This is an interesting theorem which helps illuminate the relationship between unbounded utilities and St Petersburg gambles. I particularly appreciate that you don’t make an explicit assumption that the values of gambles must be representable by real numbers which is very common, but unhelpful in a setting like this. However, I do worry a bit about the argument structure.
The St Petersburg gamble is a famously paradox-riddled case. That is, it is a very difficult case where it isn’t clear what to say, and many theories seem to produce outlandish results. When this happens, it isn’t so impressive to say that we can rule out an opposing theory because in that paradox-riddled situation it would lead to strange results. It strikes me as similar to saying that a rival theory leads to strange result in variable population-size cases so we can reject it (when actually, all theories do), or that it leads to strange results in infinite population cases (when again, all theories do).
Even if one had a proof that an alternative theory doesn’t lead to strange conclusions in the St Petersburg gamble, I don’t think this would count all that much in its favour. As it seems plausible to me that various rules of decision theory that were developed in the cleaner cases of finite possibility spaces (or well-behaved infinite spaces) need to be tweaked to account for more pathological possibility spaces. For a simple example, I’m sympathetic to the sure thing principle, but it directly implies that the St Petersburg Gamble is better than itself, because an unresolved gamble is better than a resolved one, no matter how the latter was resolved. My guess is that this means the sure thing principle needs to have its scope limited to exclude gambles whose value is higher than that of any of their resolutions.
I think that weak outcome-lottery dominance is inconsistent with transitivity + unbounded utilities in both directions (or unbounded utilities in one direction + the sure thing principle), rather than merely producing strange results. Though we could summarize “violates weak outcome-lottery dominance” as a strange result.
Violating weak outcome-lottery dominance means that a mix of gambles, each strictly better than a particular outcome X, can fail to be at least as good as X. If you give up on this property, or on transitivity, then even if you are assigning numbers you call “utilities” to actions I don’t think it’s reasonable to call them utilities in the decision-theoretic sense, and I’m comfortable saying that your procedure should no longer be described as “expected utility maximization.”
So I’d conclude that there simply don’t exist any preferences represented by unbounded utility functions (over the space of all lotteries), and that there is no patch to the notion of utility maximization that fixes this problem without giving up on some defining feature of EU maximization.
There may nevertheless be theories that are well-described as maximizing an unbounded utility function in some more limited situations. And there may well be preferences over a domain other than lotteries which are described intuitively by an unbounded utility function. (Though note that if you are only considering lotteries over a finite space then your utility function is necessarily bounded.) And although it seems somewhat less likely it could also be that in retrospect I will feel I was wrong about the defining features of EU maximization, and mixing together positive lotteries to get a negative lottery is actually consistent with its spirit.
I think it’s also worth observing that although St Petersburg cases are famously paradox-riddled, these cases seem overwhelmingly important on a conventional utilitarian view even before we consider any exotic hypotheses. Indeed, I personally became unhappy with unbounded utilities not because of impossibility results but because I tried to answer questions like “How valuable is it to accelerate technological progress?” or “How bad is it if unaligned AI takes over the world?” and immediately found that EU maximization with anything like “utility linear in population size” seemed to be unworkable in practice. I could find no sort of common-sensical regularization that let me get coherent answers out of these theories, and I’m not sure what it would look like in practice to try to use them to guide our actions.
I think the more important takeaway is that the (countable) sure thing principle and transitivity together rule out preferences allowing St. Petersburg-like lotteries, and so “unbounded” preferences.
It discusses more ways preferences allowing St. Petersburg-like lotteries seem irrational, like choosing dominated strategies, dynamic inconsistency and paying to avoid information. Furthermore, they argue that the arguments supporting the finite Sure Thing Principle are generally also arguments in favour of the Countable Sure Thing Principle, because they don’t depend on the number of possibilities in a lottery being finite. So, if you reject the Countable Sure Thing Principle, you should probably reject the finite one, too, and if you accept St. Petersburg-like lotteries, you need to in principle accept behaviour that seems irrational.
They also have a general vNM-like representation theorem, dropping the Archimedean/continuity axiom, and replacing the Independence axiom with Countable Independence, and with transitivity and completeness, they get utility functions with values in lexicographically ordered ordinal sequences of bounded real utilities. (They say the sequences can have any ordinal to order them, but that seems wrong to me, since I’d think infinite length lexicographically ordered sequences get you St. Petersburg-like lotteries and violate Limitedness, but maybe I’m misunderstanding. EDIT: I think they meant you can have a an infinite sequence of dominated components, not an infinite sequence of dominating components, so you check the most important component first, and then the second, and continue for possibly infinitely many. Well-orderedness ensures there’s always a next one to check.)
Thanks for the reference, seems better than my post and I hadn’t seen it. (I think it’s the version of the argument I allude to in this comment.)
Note that if you have unboundedly positive and unboundedly negative outcomes, then you must also violate the weaker version of the countable sure thing principle with weak inequality instead of strict inequality. Violating the weak version of the sure thing principle seems much worse to me. And I think most proponents of unbounded preferences would advocate for them to point in both directions, so they run into this stronger problem.
That said, countability and strength of the sure thing principle are orthogonal: countable weak sure thing + finite strong sure thing --> countable strong sure thing. Negative utilities are only relevant as a response to someone who is OK saying “a 1% chance of an infinitely good outcome is just as good as a sure thing,” since that view is inconsistent if the 1% chance of an infinitely good outcome could be balanced out by a 2% chance of an infinitely bad outcome.
This is an interesting theorem which helps illuminate the relationship between unbounded utilities and St Petersburg gambles. I particularly appreciate that you don’t make an explicit assumption that the values of gambles must be representable by real numbers which is very common, but unhelpful in a setting like this. However, I do worry a bit about the argument structure.
The St Petersburg gamble is a famously paradox-riddled case. That is, it is a very difficult case where it isn’t clear what to say, and many theories seem to produce outlandish results. When this happens, it isn’t so impressive to say that we can rule out an opposing theory because in that paradox-riddled situation it would lead to strange results. It strikes me as similar to saying that a rival theory leads to strange result in variable population-size cases so we can reject it (when actually, all theories do), or that it leads to strange results in infinite population cases (when again, all theories do).
Even if one had a proof that an alternative theory doesn’t lead to strange conclusions in the St Petersburg gamble, I don’t think this would count all that much in its favour. As it seems plausible to me that various rules of decision theory that were developed in the cleaner cases of finite possibility spaces (or well-behaved infinite spaces) need to be tweaked to account for more pathological possibility spaces. For a simple example, I’m sympathetic to the sure thing principle, but it directly implies that the St Petersburg Gamble is better than itself, because an unresolved gamble is better than a resolved one, no matter how the latter was resolved. My guess is that this means the sure thing principle needs to have its scope limited to exclude gambles whose value is higher than that of any of their resolutions.
I think that weak outcome-lottery dominance is inconsistent with transitivity + unbounded utilities in both directions (or unbounded utilities in one direction + the sure thing principle), rather than merely producing strange results. Though we could summarize “violates weak outcome-lottery dominance” as a strange result.
Violating weak outcome-lottery dominance means that a mix of gambles, each strictly better than a particular outcome X, can fail to be at least as good as X. If you give up on this property, or on transitivity, then even if you are assigning numbers you call “utilities” to actions I don’t think it’s reasonable to call them utilities in the decision-theoretic sense, and I’m comfortable saying that your procedure should no longer be described as “expected utility maximization.”
So I’d conclude that there simply don’t exist any preferences represented by unbounded utility functions (over the space of all lotteries), and that there is no patch to the notion of utility maximization that fixes this problem without giving up on some defining feature of EU maximization.
There may nevertheless be theories that are well-described as maximizing an unbounded utility function in some more limited situations. And there may well be preferences over a domain other than lotteries which are described intuitively by an unbounded utility function. (Though note that if you are only considering lotteries over a finite space then your utility function is necessarily bounded.) And although it seems somewhat less likely it could also be that in retrospect I will feel I was wrong about the defining features of EU maximization, and mixing together positive lotteries to get a negative lottery is actually consistent with its spirit.
I think it’s also worth observing that although St Petersburg cases are famously paradox-riddled, these cases seem overwhelmingly important on a conventional utilitarian view even before we consider any exotic hypotheses. Indeed, I personally became unhappy with unbounded utilities not because of impossibility results but because I tried to answer questions like “How valuable is it to accelerate technological progress?” or “How bad is it if unaligned AI takes over the world?” and immediately found that EU maximization with anything like “utility linear in population size” seemed to be unworkable in practice. I could find no sort of common-sensical regularization that let me get coherent answers out of these theories, and I’m not sure what it would look like in practice to try to use them to guide our actions.
I think the more important takeaway is that the (countable) sure thing principle and transitivity together rule out preferences allowing St. Petersburg-like lotteries, and so “unbounded” preferences.
I recommend https://onlinelibrary.wiley.com/doi/abs/10.1111/phpr.12704
It discusses more ways preferences allowing St. Petersburg-like lotteries seem irrational, like choosing dominated strategies, dynamic inconsistency and paying to avoid information. Furthermore, they argue that the arguments supporting the finite Sure Thing Principle are generally also arguments in favour of the Countable Sure Thing Principle, because they don’t depend on the number of possibilities in a lottery being finite. So, if you reject the Countable Sure Thing Principle, you should probably reject the finite one, too, and if you accept St. Petersburg-like lotteries, you need to in principle accept behaviour that seems irrational.
They also have a general vNM-like representation theorem, dropping the Archimedean/continuity axiom, and replacing the Independence axiom with Countable Independence, and with transitivity and completeness, they get utility functions with values in lexicographically ordered ordinal sequences of bounded real utilities. (They say the sequences can have any ordinal to order them, but that seems wrong to me, since I’d think infinite length lexicographically ordered sequences get you St. Petersburg-like lotteries and violate Limitedness, but maybe I’m misunderstanding. EDIT: I think they meant you can have a an infinite sequence of dominated components, not an infinite sequence of dominating components, so you check the most important component first, and then the second, and continue for possibly infinitely many. Well-orderedness ensures there’s always a next one to check.)
Thanks for the reference, seems better than my post and I hadn’t seen it. (I think it’s the version of the argument I allude to in this comment.)
Note that if you have unboundedly positive and unboundedly negative outcomes, then you must also violate the weaker version of the countable sure thing principle with weak inequality instead of strict inequality. Violating the weak version of the sure thing principle seems much worse to me. And I think most proponents of unbounded preferences would advocate for them to point in both directions, so they run into this stronger problem.
That said, countability and strength of the sure thing principle are orthogonal: countable weak sure thing + finite strong sure thing --> countable strong sure thing. Negative utilities are only relevant as a response to someone who is OK saying “a 1% chance of an infinitely good outcome is just as good as a sure thing,” since that view is inconsistent if the 1% chance of an infinitely good outcome could be balanced out by a 2% chance of an infinitely bad outcome.