Sometimes people refer to this relativity of utilities as “positive definite affine structure” or “invariant up to a scale and shift”, as if there were some background quantities that could scale and shift.
This is a misrepresentation of the mathematical point of view. In particular, the word “were” is misleading: when I say things like this, I am referring to a property of the map, not a property of the territory. The mathematical keyword is equivalence relation; when I say that utility functions are only well-defined up to positive affine transformations, what I mean is that “utility function” does not mean “function on outcomes,” it means “equivalence class of functions on outcomes,” where positive affine transformations define the equivalence relation. There are other equivalent ways of describing what a utility function is that don’t require working with equivalence classes, but it doesn’t matter which one you pick in the sense that the resulting mathematical theory has the same mathematical consequences.
Thanks for correcting me! I changed that paragraph. Is it less offensive to people who know what they are talking about now?
Sometimes people refer to this relativity of utilities as “positive affine structure” or “invariant up to a scale and shift”, which confuses me by making me think of a utility function as a set of things with numbers coming out, which don’t agree on the actual numbers, but can be made to agree with a linear transform, rather than a space I can measure distances in.
It’s somewhat confusing to me; you’re using words like “set,” “space,” and “measure distances” that have mathematically precise meanings but in a way which appears to disagree with those mathematically precise meanings (I don’t know what you mean when you say that a utility function is a space). It might be helpful to non-mathematicians, though.
I mean set as in set-theory. As in the utility function is a set of equivalent functions. If I’m disagreeing with math use, please correct me. (on second thought that wording is pretty bad, so I might change it anyway. Still, are my set-intuitions wrong?)
I mean space as in a 1-dimensional space (with a non-crazy metric, if crazy metrics even exist for 1d). By “measure distance” I mean go into said space with a tape measure and see how far apart things are.
I call it a space because then when I visualize it as such, it has all the right properties (scale/shift agnosticism).
If I call it a real-valued function, I imagine the real number line, which has a labeled axis, so to speak, so it tempts me to do numerology.
You can think of a utility function as defining a measure of “signed distance” on its domain.
Utilities have some similarity to distance in physical space, in that to give coordinates to all objects you need to select some origin and system of units for your coordinate system, but the physical reality is the same regardless of your coordinate system. A member of a particular utility function’s equivalence class, can then be thought of as a function that gives the coordinates of each thing in the domain (world-states, presumably), in some particular coordinate system.
For an example, if I prefer to have three ice creams over zero, three times as much as I prefer one ice cream over zero, then we can write that as a “utility function” u(no ice cream) = 0; u(one ice cream) = 1; u(three ice creams) = 3. In this case we have chosen arbitrarily no ice cream as the origin of our coordinate system, and “distance between one ice cream and none” as the basic unit of distance.
I mean set as in set-theory. As in the utility function is a set of equivalent functions. If I’m disagreeing with math use, please correct me. (on second thought that wording is pretty bad, so I might change it anyway. Still, are my set-intuitions wrong?)
Got it. This is strictly speaking true, but “equivalence class of functions” would be a more precise way of putting it.
I mean space as in a 1-dimensional space (with a non-crazy metric, if crazy metrics even exist for 1d). By “measure distance” I mean go into said space with a tape measure and see how far apart things are.
So there are some technical points I could go into here, but the short story is that most equivalence classes under positive affine transformations are 2-dimensional, not 1-dimensional, and also aren’t naturally endowed with a notion of distance.
the short story is that most equivalence classes under positive affine transformations are 2-dimensional, not 1-dimensional, and also aren’t naturally endowed with a notion of distance.
I can see how distance would be trouble in 2d affine-equivalent spaces, but distance seems to me to be a sensible concept in a 1d space, even with positive-scale and shift. And utility is 1d, so it’s safe to call it a “distance” right?
Maybe you’re referring to distance-from-A-to-B not having a meaningful value without defining some unit system? Maybe we should call them “relative distances”, except that to me, “distance” already connotes relativeness.
