Ignoring the fact that some numbers need to be non-negative and sum up to 1, the basic mathematical issue here is some well-understood subtleties involving infinite-dimensional vector spaces. More specifically, let S be an infinite set. Then we can write down the following three (real) vector spaces from S:
The vector space of finite linear combinations of elements of S.
The vector space of infinite linear combinations of elements of S such that the sum of the absolute value of the coefficients converges.
The vector space of all infinite linear combinations of elements of S.
Each of these spaces naturally sits inside of the next one. The VNM theorem should be regarded as describing certain linear functionals on the first vector space, which are uniquely determined by what they do to elements of S. Linear functionals on the second and third vector spaces are not uniquely determined by what they do to elements of S, the reason being that linear functionals only have to preserve finite linear combinations by default and this does not imply preserving infinite linear combinations. To preserve infinite linear combinations on the second vector space (a Lebesgue space ) we need some continuity hypothesis on the linear functional (Lebesgue spaces are in particular Banach spaces and have an induced topology ), which turns out to imply boundedness as a function on S essentially because of the St. Petersburg argument. The third vector space is even worse.
On the other hand, my impression is that the existence of pathological agents of the kind described in the OP (which in the language I’m using translates to pathological linear functionals on the second and third vector spaces) is independent of ZF but weaker than the axiom of choice. Similar objects such as Banach limits can be constructed using the Hahn-Banach theorem or the ultrafilter lemma. I’ve posted a math.stackexchange question about this.
I have serious doubts that any of these issues are relevant to agents in the real world.
That’s a better explanation than mine. And thanks for asking what assumptions it takes for such agents to exist where people are more likely to know the answer; I should have thought of that.
I have serious doubts that any of these issues are relevant to agents in the real world.
Ignoring the fact that some numbers need to be non-negative and sum up to 1, the basic mathematical issue here is some well-understood subtleties involving infinite-dimensional vector spaces. More specifically, let S be an infinite set. Then we can write down the following three (real) vector spaces from S:
The vector space of finite linear combinations of elements of S.
The vector space of infinite linear combinations of elements of S such that the sum of the absolute value of the coefficients converges.
The vector space of all infinite linear combinations of elements of S.
Each of these spaces naturally sits inside of the next one. The VNM theorem should be regarded as describing certain linear functionals on the first vector space, which are uniquely determined by what they do to elements of S. Linear functionals on the second and third vector spaces are not uniquely determined by what they do to elements of S, the reason being that linear functionals only have to preserve finite linear combinations by default and this does not imply preserving infinite linear combinations. To preserve infinite linear combinations on the second vector space (a Lebesgue space ) we need some continuity hypothesis on the linear functional (Lebesgue spaces are in particular Banach spaces and have an induced topology ), which turns out to imply boundedness as a function on S essentially because of the St. Petersburg argument. The third vector space is even worse.
On the other hand, my impression is that the existence of pathological agents of the kind described in the OP (which in the language I’m using translates to pathological linear functionals on the second and third vector spaces) is independent of ZF but weaker than the axiom of choice. Similar objects such as Banach limits can be constructed using the Hahn-Banach theorem or the ultrafilter lemma. I’ve posted a math.stackexchange question about this.
I have serious doubts that any of these issues are relevant to agents in the real world.
That’s a better explanation than mine. And thanks for asking what assumptions it takes for such agents to exist where people are more likely to know the answer; I should have thought of that.
Agreed.
I’m not sure about that. One relevant question is what happens if we restrict the set of priors our Bayesian utilitarians can have.