Trying to add up utilons or hedons can quickly lead to all sorts of problems, which are probably already familiar to you. However, there are all sorts of wacky and wonderful branches of non-intuitive mathematics, which may prove of more use than elementary addition. I half-remember that regular math can be treated as part of set theory, and there are various branches of set theory which can have some, but not all, of the properties of regular math—for example, being able to say that X < Y, but not necessarily that X+Z > Y. A bit of Wikipedia digging has reminded me of Cardinal numbers, which seem at least a step in the right direction: If the elements of set X has a one-to-one correspondence with the elements of set Y, then they’re equal, and if not, then they’re not. This seems to be a closer approximation of utilons than the natural numbers, such as, say, if the elements of set X being the reasons that X is good.
But I could be wrong.
I’m already well past the part of math-stuff that I understand well; I’d need to do a good bit of reading just to get my feet back under me. Does anyone here, more mathematically-inclined than I, have a better intuition of why this approach may or may not be helpful?
(I’m asking because I’m considering throwing in someone who tries to follow a cardinal-utilon-based theory of ethics in something I’m writing, as a novel change from the more commonly-presented ethical theories. But to do that, I’d need to know at least a few of the consequences of this approach might end up being. Any help would be greatly appreciated.)
I think the most mathy (and thus, best :P) way to go about this is to think of the properties that these “utility” objects have, and just define them as objects with those properties.
For starters, you can compare them for size—The relationship is either bigger, or smaller, or the same. And you can do an operation to them that is a weighted sum—if you have two utilities that are different, you can do this operation to them and get a utility that’s in between them, with a third parameter (the probability of one versus the other) distinguishing between different applications of this operation.
Actually, I think this sort of thing is pretty much what Savage did.
“The idea of cardinal utility is considered outdated except for specific contexts such as decision making under risk, utilitarian welfare evaluations, and discounted utilities for intertemporal evaluations where it is still applied. Elsewhere, such as in general consumer theory, ordinal utility with its weaker assumptions Is preferred because results that are just as strong can be derived.”
Or you could go back to the original Theory of Games proof, which I believe was ordinal- it’s going to depend on your axioms. In that document, Von Neumann definitely didn’t go so far as to treat utility as simply an integer.
When I was a freshman, I invented the electric motor! I think it’s something that just happens when you’re getting acquainted with a subject, and understand it well- you get a sense of what the good questions are, and start asking them without being told.
That’s one of the most amusing phrases on Wikipedia: “specific contexts such as decision making under risk”. In general you don’t have to make decisions and/or you can predict the future perfectly, I suppose.
It’s a tempting thought. But I think it’s hard to make the math work that way.
I have a lovely laptop here that I am going to give you. Suppose you assign some utility U to it. Now instead of giving you the laptop, I give you a lottery ticket or the like. With probability P I give you the laptop, and with probability 1 - P you get nothing. (The lottery drawing will happen immediately, so there’s no time-preference aspect here.) What utility do you attach to the lottery ticket? The natural answer is P * U, and if you accept some reasonable assumptions about preferences, you are in fact forced to that answer. (This is the basic intuition behind the von Neumann-Morgenstern Expected Utility Theorem.)
Given that probabilities are real numbers, it’s hard to avoid utilities being real numbers too.
I could try to rescue the idea by throwing in units, the way multiplying distance units by time units gives you speed units… but I’d just be trying to technobabble my way out of the corner.
I think the most that I can try to rescue from this failed hunch is that some offbeat and unexpected part of mathematics might be able to be used to generate useful, non-obvious conclusions for utilitarian-style reasoning, in parallel with math based on gambling turning out to be useful for measuring confidence-strengths more generally. Anybody have any suggestions for such a subfield which won’t make any actual mathematicians wince, should they read my story?
The von Neumann-Morgenstern theorem say that if you are uncertain about the world then you can denominate your utility in probabilities. Since probabilities are real numbers, so are utilities.
There are various ways to get infinite and infinitesimal utility. But they don’t matter in practice. Everything but the most infinite potential producer of utility will only matter as a tie breaker, which will occur with probability zero.
Cardinal numbers also wouldn’t work well even as infinite numbers go. You can’t have a set with half an element, or with a negative number of elements. And is there a difference between a 50% chance of uncountable utilons and a 100% chance?
Cardinal numbers for utilons?
I have a hunch.
Trying to add up utilons or hedons can quickly lead to all sorts of problems, which are probably already familiar to you. However, there are all sorts of wacky and wonderful branches of non-intuitive mathematics, which may prove of more use than elementary addition. I half-remember that regular math can be treated as part of set theory, and there are various branches of set theory which can have some, but not all, of the properties of regular math—for example, being able to say that X < Y, but not necessarily that X+Z > Y. A bit of Wikipedia digging has reminded me of Cardinal numbers, which seem at least a step in the right direction: If the elements of set X has a one-to-one correspondence with the elements of set Y, then they’re equal, and if not, then they’re not. This seems to be a closer approximation of utilons than the natural numbers, such as, say, if the elements of set X being the reasons that X is good.
But I could be wrong.
I’m already well past the part of math-stuff that I understand well; I’d need to do a good bit of reading just to get my feet back under me. Does anyone here, more mathematically-inclined than I, have a better intuition of why this approach may or may not be helpful?
(I’m asking because I’m considering throwing in someone who tries to follow a cardinal-utilon-based theory of ethics in something I’m writing, as a novel change from the more commonly-presented ethical theories. But to do that, I’d need to know at least a few of the consequences of this approach might end up being. Any help would be greatly appreciated.)
I think the most mathy (and thus, best :P) way to go about this is to think of the properties that these “utility” objects have, and just define them as objects with those properties.
For starters, you can compare them for size—The relationship is either bigger, or smaller, or the same. And you can do an operation to them that is a weighted sum—if you have two utilities that are different, you can do this operation to them and get a utility that’s in between them, with a third parameter (the probability of one versus the other) distinguishing between different applications of this operation.
Actually, I think this sort of thing is pretty much what Savage did.
Seems to be an established conversation around this point, see: https://en.wikipedia.org/wiki/Ordinal_utility https://en.wikipedia.org/wiki/Cardinal_utility
“The idea of cardinal utility is considered outdated except for specific contexts such as decision making under risk, utilitarian welfare evaluations, and discounted utilities for intertemporal evaluations where it is still applied. Elsewhere, such as in general consumer theory, ordinal utility with its weaker assumptions Is preferred because results that are just as strong can be derived.”
Or you could go back to the original Theory of Games proof, which I believe was ordinal- it’s going to depend on your axioms. In that document, Von Neumann definitely didn’t go so far as to treat utility as simply an integer.
Well, I guess coming up with an idea a century-ish old could be considered better than /not/ having come up with something that recent...
When I was a freshman, I invented the electric motor! I think it’s something that just happens when you’re getting acquainted with a subject, and understand it well- you get a sense of what the good questions are, and start asking them without being told.
That’s one of the most amusing phrases on Wikipedia: “specific contexts such as decision making under risk”. In general you don’t have to make decisions and/or you can predict the future perfectly, I suppose.
It’s a tempting thought. But I think it’s hard to make the math work that way.
I have a lovely laptop here that I am going to give you. Suppose you assign some utility U to it. Now instead of giving you the laptop, I give you a lottery ticket or the like. With probability P I give you the laptop, and with probability 1 - P you get nothing. (The lottery drawing will happen immediately, so there’s no time-preference aspect here.) What utility do you attach to the lottery ticket? The natural answer is P * U, and if you accept some reasonable assumptions about preferences, you are in fact forced to that answer. (This is the basic intuition behind the von Neumann-Morgenstern Expected Utility Theorem.)
Given that probabilities are real numbers, it’s hard to avoid utilities being real numbers too.
If we are going into VNM utility, it is defined as the output of the utility function and the utility function is defined as returning real numbers.
I could try to rescue the idea by throwing in units, the way multiplying distance units by time units gives you speed units… but I’d just be trying to technobabble my way out of the corner.
I think the most that I can try to rescue from this failed hunch is that some offbeat and unexpected part of mathematics might be able to be used to generate useful, non-obvious conclusions for utilitarian-style reasoning, in parallel with math based on gambling turning out to be useful for measuring confidence-strengths more generally. Anybody have any suggestions for such a subfield which won’t make any actual mathematicians wince, should they read my story?
The von Neumann-Morgenstern theorem say that if you are uncertain about the world then you can denominate your utility in probabilities. Since probabilities are real numbers, so are utilities.
There are various ways to get infinite and infinitesimal utility. But they don’t matter in practice. Everything but the most infinite potential producer of utility will only matter as a tie breaker, which will occur with probability zero.
Cardinal numbers also wouldn’t work well even as infinite numbers go. You can’t have a set with half an element, or with a negative number of elements. And is there a difference between a 50% chance of uncountable utilons and a 100% chance?
I don’t think that non-additivity is the only thing that matters about utilons: sometimes they do add, after all.
Besides that, yes, infinite cardinal numbers can have the property you cite: since for them
X + Z = max(X,Z)
if X < Y and Z < Y, it results
X + Z < Y