.)
Yes, this seems almost certainly true (and I think is even necessary if you want to satisfy the VNM axioms, otherwise you violate the continuity axiom).
An unbounded function is one that can take arbitrarily large finite values, not necessarily one that actually evaluates to infinity somewhere.
Yes I’m quite aware… note that if there’s a sequence of outcomes whose values increase without bound, then you could construct a lottery that has infinite value by appropriately mixing the lotteries together, e.g. put probability 2^-k on the outcome with value 2^k. Then this lottery would be problematic from the perspective of continuity (or even having an evaluable utliity function).
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).
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, try “Follmer H., and Schied A., Stochastic Finance: An Introduction in Discrete Time, de Gruyter, Berlin, 2004”
EDIT: It’s online.
See also Kreps, Notes on the Theory of Choice. Note that one of these two restrictions are required in order to specifically prevent infinite expected utility. So if a lottery spits out infinite expected utility, you broke something in the VNM axioms.
For anyone who’s interested, a quick and dirty explanation is that the preference relation is primitive, and we’re trying to come up with an index (a utility function) that reproduces the preference relation. In the case of certainty, we want a function U:O->R where O is the outcome space and R is the real numbers such that U(o1) > U(o2) if and only if o1 is preferred to o2. In the case of uncertainty, U is defined on the set of probability distributions over O, i.e. U:M(O) → R. With the VNM axioms, we get U(L) = E_L[u(o)] where L is some lottery (i.e. a probability distribution over O). U is strictly prohibited from taking the value of infinity in these definition. Now you probably could extend them a little bit to allow for such infinities (at the cost of VNM utility perhaps), but you would need every lottery with infinite expected value to be tied for the best lottery according to the preference relation.
I’m not sure, although I would expect VNM to invoke the Hahn-Banach 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).