Maybe you’re referring to distance-from-A-to-B not having a meaningful value without defining some unit system? Maybe we should call them “relative distances”, except that to me, “distance” already connotes relativeness.
This is a totally sensible point of view but disagrees with the mathematical definition. It also doesn’t apply directly to the 2-dimensional equivalence classes, as far as I can tell. For example, suppose we’re talking about utilities over two possible outcomes {heads, tails}. There are three equivalence classes here, which are u(heads) > u(tails), u(heads) = u(tails), and u(heads) < u(tails). The first and third equivalence classes are 2-dimensional. What is the distance between the two functions (u(heads) = 2, u(tails) = 1) and (u(heads) = 3, u(tails) = 2) in the first case, even in a relative sense?
Ohhhhhhh, do you mean 2d as in 2 degrees of freedom? I mean it as in spatial coordinates.
As an aside, I just realized that “displacement” is more accurate for what I’m getting at than “distance”. The thing I’m talking about can be negative.
And distance/displacement isn’t between equivalent utility functions, it’s between two outcomes in one utility function. “X is 5 tasty sandwiches better than Y” is what I’m referring to as a displacement.
And the displacement numbers will be the same for the entire equivalence class, which is why I prefer it to picking one of the equivalent functions out of a hat. If you only ever talk about measured distances, there is only one utility function in the equivalence class, because all the scales and shifts cancel out:
This way, the utility function can scale and shift all it wants, and my numbers will always be the same. Equivalently, all agents that share my preferences will always agree that a day as a whale is “400 orgasms better than a normal day”, even if they use another basis themselves.
Was that less clear than I thought?
If there are only two points in a space, you can’t get a relative distance because there’s nothing to make the distance relative to. For that problem I would define U(heads) = 1 and U(tails) = 0, as per my dimensionless scheme.
Ohhhhhhh, do you mean 2d as in 2 degrees of freedom? I mean it as in spatial coordinates.
What’s the difference?
And distance/displacement isn’t between equivalent utility functions, it’s between two outcomes in one utility function. “X is 5 tasty sandwiches better than Y” is what I’m referring to as a displacement.
Your use of the word “in” here disagrees with my usage of the word “utility function.” Earlier you said something like “a utility function is a space” and I defined “utility function” to mean “equivalence class of functions over outcomes,” so I thought you were referring to the equivalence class. Now it looks like you’re referring to the space of (probability distributions over) outcomes, which is a different thing. Among other things, I can talk about this space without specifying a utility function. A choice of utility function allows you to define a ternary operation on this space which I suppose could reasonably be called “relative displacement,” but it’s important to distinguish between a mathematical object and a further mathematical object you can construct from it.
Your use of the word “in” here disagrees with my usage of the word “utility function.”
Yes, it does. You seem to understand what I’m getting at.
it’s important to distinguish between a mathematical object and a further mathematical object you can construct from it.
I don’t think anyone is making mathematical errors in the actual model, we are just using different words which makes it impossible to communicate. If you dereference my words in your model, you will see errors, and likewise the other way.
Is there a resource where I could learn the correct terminology?
I don’t think anyone is making mathematical errors in the actual model, we are just using different words which makes it impossible to communicate. If you dereference my words in your model, you will see errors, and likewise the other way.
Yep.
Is there a resource where I could learn the correct terminology?
My conventions for describing mathematical objects comes from a somewhat broad range of experiences and I’m not sure I could recommend a specific resource that would duplicate the effect of all of those experiences. Recommending a range of resources would entail learning much more than just a few conventions for describing mathematical objects, and you may not feel that this is a good use of your time, and I might agree. I can at least broadly indicate that some useful mathematical subjects to read up on might be real analysis and topology, although most of the content of these subjects is not directly relevant; what’s relevant is the conventions you’ll pick up for describing mathematical objects.
Sometime soon I might write a Discussion post about mathematics for rationalists which will hopefully address these and other concerns.
I feel confused. “a space I can measure distances in” is a strong property of a value, and it does not follow from your initial 5 axioms, and seems contrary to the 5th axiom.
