You can double the real numbers representing them, but the results of this won’t be preserved under affine transformations. So you can have two people whose utility functions are the same, tell them both “double your utility assigned to X” and get different results.
I think it would be very useful to explicitly state what the consequences of utility functions being only defined up to an “affine transformation” are in the grandparent post instead of assuming that everyone knows this at a 5-second level. My immediate reaction to the parent post was to look at the wikipedia article for affine transformations without much enlightenment.
Temperature is pretty useless as an analogy, because everyone knows that 2*40 degrees = 80 degrees; you have to think about moving between Celsius and Fahrenheit to actually get something useful out of the analogy. Voltage is even less helpful, because all it depends on is having a fixed reference point (i.e., differences between two voltages are always preserved, even if you change zero points), while distances aren’t preserved in general under affine transformations.
The downvoted post and your response are the most valuable in this entire thread, as they were the only ones that clearly communicated what the actual ramifications of utility functions only being defined up to an affine transformation were.
Ugh. Sorry if this came across as snarky, but the grandparent post came across as “If this young man expresses himself in terms too deep for me,/Why, what a very singularly deep young man/this deep young man must be!”
Except 2*40 degrees isn’t 80 degrees; the operation “2*40 degrees” is simply meaningless in the first place. (I mean, unless that’s a temperature difference of 40 degrees.)
(I mean, OK, you can strictly speaking double a temperature, but in order to do so you need to know what absolute zero is.)
You can double the real numbers representing them, but the results of this won’t be preserved under affine transformations. So you can have two people whose utility functions are the same, tell them both “double your utility assigned to X” and get different results.
Adding utilities is a totally routine operation. It is not necessarily a mistake.
Sure, you cam make mistakes by adding utilities—but that’s quite a different topic.
It looks to me like you haven’t gotten your head around the notion of an affine space.
In short: Utilites are not being added there. They are being linearly combined with coefficients summing to 1. I.e. you are taking an affine combination of them. Not a general linear combination, such as adding them (if you were adding two of them, the coefficients would sum to 2.)
If x and y are utilities, (x+y)/2 is meaningful, as is x/3+2y/3, as is 2x-y, etc. x+y is not meaningful, nor is 2x, or -x, or 3x-y.
Edit: To be clear, by “meaningful” here I mean meaningful as utilities, not meaningful as absolute numbers. Obviously none of these are meaningful as absolute numbers—to accomplish that, you need something like (x-y)/|z-w|.
Utilities are elements of an (ordered) affine space—not a vector space. Hence why (if represented by real numbers) they are only unique up to affine transformation.
I am not going to explain why, because this is bog-standard VNM. I am just hoping that by presenting a missing concept (that of an affine space) I might clear up some confusion.
You might also want to read John Baez’s thing on torsors that satt linked.
It looks to me like you haven’t gotten your head around the notion of an affine space.
In short: Utilites are not being added there. They are being linearly combined with coefficients summing to 1.
My goodness, this is getting ridiculous! Do I have to explain the concept of adding now? See those “plus” symbols? They represent the standard mathematical operation of addition. Utilites are being added there. For another example of adding utilities (with slightly less potentially-confusing math nearby) see the very next section of the same document. As I already explained, adding utilities is totally routine and is not necessarily a mistake.
...I wasn’t going to reply to this—I can’t expect to quickly correct the macro-mistake you’re making—but in this case the micro-mistake you’re making is a very simple one, so I can point it out.
Yes, the plus sign represents the operation of addition. But it isn’t utilities being added; pay attention to what the summands are! The summands are not utilities, but probabilities multiplied by utilities. At no point in anything you have pointed to are two utilities added without first being multiplied by probabilities (because that would be a meaningless operation).
Now, it’s true that if x and why are utilities, then x+y is the same as 1x+1y, and so this is a special case of “multiplying by probabilities and then summing”. But in fact the general operation of “multiply by probabilities and sum” is not meaningful; it is meaningful only in the special case when the probabilities in question sum to 1. (Though as I said above, it’s slightly more general than that, in that they don’t have to be probabilities—they can be any real number.)
Every “sum of utilities” you’ve pointed me to—which, as I’ve said, have not been sums of utilities, but rather sums of utilities scaled by real numbers—has taken this restricted form. Which is good, because otherwise the reasult would be meaningless. (Well, meaningless as a utility, anyway—we could consider something like x/3 + y/3 + z/3, where x, y, and z are utilities. Then you could point out that this contains the subsum x/3+y/3, which is not of the right form. But while meaningless as a utility, it’s a perfectly valid 2/3-of-a-utility. You could mulitply it by 3⁄2, or add another third-of-a-utility, or or have it and then add a 2/3-of-a-utility, and get a meaningful utility.)
(If I really wanted to pick nits, I could point out that “+” only really stands for addition if we first embed the affine space of utilities in a vector space, or assign particular real numbers to utilities; otherwise it’s just a notational shorthand. But in the case that we do assign real numbers to utilities, yes, it’s standard addition. Just not utilities that are being added.)
Yes, the plus sign represents the operation of addition. But it isn’t utilities being added; pay attention to what the summands are! The summands are not utilities, but probabilities multiplied by utilities.
Your mistake is even simpler. Probabilities are unit-free. They are numbers between 0 and 1. As such, they are dimensionless. So: a utility multiplied by a probability is still a utility.
If it helps any, I have a degree in mathematics. I do actually know what I am talking about here.
That would be a sensible inference, yes, if utilities were elements of a vector space. This is exactly why I said the terminology “utils” was so misleading—it suggests that they work like meters, seconds, etc; that we can think in terms of units and dimensions. But that doesn’t work here!
Dimensional analysis is basically analysis of scaling symmetries. It works when the only symmetries are scaling symmetries. But utilities, being an affine thing rather than a linear thing, have more symmetries than that! They have translation symmetries too! Units and dimensions are a very useful tool but they are not universal and they can’t handle something like this, unless you make sure you carefully distinguish between utilities and utility differences.
(That last one seems to be the usual solution to this sort of problem—not in the case of utitilities but more generally. E.g. we don’t hesitate to talk about points in time as quantities of seconds, even though time is translation-invariant, but really it’s only durations that are quantities of seconds. When we describe a point in time as being at “5 seconds”, we really mean “5 seconds after some agreed upon starting point”. And so while adding or halving durations is meaningful, adding or halving positions in time is not, because what’s going on with the starting point? (By contrast, averaging positions in time is meaningful, as is 2*t_1 - t_2, etc.) But time is familiar, so people don’t tend to make that sort of mistake, of forgetting that times are measured relative to some implicit arbitrary baseline; whereas utility is not so familiar, and, well, you’re making that mistake right now.)
If you like, you can imagine—as I’ve essentially done in my post above—that this affine space is embedded in some larger vector space, like the line x+y=1 embedded in R^2, and that elements of x+y=k have type “k*utility”.
But this is becoming stupid. This is a hell of a lot of words; the fact of the matter is that if you were right, then
we could take two outcomes a and b, with u(a)=1, u(b)=2, and observe that 3u(a) > u(b); then define an equivalent utility function v by v(x)=u(x)-2, and observe that now 3v(a) < v(b), so apparently in fact v was not equivalent. I.e. if you were right, then utility functions would only be unique up to positive scaling, not up to general positive affine transformations.
Only one of the following can be true:
1) It is meaningful to take non-affine combinations of utility functions
2) Two utility functions related by a positive affine transformation are equivalent
And it’s the latter. Why? Because if you look at the definition of what it means for a function u to be the utility function of a given agent, you’ll notice it only involves comparisons of affine combinations of values of u, not general linear combinations. Hence, applying any positive affine transformation will not change the comparisions, and the result will again be a utility function for the given agent.
And this is why everyone says that they’re only unique up to positive affine transformation, and correct in saying so. If the definition of a utility function relied on more general linear combinations of utilities, then that would restrict the symmetries further, and it would probably result in there only being scaling symmetries, in which case you would be right.
It seems as though you would agree that it is possible to add utility differences. The thing is, whenever anyone discusses utilities at all they are normally discussing utility differences. It’s the utility of having a banana over the utility over not having a banana. Or the utility of taking a medicine over not taking it. Such differences are the things that are being added together by those who add utilities.
Describing utility differences by using the term “utility” is like calling an elapsed time a “time”—both are commonplace. You can add utilities in much the same way that you can add times and distances.
Yes, of course you can add utility differences. Utilities form an affine space, their differences lie in the vector space acting on this affine space.
I disagree that discussion of utilities is normally discussion of utility differences, but, whatever. I’m not going to spend any more karma arguing over this. Regardless, it is important to recognize the difference between the two and keep the distinction clear, just as it is with positions in time vs. durations, positions vs. displacements, etc.
If people are only going to talk about utility differences rather than utilities, then sure, “utils” is fine. I feel like I’ve seen enough cases of trying to add utilities (not utility differences) that I think this is a bad idea, but, whatever; I’m not going to argue about that. (And it is possible I misunderstood what they were saying because it didn’t occur to me that maybe they meant utility differences and I wasn’t trying to read charitably. If that is the case, that might explain why some people thought my suggestion was so unnecessary...)
You can double the real numbers representing them, but the results of this won’t be preserved under affine transformations. So you can have two people whose utility functions are the same, tell them both “double your utility assigned to X” and get different results.
I think it would be very useful to explicitly state what the consequences of utility functions being only defined up to an “affine transformation” are in the grandparent post instead of assuming that everyone knows this at a 5-second level. My immediate reaction to the parent post was to look at the wikipedia article for affine transformations without much enlightenment.
Temperature is pretty useless as an analogy, because everyone knows that 2*40 degrees = 80 degrees; you have to think about moving between Celsius and Fahrenheit to actually get something useful out of the analogy. Voltage is even less helpful, because all it depends on is having a fixed reference point (i.e., differences between two voltages are always preserved, even if you change zero points), while distances aren’t preserved in general under affine transformations.
The downvoted post and your response are the most valuable in this entire thread, as they were the only ones that clearly communicated what the actual ramifications of utility functions only being defined up to an affine transformation were.
Ugh. Sorry if this came across as snarky, but the grandparent post came across as “If this young man expresses himself in terms too deep for me,/Why, what a very singularly deep young man/this deep young man must be!”
Except 2*40 degrees isn’t 80 degrees; the operation “2*40 degrees” is simply meaningless in the first place. (I mean, unless that’s a temperature difference of 40 degrees.)
(I mean, OK, you can strictly speaking double a temperature, but in order to do so you need to know what absolute zero is.)
Adding utilities is a totally routine operation. It is not necessarily a mistake.
Sure, you cam make mistakes by adding utilities—but that’s quite a different topic.
It looks to me like you haven’t gotten your head around the notion of an affine space.
In short: Utilites are not being added there. They are being linearly combined with coefficients summing to 1. I.e. you are taking an affine combination of them. Not a general linear combination, such as adding them (if you were adding two of them, the coefficients would sum to 2.)
If x and y are utilities, (x+y)/2 is meaningful, as is x/3+2y/3, as is 2x-y, etc. x+y is not meaningful, nor is 2x, or -x, or 3x-y.
Edit: To be clear, by “meaningful” here I mean meaningful as utilities, not meaningful as absolute numbers. Obviously none of these are meaningful as absolute numbers—to accomplish that, you need something like (x-y)/|z-w|.
Utilities are elements of an (ordered) affine space—not a vector space. Hence why (if represented by real numbers) they are only unique up to affine transformation.
I am not going to explain why, because this is bog-standard VNM. I am just hoping that by presenting a missing concept (that of an affine space) I might clear up some confusion.
You might also want to read John Baez’s thing on torsors that satt linked.
My goodness, this is getting ridiculous! Do I have to explain the concept of adding now? See those “plus” symbols? They represent the standard mathematical operation of addition. Utilites are being added there. For another example of adding utilities (with slightly less potentially-confusing math nearby) see the very next section of the same document. As I already explained, adding utilities is totally routine and is not necessarily a mistake.
...I wasn’t going to reply to this—I can’t expect to quickly correct the macro-mistake you’re making—but in this case the micro-mistake you’re making is a very simple one, so I can point it out.
Yes, the plus sign represents the operation of addition. But it isn’t utilities being added; pay attention to what the summands are! The summands are not utilities, but probabilities multiplied by utilities. At no point in anything you have pointed to are two utilities added without first being multiplied by probabilities (because that would be a meaningless operation).
Now, it’s true that if x and why are utilities, then x+y is the same as 1x+1y, and so this is a special case of “multiplying by probabilities and then summing”. But in fact the general operation of “multiply by probabilities and sum” is not meaningful; it is meaningful only in the special case when the probabilities in question sum to 1. (Though as I said above, it’s slightly more general than that, in that they don’t have to be probabilities—they can be any real number.)
Every “sum of utilities” you’ve pointed me to—which, as I’ve said, have not been sums of utilities, but rather sums of utilities scaled by real numbers—has taken this restricted form. Which is good, because otherwise the reasult would be meaningless. (Well, meaningless as a utility, anyway—we could consider something like x/3 + y/3 + z/3, where x, y, and z are utilities. Then you could point out that this contains the subsum x/3+y/3, which is not of the right form. But while meaningless as a utility, it’s a perfectly valid 2/3-of-a-utility. You could mulitply it by 3⁄2, or add another third-of-a-utility, or or have it and then add a 2/3-of-a-utility, and get a meaningful utility.)
(If I really wanted to pick nits, I could point out that “+” only really stands for addition if we first embed the affine space of utilities in a vector space, or assign particular real numbers to utilities; otherwise it’s just a notational shorthand. But in the case that we do assign real numbers to utilities, yes, it’s standard addition. Just not utilities that are being added.)
Your mistake is even simpler. Probabilities are unit-free. They are numbers between 0 and 1. As such, they are dimensionless. So: a utility multiplied by a probability is still a utility.
If it helps any, I have a degree in mathematics. I do actually know what I am talking about here.
That would be a sensible inference, yes, if utilities were elements of a vector space. This is exactly why I said the terminology “utils” was so misleading—it suggests that they work like meters, seconds, etc; that we can think in terms of units and dimensions. But that doesn’t work here!
Dimensional analysis is basically analysis of scaling symmetries. It works when the only symmetries are scaling symmetries. But utilities, being an affine thing rather than a linear thing, have more symmetries than that! They have translation symmetries too! Units and dimensions are a very useful tool but they are not universal and they can’t handle something like this, unless you make sure you carefully distinguish between utilities and utility differences.
(That last one seems to be the usual solution to this sort of problem—not in the case of utitilities but more generally. E.g. we don’t hesitate to talk about points in time as quantities of seconds, even though time is translation-invariant, but really it’s only durations that are quantities of seconds. When we describe a point in time as being at “5 seconds”, we really mean “5 seconds after some agreed upon starting point”. And so while adding or halving durations is meaningful, adding or halving positions in time is not, because what’s going on with the starting point? (By contrast, averaging positions in time is meaningful, as is 2*t_1 - t_2, etc.) But time is familiar, so people don’t tend to make that sort of mistake, of forgetting that times are measured relative to some implicit arbitrary baseline; whereas utility is not so familiar, and, well, you’re making that mistake right now.)
If you like, you can imagine—as I’ve essentially done in my post above—that this affine space is embedded in some larger vector space, like the line x+y=1 embedded in R^2, and that elements of x+y=k have type “k*utility”.
But this is becoming stupid. This is a hell of a lot of words; the fact of the matter is that if you were right, then we could take two outcomes a and b, with u(a)=1, u(b)=2, and observe that 3u(a) > u(b); then define an equivalent utility function v by v(x)=u(x)-2, and observe that now 3v(a) < v(b), so apparently in fact v was not equivalent. I.e. if you were right, then utility functions would only be unique up to positive scaling, not up to general positive affine transformations.
Only one of the following can be true: 1) It is meaningful to take non-affine combinations of utility functions 2) Two utility functions related by a positive affine transformation are equivalent
And it’s the latter. Why? Because if you look at the definition of what it means for a function u to be the utility function of a given agent, you’ll notice it only involves comparisons of affine combinations of values of u, not general linear combinations. Hence, applying any positive affine transformation will not change the comparisions, and the result will again be a utility function for the given agent.
And this is why everyone says that they’re only unique up to positive affine transformation, and correct in saying so. If the definition of a utility function relied on more general linear combinations of utilities, then that would restrict the symmetries further, and it would probably result in there only being scaling symmetries, in which case you would be right.
It seems as though you would agree that it is possible to add utility differences. The thing is, whenever anyone discusses utilities at all they are normally discussing utility differences. It’s the utility of having a banana over the utility over not having a banana. Or the utility of taking a medicine over not taking it. Such differences are the things that are being added together by those who add utilities.
Describing utility differences by using the term “utility” is like calling an elapsed time a “time”—both are commonplace. You can add utilities in much the same way that you can add times and distances.
Yes, of course you can add utility differences. Utilities form an affine space, their differences lie in the vector space acting on this affine space.
I disagree that discussion of utilities is normally discussion of utility differences, but, whatever. I’m not going to spend any more karma arguing over this. Regardless, it is important to recognize the difference between the two and keep the distinction clear, just as it is with positions in time vs. durations, positions vs. displacements, etc.
If people are only going to talk about utility differences rather than utilities, then sure, “utils” is fine. I feel like I’ve seen enough cases of trying to add utilities (not utility differences) that I think this is a bad idea, but, whatever; I’m not going to argue about that. (And it is possible I misunderstood what they were saying because it didn’t occur to me that maybe they meant utility differences and I wasn’t trying to read charitably. If that is the case, that might explain why some people thought my suggestion was so unnecessary...)