Maybe worth noting that ordinal preferences and a probability distribution over empirical outcomes (the setup in “BR under empirical uncertainty”) are used to generate cardinal preferences in the vNM utility theorem.
Interesting. I hadn’t explicitly made that connection, but it does seem worth thinking about.
I don’t know if this is what you were implying, but that made me wonder about whether what I wrote in this post effectively entails that we could “cardinalise” the preferences of the ordinal theories under consideration. My first impression is that we probably still can’t/shouldn’t, but I’m actually not sure about that, so here’s some long-winded thinking-aloud on the matter.
In MacAskill’s thesis, he discusses a related matter:
Many theories do provide cardinally measurable choice-worthiness: in general, if a theory orders empirically uncertain prospects in terms of their choice-worthiness, such that the choice-worthiness relation satisfies the axioms of expected utility theory, then the theory provides cardinally measurable choice-worthiness. Many theories satisfy this condition. Consider, for example, decision-theoretic utilitarianism, according to which one should maximise expected wellbeing (and which therefore satisfies the axioms of expected utility theory). If, according to decision-theoretic utilitarianism, a guarantee of saving Person A is equal to a 50% chance of saving no-one and a 50% chance of saving both Persons B and C, then we would know that, according to decision-theoretic utilitarianism, the difference in choice-worthiness between saving person B and C and saving person A is the same as the difference in choice-worthiness between saving person A and saving no-one. We give meaning to the idea of ‘how much’ more choice-worthy one option is than another by appealing to what the theory says in cases of uncertainty.
However, this method cannot be applied to all theories, for two reasons. First, if the theory does not order empirically uncertain prospects, then the axioms of expected utility theory are inapplicable. This problem arises even for some consequentialist theories: if the theory orders options by the value of the consequences the option actually produces, rather than the value of the consequences it is expected to produce, then the theory has not given enough structure such that we can use probabilities to measure choice-worthiness on an interval scale. For virtue-ethical theories, or theories that focus on the intention of the agent, this problem looms even larger.
Second, the axioms of expected utility theory sometimes clash with common-sense intuition, such as in the Allais paradox. If a theory is designed to cohere closely with common-sense intuition, as many non-consequentialist theories are, then it may violate these axioms. And if the theory does violate these axioms, then, again, we cannot use probabilities in order to make sense of cardinal choice-worthiness. Plausibly, Kant’s ethical theory is an example of a merely ordinally measurable theory. According to Kant, murder is less choiceworthy than lying, which is less choice-worthy than failing to aid someone in need. But I don’t think it makes sense to say, even roughly, that on Kant’s view the difference in choice-worthiness between murder and lying is greater than or less than the difference in choice-worthiness between lying and failing to aid someone in need. So someone who has non-zero credence in Kant’s ethical theory simply can’t use expected choiceworthiness maximization over all theories in which she had credence. (line break added)
And later he adds:
Often, in responses to my work on taking into account normative uncertainty over merely ordinal theories, people make the following objection. They claim that we know that under empirical uncertainty, that expected utility theory or some variant is the correct decision theory. And we should treat normative uncertainty in the same way as empirical uncertainty. So if we encounter a merely ordinal theory, over which one cannot take an expectation, we should either ignore it or we should force some cardinalisation upon it. To this objection I replied that, under empirical uncertainty we rarely or never face merely ordinal choice-worthiness. This is a genuine disanalogy with empirical uncertainty. And to simply force merely ordinal theories to fit into the framework of expected utility theory, rather than to consider how to aggregate merely ordinal theories, is simply not to take one’s credence in those merely ordinal theories sufficiently seriously.
So it seems to me that he’s arguing that we should respect that the theory is really meant to be ordinal, and we shouldn’t force cardinality upon it.
Which leaves me with an initial, unclear thought along the lines of:
We can still do things as I suggested in this post.
If an ordinal moral theory really does only care about what action you take and not what it causes, then, as noted in this post, we can either (a) ignore empirical uncertainty or (b) set the probabilities to 100% (because the action is guaranteed to lead to the “outcome”, which is that the outcome was taken); either way, we then use the Borda Rule as per usual.
But if an ordinal moral theory does care about outcomes, as most plausible theories do at least in part, then, as suggested in this post, we first look at the probabilities of each action under consideration leading to each outcome this theory “cares about”. We then work out how each theory would rank these actions, with these probabilities of causing those outcomes in mind. We then use the Borda Rule on those rankings.
But we still haven’t said the theories can tell us “how much” better one action is than another. And we haven’t had to assume that the theories have a sufficiently complete/coherent/whatever [I only have a layperson’s knowledge of the vNM utility theorem] set of preferences under empirical uncertainty that we can work out its cardinal preferences. It could have quite fuzzy or inconsistent ideas about what it would prefer in various situations of uncertainty, or it might very often consider an x% chance of A and a y% chance of A and B basically “equal” or “incomparable” or something like that.
But to be honest this seems to run into a bit of a roadblock related to me not really understanding how ordinal moral theories are really meant to work. I think that’s a hard issue to avoid in general when thinking about moral uncertainty. There are these theories that seem like they just can’t really be made to give us consistent, coherent preferences or follow axioms of rationality or whatever. But some of these theories are also very popular, including among professional philosophers, so, given epistemic humility, it does seem like it’s worth trying to take them seriously—and trying to take them seriously for what they claim themselves to be (i.e., ordinal and arguably irrational).
(Plus there’s the roadblock of me not having in-depth understanding of how the vNM utility theorem is meant to work.)
Additionally, in any case, it’s also possible that MacAskill’s Borda Rule effectively does implicitly cardinalize the theories. Tarsney seems to argue this, e.g.:
As I will argue at more length shortly, it seems to me that MacAskill’s approach fails to genuinely respect the phenomenon of merely ordinal theories, since the Borda counting approach (and MacAskill’s version of it in particular) is non-arbitrary only on the assumption that there are “hidden” cardinal values underlying the ordinal rankings of merely-ordinal theories.
If I’m interpreting Tarsney correctly, and if he’s right, then that may be why when you poke around and consider empirical uncertainties it starts to look a lot like a typical method for getting cardinal preferences from preference orderings. But I haven’t read that section of Tarsney’s thesis properly, so I’m not sure.
(I may later write a post about Tarsney’s suggested approach for making decision under moral uncertainty, which seems to have some advantages, and may also more fully respect ordinal theories ordinality.)
not really understanding how ordinal moral theories are really meant to work
Yeah, I think this is where I’m at too. It seems inescapable that ordinal preferences have cardinal implications when combined with empirical uncertainty (e.g. if I prefer a 20% chance of A to an 80% chance of B, that implies I like A at least four times as much). The only choice we really have is whether the corresponding cardinal implications are well-formed (e.g. Dutch bookable). The best distinctions I can come up with are:
In a purely deterministic world without lotteries, there wouldn’t be an obvious mechanism forcing the cardinalization of ordinal preferences. So their overlap is only a contingent feature of the world we find ourselves in. (Though see A Theory of Experienced Utility and Utilitarianism for an alternate basis for cardinalization.)
Ordinal preferences only specify a unique cardinalization in the limit of an infinite sequence of choices. Since we aren’t likely to face an infinite sequence of choices any time soon, they’re distinct in practice.
P.S. Thanks for the Tarsney link. I have it open in a tab and should get around to reading it at some point.
Plus there’s the roadblock of me not having in-depth understanding of how the vNM utility theorem is meant to work.
Not sure if it’ll help but I have a short explanation and interactive widget trying to explain it here.
Those are two interesting distinctions. I don’t have anything to add on that, but thanks for sharing those thoughts.
Not sure if it’ll help but I have a short explanation and interactive widget trying to explain it here.
Oh, you’re the person who made this value of information widget! I stumbled upon that earlier somehow, and am likely to link to it in a later post on applying VoI ideas to moral uncertainty.
Thanks for sharing the vNM widget; I intend to look at that soon.
Maybe worth noting that ordinal preferences and a probability distribution over empirical outcomes (the setup in “BR under empirical uncertainty”) are used to generate cardinal preferences in the vNM utility theorem.
Interesting. I hadn’t explicitly made that connection, but it does seem worth thinking about.
I don’t know if this is what you were implying, but that made me wonder about whether what I wrote in this post effectively entails that we could “cardinalise” the preferences of the ordinal theories under consideration. My first impression is that we probably still can’t/shouldn’t, but I’m actually not sure about that, so here’s some long-winded thinking-aloud on the matter.
In MacAskill’s thesis, he discusses a related matter:
And later he adds:
So it seems to me that he’s arguing that we should respect that the theory is really meant to be ordinal, and we shouldn’t force cardinality upon it.
Which leaves me with an initial, unclear thought along the lines of:
But to be honest this seems to run into a bit of a roadblock related to me not really understanding how ordinal moral theories are really meant to work. I think that’s a hard issue to avoid in general when thinking about moral uncertainty. There are these theories that seem like they just can’t really be made to give us consistent, coherent preferences or follow axioms of rationality or whatever. But some of these theories are also very popular, including among professional philosophers, so, given epistemic humility, it does seem like it’s worth trying to take them seriously—and trying to take them seriously for what they claim themselves to be (i.e., ordinal and arguably irrational).
(Plus there’s the roadblock of me not having in-depth understanding of how the vNM utility theorem is meant to work.)
Additionally, in any case, it’s also possible that MacAskill’s Borda Rule effectively does implicitly cardinalize the theories. Tarsney seems to argue this, e.g.:
If I’m interpreting Tarsney correctly, and if he’s right, then that may be why when you poke around and consider empirical uncertainties it starts to look a lot like a typical method for getting cardinal preferences from preference orderings. But I haven’t read that section of Tarsney’s thesis properly, so I’m not sure.
(I may later write a post about Tarsney’s suggested approach for making decision under moral uncertainty, which seems to have some advantages, and may also more fully respect ordinal theories ordinality.)
Yeah, I think this is where I’m at too. It seems inescapable that ordinal preferences have cardinal implications when combined with empirical uncertainty (e.g. if I prefer a 20% chance of A to an 80% chance of B, that implies I like A at least four times as much). The only choice we really have is whether the corresponding cardinal implications are well-formed (e.g. Dutch bookable). The best distinctions I can come up with are:
In a purely deterministic world without lotteries, there wouldn’t be an obvious mechanism forcing the cardinalization of ordinal preferences. So their overlap is only a contingent feature of the world we find ourselves in. (Though see A Theory of Experienced Utility and Utilitarianism for an alternate basis for cardinalization.)
Ordinal preferences only specify a unique cardinalization in the limit of an infinite sequence of choices. Since we aren’t likely to face an infinite sequence of choices any time soon, they’re distinct in practice.
P.S. Thanks for the Tarsney link. I have it open in a tab and should get around to reading it at some point.
Not sure if it’ll help but I have a short explanation and interactive widget trying to explain it here.
Those are two interesting distinctions. I don’t have anything to add on that, but thanks for sharing those thoughts.
Oh, you’re the person who made this value of information widget! I stumbled upon that earlier somehow, and am likely to link to it in a later post on applying VoI ideas to moral uncertainty.
Thanks for sharing the vNM widget; I intend to look at that soon.