This makes Savage a better comparison point, since the Savage axioms are more similar to the VNM framework while also trying to construct probability and utility together with one representation theorem.
Sure, I guess I just always talk about VNM instead of Savage because I never bothered to learn how Savage’s version works. Perhaps I should.
As a representation theorem, this makes VNM weaker and JB stronger: VNM requires stronger assumptions (it requires that the preference structure include information about all these probability-distribution comparisons), where JB only requires preference comparison of events which the agent sees as real possibilities.
This might be true if we were idealized agents who do Bayesian updating perfectly without any computational limitations, but as it is, it seems to me that the assumption that there is a fixed prior is unreasonably demanding. People sometimes update probabilities based purely on further thought, rather than empirical evidence, and a framework in which there is a fixed prior which gets conditioned on events, and banishes discussion of any other probability distributions, would seem to have some trouble handling this.
Doesn’t pointless topology allow for some distinctions which aren’t meaningful in pointful topology, though?
Sure, for instance, there are many distinct locales that have no points (only one of which is the empty locale), whereas there is only one ordinary topological space with no points.
Isn’t the approach you mention pretty close to JB? You’re not modeling the VNM/Savage thing of arbitrary gambles; you’re just assigning values (and probabilities) to events, like in JB.
Assuming you’re referring to “So a similar thing here would be to treat a utility function as a function from some lattice of subsets of R (the Borel subsets, for instance) to the lattice of events”, no. In JB, the set of events is the domain of the utility function, and in what I said, it is the codomain.
Sure, I guess I just always talk about VNM instead of Savage because I never bothered to learn how Savage’s version works. Perhaps I should.
This might be true if we were idealized agents who do Bayesian updating perfectly without any computational limitations, but as it is, it seems to me that the assumption that there is a fixed prior is unreasonably demanding. People sometimes update probabilities based purely on further thought, rather than empirical evidence, and a framework in which there is a fixed prior which gets conditioned on events, and banishes discussion of any other probability distributions, would seem to have some trouble handling this.
Sure, for instance, there are many distinct locales that have no points (only one of which is the empty locale), whereas there is only one ordinary topological space with no points.
Assuming you’re referring to “So a similar thing here would be to treat a utility function as a function from some lattice of subsets of R (the Borel subsets, for instance) to the lattice of events”, no. In JB, the set of events is the domain of the utility function, and in what I said, it is the codomain.