People like talking about extensions of the real numbers, but those don’t help you avoid any of the contradictions above. For example, if you want to extend < to a preference order over hyperreal lotteries, it’s just even harder for it to be consistent.
I’m a recent proponent of hyperreal utilities. I totally agree that hyperreals don’t solve issues with divergent / St. Petersburg-style lotteries. I just think hyperreals are perfect for describing and comparing potentially-infinite-utility universes, though not necessarily lotteries over those universes. (This doesn’t contradict Paul; I’m just clarifying.)
Separately, while cases like this do make it feel like we “should give up on probabilities as an abstraction for describing uncertainty,” this conclusion makes me feel quite nihilistic about decision-under-uncertainty; I will be utterly shocked if “a totally different kind of machinery to understand or analyze those preferences” is satisfactory.
My main concern is that unbounded utilities (and hence I assume also hyperreal utilities, unless you are just using them to express simple lexical preferences?) have a really hard time playing nicely with lotteries. But then if you aren’t describing preferences over lotteries, why do you want to have scalar utilities at all?
Separately, while cases like this do make it feel like we “should give up on probabilities as an abstraction for describing uncertainty,” this conclusion makes me feel quite nihilistic about decision-under-uncertainty; I will be utterly shocked if “a totally different kind of machinery to understand or analyze those preferences” is satisfactory.
I think I’d be less shocked by a totally different framework than probability, but I do agree that it looks kind of bleak. I would love to just reject unbounded utilities out of hand based on this kind of argument, and personally I don’t find unbounded utilities very appealing, but if someone swings that way I don’t feel like you can very well tell them to just change their preferences.
I’m also a fan of hyperreal probabilities/utilities—not that I think humans (do/should) use them, particularly, but that I think they’re not ruled out by appealing rationality principles.
My main concern is that unbounded utilities [...] have a really hard time playing nicely with lotteries. But then if you aren’t describing preferences over lotteries, why do you want to have scalar utilities at all?
I think the Jeffrey-Bolker axioms are more appealing than lottery-based elucidations of utility theory, and don’t have the sorts of problems you’re pointing to. In particular, you can just drop their version of the continuity axiom, which Jeffrey also feels is unmotivated.
Preferences can be represented by a probability distribution and an expected-utility distribution, but you can’t necessarily go the other way, from arbitrary probabilities and utilities to coherent preferences. So you can’t just define arbitrary lotteries like you do in the OP. The agent has to actually believe these to be possible.
So, rather than an impossibility result for unbounded utilities, I think you get that unbounded utilities are only consistent with specific beliefs.
EDIT: I still have to think about this more, but I think I misrepresented things a little. If we want expectations to be real-valued, but unbounded, then I think we can keep all the axioms including continuity, but the agent can’t believe in the possibility of lotteries corresponding to divergent sums. If we are OK with hyperreal values, then I think we drop continuity, and belief in lotteries with divergent sums are OK (but we still don’t need to deal with arbitrary lotteries, because that’s just not a very natural thing to do in the JB framework—we only want to work with what an agent believes is possible). However, it’s possible that your Dominance axiom gets violated. Intuitively, I don’t think this is a necessary thing (IE, it seems like we could add a Dominance-like axiom). In particular, your arguments don’t go through, since we don’t get to construct arbitrary lotteries; we only get to examine what the agent actually believes in and has preferences about. (So, to work with arbitrary lotteries, you have to add axioms asserting that events exist in the event-algebra corresponding to the desired lotteries.)
I’m really interested in this direction (largely because I’m already interested in pointless-topology/geometric-logic approaches to world-modeling), but I have a couple concerns off the bat. (Maybe if I read more about Jeffrey-Bolker and the surrounding literature I can answer my own questions here, but I thought I’d ask now anyway.)
One of the neat things about the standard interpretation of vNM is that it gives me an algorithmic recipe for (a) eliciting my beliefs-and-preferences about simple events, and then (b) deducing uniquely-valid consequences about what my preferences about complex events have to be (by computing integrals), so I don’t have to think about the complex events directly. Is there anything analogous to this in the Jeffrey-Bolker world?
If there is, can I apply it to ordinary lotteries that I definitely believe are possible, like prediction-market payouts?
If so, what does it look like mechanistically when I try to apply it to a St. Petersburg lottery? Where exactly are the guardrails between ordinary lotteries and impossible lotteries, and how might I come to realize (from within the Jeffrey-Bolker framework) that I’m not allowed to believe that a St. Petersburg lottery is real?
Jeffrey does talk about this in his book! Denote the probability of an event as P(E), and the expected value as V(E). Now suppose we cut an event A into parts B and C. We must have that V(A) = V(B)P(B|A) + V(C)P(C|A). Using this, we can cut the world up into small events which we’re comfortable assigning values to, and then put things back together into expected values for larger events. Basically exactly what you’d do normally, but no events are distinguished as “outcomes” so you can start wherever you want.
The JB axioms don’t assume anything like countable additivity, so an event like “St. Petersburg Lottery” needs a valuation which is consistent (in the above-mentioned sense) with the value of all other events, but there isn’t (necessarily) a computation which tells you the value of the infinite sum, the way there is for finite sums. We can add axioms which constrain values in situations like that, to avoid absurd things; but since it isn’t clear how to evaluate infinite sums in general, it makes sense to keep those axioms weak.
I think of this as a true result: the value of infinite lotteries is subjective (even after we know how to value all their finite parts). It has a lot of coherence constraints, but not enough to fully pin down a value. This means we don’t have to worry about all the messy nonsense of trying to evaluate divergent sums.
However, I think if we assume that any sub-event of the st petersburg lottery in which we still have a chance of winning something has a positive value (which seems very reasonable), then we can prove that the total value of the lottery is not any real number, by splitting off more and more of the finite sub-lotteries and arguing that the total value must exceed each.
Where exactly are the guardrails between ordinary lotteries and impossible lotteries, and how might I come to realize (from within the Jeffrey-Bolker framework) that I’m not allowed to believe that a St. Petersburg lottery is real?
The continuity axiom is what stops you from having non-archimedean values (just like with VNM), so that’s where the buck stops. If our preferences respect continuity, then we have to choose between believing St-petersburg-like lotteries are possible, vs believing enough niceness axioms about the values of infinite lotteries to prove that St. Petersburg has a value greater than all the reals.
For me, it seems like a pretty obvious choice to reject continuity (as a rationality condition that must apply to all rational minds—it could very well apply to humans in a practical sense). Continuity seems poorly-motivated in the first place.
I think assigning real (or hyperreal) values to possible universes can give really aesthetic properties (edit: at least to me; probably much less aesthetic for those who “don’t find unbounded utilities very appealing”) that I’d roughly call “additivity” or “linearity,” like: if A and B are systems, U(universe containing A) + U(universe containing B) = U(universe containing A and B). (This assumes that value is local, or something, which seems reasonable.) Utilities contain more than just lexical-ordering information if we can use them to describe the utility of new possible universes.
Perhaps more importantly, real and hyperreal utilities seem to play nice with finite lotteries, which seems quite desirable (and quite enough reason to have scalar utilities), even though it isn’t as strong as we’d hope.
I’m a recent proponent of hyperreal utilities. I totally agree that hyperreals don’t solve issues with divergent / St. Petersburg-style lotteries. I just think hyperreals are perfect for describing and comparing potentially-infinite-utility universes, though not necessarily lotteries over those universes. (This doesn’t contradict Paul; I’m just clarifying.)
Separately, while cases like this do make it feel like we “should give up on probabilities as an abstraction for describing uncertainty,” this conclusion makes me feel quite nihilistic about decision-under-uncertainty; I will be utterly shocked if “a totally different kind of machinery to understand or analyze those preferences” is satisfactory.
My main concern is that unbounded utilities (and hence I assume also hyperreal utilities, unless you are just using them to express simple lexical preferences?) have a really hard time playing nicely with lotteries. But then if you aren’t describing preferences over lotteries, why do you want to have scalar utilities at all?
I think I’d be less shocked by a totally different framework than probability, but I do agree that it looks kind of bleak. I would love to just reject unbounded utilities out of hand based on this kind of argument, and personally I don’t find unbounded utilities very appealing, but if someone swings that way I don’t feel like you can very well tell them to just change their preferences.
I’m also a fan of hyperreal probabilities/utilities—not that I think humans (do/should) use them, particularly, but that I think they’re not ruled out by appealing rationality principles.
I think the Jeffrey-Bolker axioms are more appealing than lottery-based elucidations of utility theory, and don’t have the sorts of problems you’re pointing to. In particular, you can just drop their version of the continuity axiom, which Jeffrey also feels is unmotivated.
Preferences can be represented by a probability distribution and an expected-utility distribution, but you can’t necessarily go the other way, from arbitrary probabilities and utilities to coherent preferences. So you can’t just define arbitrary lotteries like you do in the OP. The agent has to actually believe these to be possible.
So, rather than an impossibility result for unbounded utilities, I think you get that unbounded utilities are only consistent with specific beliefs.
EDIT: I still have to think about this more, but I think I misrepresented things a little. If we want expectations to be real-valued, but unbounded, then I think we can keep all the axioms including continuity, but the agent can’t believe in the possibility of lotteries corresponding to divergent sums. If we are OK with hyperreal values, then I think we drop continuity, and belief in lotteries with divergent sums are OK (but we still don’t need to deal with arbitrary lotteries, because that’s just not a very natural thing to do in the JB framework—we only want to work with what an agent believes is possible). However, it’s possible that your Dominance axiom gets violated. Intuitively, I don’t think this is a necessary thing (IE, it seems like we could add a Dominance-like axiom). In particular, your arguments don’t go through, since we don’t get to construct arbitrary lotteries; we only get to examine what the agent actually believes in and has preferences about. (So, to work with arbitrary lotteries, you have to add axioms asserting that events exist in the event-algebra corresponding to the desired lotteries.)
I’m really interested in this direction (largely because I’m already interested in pointless-topology/geometric-logic approaches to world-modeling), but I have a couple concerns off the bat. (Maybe if I read more about Jeffrey-Bolker and the surrounding literature I can answer my own questions here, but I thought I’d ask now anyway.)
One of the neat things about the standard interpretation of vNM is that it gives me an algorithmic recipe for (a) eliciting my beliefs-and-preferences about simple events, and then (b) deducing uniquely-valid consequences about what my preferences about complex events have to be (by computing integrals), so I don’t have to think about the complex events directly. Is there anything analogous to this in the Jeffrey-Bolker world?
If there is, can I apply it to ordinary lotteries that I definitely believe are possible, like prediction-market payouts?
If so, what does it look like mechanistically when I try to apply it to a St. Petersburg lottery? Where exactly are the guardrails between ordinary lotteries and impossible lotteries, and how might I come to realize (from within the Jeffrey-Bolker framework) that I’m not allowed to believe that a St. Petersburg lottery is real?
Jeffrey does talk about this in his book! Denote the probability of an event as P(E), and the expected value as V(E). Now suppose we cut an event A into parts B and C. We must have that V(A) = V(B)P(B|A) + V(C)P(C|A). Using this, we can cut the world up into small events which we’re comfortable assigning values to, and then put things back together into expected values for larger events. Basically exactly what you’d do normally, but no events are distinguished as “outcomes” so you can start wherever you want.
The JB axioms don’t assume anything like countable additivity, so an event like “St. Petersburg Lottery” needs a valuation which is consistent (in the above-mentioned sense) with the value of all other events, but there isn’t (necessarily) a computation which tells you the value of the infinite sum, the way there is for finite sums. We can add axioms which constrain values in situations like that, to avoid absurd things; but since it isn’t clear how to evaluate infinite sums in general, it makes sense to keep those axioms weak.
I think of this as a true result: the value of infinite lotteries is subjective (even after we know how to value all their finite parts). It has a lot of coherence constraints, but not enough to fully pin down a value. This means we don’t have to worry about all the messy nonsense of trying to evaluate divergent sums.
However, I think if we assume that any sub-event of the st petersburg lottery in which we still have a chance of winning something has a positive value (which seems very reasonable), then we can prove that the total value of the lottery is not any real number, by splitting off more and more of the finite sub-lotteries and arguing that the total value must exceed each.
The continuity axiom is what stops you from having non-archimedean values (just like with VNM), so that’s where the buck stops. If our preferences respect continuity, then we have to choose between believing St-petersburg-like lotteries are possible, vs believing enough niceness axioms about the values of infinite lotteries to prove that St. Petersburg has a value greater than all the reals.
For me, it seems like a pretty obvious choice to reject continuity (as a rationality condition that must apply to all rational minds—it could very well apply to humans in a practical sense). Continuity seems poorly-motivated in the first place.
I think assigning real (or hyperreal) values to possible universes can give really aesthetic properties (edit: at least to me; probably much less aesthetic for those who “don’t find unbounded utilities very appealing”) that I’d roughly call “additivity” or “linearity,” like: if A and B are systems, U(universe containing A) + U(universe containing B) = U(universe containing A and B). (This assumes that value is local, or something, which seems reasonable.) Utilities contain more than just lexical-ordering information if we can use them to describe the utility of new possible universes.
Perhaps more importantly, real and hyperreal utilities seem to play nice with finite lotteries, which seems quite desirable (and quite enough reason to have scalar utilities), even though it isn’t as strong as we’d hope.