I strongly agree that the key problem with St. Petersburg (and Pasadena) paradoxes is utility not being integrable with respect to the lotteries/​probabilities. Non-integrability is precisely what makes 𝔼U undefined (as a real number), whereas unboundedness of U alone does not.
However, it’s also worth pointing out that the space of functions which are guaranteed to be integrable with respect to any probability measure is exactly the space of bounded (measurable) functions. So if one wants to save utilities’ unboundedness by arguing from integrability, that requires accepting some constraints on one’s beliefs (e.g., that they be finitely supported). If one doesn’t want to accept any constraints on beliefs, then accepting a boundedness constraint on utility looks like a very natural alternative.
I agree with your last paragraph. If the state space of the world is â„ť, and the utility function is the identity function, then the induced preferences over finitely-supported lotteries can only be represented by unbounded utility functions, but are also consistent and closed under finite mixtures.
Finite support feels like a really harsh constraint on beliefs. I wonder if there are some other natural ways to constrain probability measures and utility functions. For example, if we have a topology on our state-space, we can require that our beliefs be compactly supported and our utilities be continuous. “Compactly supported” is way less strict than “finitely supported,” and “continuous” feels more natural than “bounded.” What are some other pairs of “compromise” conditions such that any permissible utility function is integrable with respect to any permissible belief distribution? (Perhaps it would be nice to have one that allows Gaussian beliefs, say, which are neither finitely nor compactly supported.)
Note that if P dominates Q in the sense that there is a c>0 such that for all events E, P(E)>câ‹…Q(E), U is integrable wrt P, then I think U is integrable wrt Q. I propose the space of all probability distribution dominated by a given distribution P.
Conveniently, if we move to semi-measures, we can take P to be the universal semi-measure. I think we can have our space of utility functions be anything integrable WRT the universal semi-measure, and our space of probabilities be anything lower semi-computable, and everything will work out nicely.
I think bounded functions are the only computable functions that are integrable WRT the universal semi-measure. I think this is equivalent to de Blanc 2007?
The construction is just the obvious one: for any unbounded computable utility function, any universal semi-measure must assign reasonable probability to a St Petersburg game for that utility function (since we can construct a computable St Petersburg game by picking a utility randomly then looping over outcomes until we find one of at least the desired utility).
Compact support still seems like an unreasonably strict constraint to me, not much less so than finite support. Compactness can be thought of as a topological generalization of finiteness, so, on a noncompact space, compact support means assigning probability 1 to a subset that’s infinitely tiny compared to its complement.
I strongly agree that the key problem with St. Petersburg (and Pasadena) paradoxes is utility not being integrable with respect to the lotteries/​probabilities. Non-integrability is precisely what makes 𝔼U undefined (as a real number), whereas unboundedness of U alone does not.
However, it’s also worth pointing out that the space of functions which are guaranteed to be integrable with respect to any probability measure is exactly the space of bounded (measurable) functions. So if one wants to save utilities’ unboundedness by arguing from integrability, that requires accepting some constraints on one’s beliefs (e.g., that they be finitely supported). If one doesn’t want to accept any constraints on beliefs, then accepting a boundedness constraint on utility looks like a very natural alternative.
I agree with your last paragraph. If the state space of the world is â„ť, and the utility function is the identity function, then the induced preferences over finitely-supported lotteries can only be represented by unbounded utility functions, but are also consistent and closed under finite mixtures.
Finite support feels like a really harsh constraint on beliefs. I wonder if there are some other natural ways to constrain probability measures and utility functions. For example, if we have a topology on our state-space, we can require that our beliefs be compactly supported and our utilities be continuous. “Compactly supported” is way less strict than “finitely supported,” and “continuous” feels more natural than “bounded.” What are some other pairs of “compromise” conditions such that any permissible utility function is integrable with respect to any permissible belief distribution? (Perhaps it would be nice to have one that allows Gaussian beliefs, say, which are neither finitely nor compactly supported.)
Note that if P dominates Q in the sense that there is a c>0 such that for all events E, P(E)>câ‹…Q(E), U is integrable wrt P, then I think U is integrable wrt Q. I propose the space of all probability distribution dominated by a given distribution P.
Conveniently, if we move to semi-measures, we can take P to be the universal semi-measure. I think we can have our space of utility functions be anything integrable WRT the universal semi-measure, and our space of probabilities be anything lower semi-computable, and everything will work out nicely.
I think bounded functions are the only computable functions that are integrable WRT the universal semi-measure. I think this is equivalent to de Blanc 2007?
The construction is just the obvious one: for any unbounded computable utility function, any universal semi-measure must assign reasonable probability to a St Petersburg game for that utility function (since we can construct a computable St Petersburg game by picking a utility randomly then looping over outcomes until we find one of at least the desired utility).
Compact support still seems like an unreasonably strict constraint to me, not much less so than finite support. Compactness can be thought of as a topological generalization of finiteness, so, on a noncompact space, compact support means assigning probability 1 to a subset that’s infinitely tiny compared to its complement.