In fact, your own examples given further seem to provide a counterexample—i.e., if someone prefers being a whale to 400 actual orgasms, but prefers 1⁄400 of being a whale to 1 orgasm, then both “being a whale” and “orgasm” have some utility value, but they cannot be used as units to measure distance.
If you’re in a reality where a>b and 2a<2b, then you’re not allowed to use classic arithmetic simply because some of your items look like numbers, since they don’t behave like numbers.
“Hawaii” can’t be used as a unit to measure distance, nor can “the equator”, but “the distance from Hawaii to the equator” can. Similarly, “the difference between 0 orgasms and 1 orgasm” can be used as a unit to measure utilities (you could call this unit “1 orgasm”, but that would be confusing and silly if you had nonlinear utility in orgasms: 501 orgasms could be less than or more than “1 orgasm” better than 500).
Also, did you mean to have these the other way around?:
This is a misrepresentation of the mathematical point of view. In particular, the word “were” is misleading: when I say things like this, I am referring to a property of the map, not a property of the territory. The mathematical keyword is equivalence relation; when I say that utility functions are only well-defined up to positive affine transformations, what I mean is that “utility function” does not mean “function on outcomes,” it means “equivalence class of functions on outcomes,” where positive affine transformations define the equivalence relation. There are other equivalent ways of describing what a utility function is that don’t require working with equivalence classes, but it doesn’t matter which one you pick in the sense that the resulting mathematical theory has the same mathematical consequences.
Thanks for correcting me! I changed that paragraph. Is it less offensive to people who know what they are talking about now?
It’s somewhat confusing to me; you’re using words like “set,” “space,” and “measure distances” that have mathematically precise meanings but in a way which appears to disagree with those mathematically precise meanings (I don’t know what you mean when you say that a utility function is a space). It might be helpful to non-mathematicians, though.
I mean set as in set-theory. As in the utility function is a set of equivalent functions. If I’m disagreeing with math use, please correct me. (on second thought that wording is pretty bad, so I might change it anyway. Still, are my set-intuitions wrong?)
I mean space as in a 1-dimensional space (with a non-crazy metric, if crazy metrics even exist for 1d). By “measure distance” I mean go into said space with a tape measure and see how far apart things are.
I call it a space because then when I visualize it as such, it has all the right properties (scale/shift agnosticism).
If I call it a real-valued function, I imagine the real number line, which has a labeled axis, so to speak, so it tempts me to do numerology.
You can think of a utility function as defining a measure of “signed distance” on its domain.
Utilities have some similarity to distance in physical space, in that to give coordinates to all objects you need to select some origin and system of units for your coordinate system, but the physical reality is the same regardless of your coordinate system. A member of a particular utility function’s equivalence class, can then be thought of as a function that gives the coordinates of each thing in the domain (world-states, presumably), in some particular coordinate system.
For an example, if I prefer to have three ice creams over zero, three times as much as I prefer one ice cream over zero, then we can write that as a “utility function”
u(no ice cream) = 0; u(one ice cream) = 1; u(three ice creams) = 3. In this case we have chosen arbitrarilyno ice creamas the origin of our coordinate system, and “distance betweenone ice creamandnone” as the basic unit of distance.Is this what you mean by a 1-dimensional space?
That’s exactly what I mean.
Four servings of ice cream would have me ill.
Got it. This is strictly speaking true, but “equivalence class of functions” would be a more precise way of putting it.
So there are some technical points I could go into here, but the short story is that most equivalence classes under positive affine transformations are 2-dimensional, not 1-dimensional, and also aren’t naturally endowed with a notion of distance.
I can see how distance would be trouble in 2d affine-equivalent spaces, but distance seems to me to be a sensible concept in a 1d space, even with positive-scale and shift. And utility is 1d, so it’s safe to call it a “distance” right?
Maybe you’re referring to distance-from-A-to-B not having a meaningful value without defining some unit system? Maybe we should call them “relative distances”, except that to me, “distance” already connotes relativeness.
