I am not a fan of unbounded utilities, but it is worth noting that most (all?) the problems with unbounded utilties are actually a problem with utility functions that are not integrable with respect to your probabilities. It feels basically okay to me to have unbounded utilities as long as extremely good/bad events are also sufficiently unlikely.
The space of allowable probability functions that go with an unbounded utility can still be closed under finite mixtures and conditioning on positive probability events.
Indeed, if you think of utility functions as coming from VNM, and you a space of lotteries closed under finite mixtures but not arbitrary mixtures, I think there are VNM preferences that can only correspond to unbounded utility functions, and the space of lotteries is such that you can’t make St. Petersburg paradoxes. (I am guessing, I didn’t check this.)
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 ℝ, 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)>c⋅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 observe that I probably miscommunicated. I think multiple people took me to be arguing for a space of lotteries with finite support. That is NOT what I meant. That is sufficient, but I meant something more general when I said “lotteries closed under finite mixtures” I did not mean there only finitely many atomic worlds in the lottery. I only meant that there is a space of lotteries, some of which maybe have infinite support if you want to think about atomic worlds, and for any finite set of lotteries, you can take a finite mixture of those lotteries to get a new lottery in the space. The space of lotteries has to be closed under finite mixtures for VNM to make sense, but the emphasis is on the fact that it is not closed under all possible countable mixtures, not that the mixtures have finite support.
Like, I agree that you can have gambles closed under finite but not countable gambling and the math works. But it seems like reality is a countably-additive sort of a place. E.g. if these different outcomes of a lottery are physical states of some system, QM is going to tell you to take some infinite sums. I’m just generally having trouble getting a grasp on what the world (and our epistemic state re. the world) would look like for this finite gambles stuff to make sense.
Note that you can take infinite sums, without being able to take all possible infinite sums.
I suspect it looks like you have a prior distribution, and the allowable probability distributions are those that you can get to from this distribution using finitely many bits of evidence.
I am not a fan of unbounded utilities, but it is worth noting that most (all?) the problems with unbounded utilties are actually a problem with utility functions that are not integrable with respect to your probabilities. It feels basically okay to me to have unbounded utilities as long as extremely good/bad events are also sufficiently unlikely.
The space of allowable probability functions that go with an unbounded utility can still be closed under finite mixtures and conditioning on positive probability events.
Indeed, if you think of utility functions as coming from VNM, and you a space of lotteries closed under finite mixtures but not arbitrary mixtures, I think there are VNM preferences that can only correspond to unbounded utility functions, and the space of lotteries is such that you can’t make St. Petersburg paradoxes. (I am guessing, I didn’t check this.)
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 ℝ, 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)>c⋅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 observe that I probably miscommunicated. I think multiple people took me to be arguing for a space of lotteries with finite support. That is NOT what I meant. That is sufficient, but I meant something more general when I said “lotteries closed under finite mixtures” I did not mean there only finitely many atomic worlds in the lottery. I only meant that there is a space of lotteries, some of which maybe have infinite support if you want to think about atomic worlds, and for any finite set of lotteries, you can take a finite mixture of those lotteries to get a new lottery in the space. The space of lotteries has to be closed under finite mixtures for VNM to make sense, but the emphasis is on the fact that it is not closed under all possible countable mixtures, not that the mixtures have finite support.
Hm, what would that last thing look like?
Like, I agree that you can have gambles closed under finite but not countable gambling and the math works. But it seems like reality is a countably-additive sort of a place. E.g. if these different outcomes of a lottery are physical states of some system, QM is going to tell you to take some infinite sums. I’m just generally having trouble getting a grasp on what the world (and our epistemic state re. the world) would look like for this finite gambles stuff to make sense.
Note that you can take infinite sums, without being able to take all possible infinite sums.
I suspect it looks like you have a prior distribution, and the allowable probability distributions are those that you can get to from this distribution using finitely many bits of evidence.