Savage chooses not to define probabilities on a sigma-algebra. I haven’t seen any decision-theorist who prefers to use sigma-algebras yet. Similarly, he only derives finite additivity, not countable additivity; this also seems common among decision theorists.
This is annoying. Does anyone here know why they do this? My guess is that it’s because their nice theorems about the finite case don’t have straightforward generalizations that refer to sigma-algebras (I’m guessing this mainly because it appears to be the case for the VNM theorem, which only works if lotteries can only assign positive probability to finitely many outcomes).
Is it indeed the case that the VNM theorem cannot be generalized to the measure-theoretic setting?
Hypothesis: Consider X a compact Polish space. Let R⊆P(X)×P(X) be closed in the weak topology and satisfy the VNM axioms (in the sense that μ≤ν iff (μ,ν)∈R). Then, there exists u:X→R continuous s.t.(μ,ν)∈R iff Eμ[u]≤Eν[u].
Counterexamples?
One is also tempted to conjecture a version of the above where X is just a measurable space, R is closed in the strong convergence topology and u is just measurable. However, there’s the issue that if u is not bounded from either direction, there will be μ s.t.Eμ[u] is undefined. Does it mean u automatically comes out bounded from one direction? Or that we need to add an additional axiom, e.g. that there exists μ which is a global minimum (or maximum) in the preference ordering?
Both of your conjectures are correct. In the measurable / strong topology case, u will necessarily be bounded (from both directions), though it does not follow that the bounds are achievable by any probability distribution.
I described the VNM theorem as failing on sigma-algebras because the preference relation being closed (in the weak or strong topologies) is an additional assumption, which seems much more poorly motivated than the VNM axioms (in Abram’s terminology, the assumption is purely structural).
I think that one can argue that a computationally bounded agent cannot reason about probabilities with infinite precision, and that therefore preferences have to depend on probabilities in a way which is in some sense sufficiently regular, which can justify the topological condition. It would be nice to make this idea precise. Btw, it seems that the topological condition implies the continuity axiom.
Is it indeed the case that the VNM theorem cannot be generalized to the measure-theoretic setting?
Hypothesis: Consider X a compact Polish space. Let R⊆P(X)×P(X) be closed in the weak topology and satisfy the VNM axioms (in the sense that μ≤ν iff (μ,ν)∈R). Then, there exists u:X→R continuous s.t.(μ,ν)∈R iff Eμ[u]≤Eν[u].
Counterexamples?
One is also tempted to conjecture a version of the above where X is just a measurable space, R is closed in the strong convergence topology and u is just measurable. However, there’s the issue that if u is not bounded from either direction, there will be μ s.t.Eμ[u] is undefined. Does it mean u automatically comes out bounded from one direction? Or that we need to add an additional axiom, e.g. that there exists μ which is a global minimum (or maximum) in the preference ordering?
Both of your conjectures are correct. In the measurable / strong topology case, u will necessarily be bounded (from both directions), though it does not follow that the bounds are achievable by any probability distribution.
I described the VNM theorem as failing on sigma-algebras because the preference relation being closed (in the weak or strong topologies) is an additional assumption, which seems much more poorly motivated than the VNM axioms (in Abram’s terminology, the assumption is purely structural).
I think that one can argue that a computationally bounded agent cannot reason about probabilities with infinite precision, and that therefore preferences have to depend on probabilities in a way which is in some sense sufficiently regular, which can justify the topological condition. It would be nice to make this idea precise. Btw, it seems that the topological condition implies the continuity axiom.