I’m not sure what you mean by this.
This is a totally sensible point of view but disagrees with the mathematical definition. It also doesn’t apply directly to the 2-dimensional equivalence classes, as far as I can tell. For example, suppose we’re talking about utilities over two possible outcomes {heads, tails}. There are three equivalence classes here, which are u(heads) > u(tails), u(heads) = u(tails), and u(heads) < u(tails). The first and third equivalence classes are 2-dimensional. What is the distance between the two functions (u(heads) = 2, u(tails) = 1) and (u(heads) = 3, u(tails) = 2) in the first case, even in a relative sense?
Ohhhhhhh, do you mean 2d as in 2 degrees of freedom? I mean it as in spatial coordinates.
As an aside, I just realized that “displacement” is more accurate for what I’m getting at than “distance”. The thing I’m talking about can be negative.
And distance/displacement isn’t between equivalent utility functions, it’s between two outcomes in one utility function. “X is 5 tasty sandwiches better than Y” is what I’m referring to as a displacement.
And the displacement numbers will be the same for the entire equivalence class, which is why I prefer it to picking one of the equivalent functions out of a hat. If you only ever talk about measured distances, there is only one utility function in the equivalence class, because all the scales and shifts cancel out:
Was that less clear than I thought?
If there are only two points in a space, you can’t get a relative distance because there’s nothing to make the distance relative to. For that problem I would define U(heads) = 1 and U(tails) = 0, as per my dimensionless scheme.
What’s the difference?
Your use of the word “in” here disagrees with my usage of the word “utility function.” Earlier you said something like “a utility function is a space” and I defined “utility function” to mean “equivalence class of functions over outcomes,” so I thought you were referring to the equivalence class. Now it looks like you’re referring to the space of (probability distributions over) outcomes, which is a different thing. Among other things, I can talk about this space without specifying a utility function. A choice of utility function allows you to define a ternary operation on this space which I suppose could reasonably be called “relative displacement,” but it’s important to distinguish between a mathematical object and a further mathematical object you can construct from it.
Yes, it does. You seem to understand what I’m getting at.
I don’t think anyone is making mathematical errors in the actual model, we are just using different words which makes it impossible to communicate. If you dereference my words in your model, you will see errors, and likewise the other way.
Is there a resource where I could learn the correct terminology?
Yep.
My conventions for describing mathematical objects comes from a somewhat broad range of experiences and I’m not sure I could recommend a specific resource that would duplicate the effect of all of those experiences. Recommending a range of resources would entail learning much more than just a few conventions for describing mathematical objects, and you may not feel that this is a good use of your time, and I might agree. I can at least broadly indicate that some useful mathematical subjects to read up on might be real analysis and topology, although most of the content of these subjects is not directly relevant; what’s relevant is the conventions you’ll pick up for describing mathematical objects.
Sometime soon I might write a Discussion post about mathematics for rationalists which will hopefully address these and other concerns.
Upvoted for promise of Mathematics for Rationalists.
I feel confused. “a space I can measure distances in” is a strong property of a value, and it does not follow from your initial 5 axioms, and seems contrary to the 5th axiom.
In fact, your own examples given further seem to provide a counterexample—i.e., if someone prefers being a whale to 400 actual orgasms, but prefers 1⁄400 of being a whale to 1 orgasm, then both “being a whale” and “orgasm” have some utility value, but they cannot be used as units to measure distance.
If you’re in a reality where a>b and 2a<2b, then you’re not allowed to use classic arithmetic simply because some of your items look like numbers, since they don’t behave like numbers.
“Hawaii” can’t be used as a unit to measure distance, nor can “the equator”, but “the distance from Hawaii to the equator” can. Similarly, “the difference between 0 orgasms and 1 orgasm” can be used as a unit to measure utilities (you could call this unit “1 orgasm”, but that would be confusing and silly if you had nonlinear utility in orgasms: 501 orgasms could be less than or more than “1 orgasm” better than 500).
Also, did you mean to have these the other way around?: