# 14

A common mistake people make with utility functions is taking individual utility numbers as meaningful, and performing operations such as adding them or doubling them.  But utility functions are only defined up to positive affine transformation.

Talking about "utils" seems like it would encourage this sort of mistake; it makes it sound like some sort of quantity of stuff, that can be meaningfully added, scaled, etc.  Now the use of a unit -- "utils" -- instead of bare real numbers does remind us that the scale we've picked is arbitrary, but it doesn't remind us that the zero we've picked is also arbitrary, and encourages such illegal operations as addition and scaling.  It suggests linear, not affine.

But there is a common everyday quantity which we ordinarily measure with an affine scale, and that's temperature.  Now, in fact, temperatures really do have an absolute zero (and if you make sufficient use natural units, they have an absolute scale, as well), but generally we measure temperature with scales that were invented before that fact was recognized.  And so while we may have Kelvins, we have "degrees Fahrenheit" or "degrees Celsius".

If you've used these scales long enough you recognize that it is meaningless to e.g. add things measured on these scales, or to multiply them by scalars.  So I think it would be a helpful cognitive reminder to say something like "degrees utility" instead of "utils", to suggest an affine scale like we use for temperature, rather than a linear scale like we use for length or time or mass.

The analogy isn't entirely perfect, because as I've mentioned above, temperature actually can be measured on a linear scale (and with sufficient use of natural units, an absolute scale); but the point is just to prompt the right style of thinking, and in everyday life we usually think of temperature as an (ordered) affine thing, like utility.

As such I recommend saying "degrees utility" instead of "utils".  If there is some other familiar quantity we also tend to use an affine scale for, perhaps an analogy with that could be used instead or as well.

# 14

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This post seems much more appropriate for the Discussion section.

Agreed. Moved.

But there is a common everyday quantity which we ordinarily measure with an affine scale, and that's temperature.

Also time (there's the Big Bang, but no-one uses it as the zero in everyday usage); for broader values of “everyday”, voltage and energy, too.

[-]satt80

This all reminds me of torsors.

That's true. People don't seem to mess those up as often as "utils". I wonder why?

Hypothesis: For energy and voltage, it's becaue these are mostly only used by people who know what they're talking about in the first place. For time, it's because we usually measure time as "12:00", etc.; the only people saying "the time is 5 seconds" are people who know what they're doing.

...except that explanation doesn't quite work, because it doesn't explain years. But then, with years we usually use a bare number... hm, this is sounding pretty contrived.

Better hypothesis: Time is familiar enough that people know not to do that, utility isn't.

People don't seem to mess those up as often as "utils".

(OTOH, people saying stuff like “X is twice as hot as Y” when X is 80 °C and Y is 40 °C aren't totally unheard of.)

An advantage of using Fahrenheit -- the zero is clearly arbitrary! :)

Another option is to replace "50 utils" or "50 degrees utility" with "50 utils". Yes, always. The wiki link would have to be updated to address the affine caveat (as well as some others) but it might be worth it.

Edit: yet another option is to explicitly include a constant: "50+C utils". This has a fine tradition stemming from calculus. If necessary, it could be combined with my previous suggestion.

[-][anonymous]40

(Moving a bit more to the concrete side for the sake of those who fall closer to the engineer perspective on the mathematician-engineer continuum):

Affine transformation == Linear transformation + translation.

It preserves ratios of distances, but not (necessarily) angles or distances themselves. Utility functions are only defined up to an affine transformation, which means that preferences are preserved, but that "doubling the utility assigned to an action" will not necessarily give the same results even if two people have the same utility function. (Just like doubling 15 degrees C == 30 degrees C, but the corresponding 59 degrees F and 86 degrees F are not doubles of each other.)

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[-]Jack30

I guess don't really see how "utils" encourages this mistake nor am I sure it is that common. I mean, the idea that most, if not all goods are not linear is 101-level stuff.

Meanwhile, "degrees of utility" is clumsier and "degree" has other meanings that don't involve scales. I am also against replacing adequate terminology used in academia with new terms unless there is very good reason. It is a bad idea to artificially increase the apparent distance between the work done here on decision theory and the work done elsewhere. There is way too much of that already. It makes for bad press and is phyg-like.

I guess don't really see how "utils" encourages this mistake nor am I sure it is that common. I mean, the idea that most, if not all goods are not linear is 101-level stuff.

I don't understand this comment. I'm assuming the linearity of a good refers to whether its utility is a linear function of how many of it you have? In that sense, this is unrelated; this is a much broader issue, having nothing to do with how the utility of something varies with having multiple of it.

Although I suppose it is related in that, if "linear good" means u(kx)=ku(x) (where here u(x) means the utility of having x of the good), then no good can be linear in that strong sense, because the equation isn't even meaningful! Edit: But, as I should have realized earlier, this is really a silly equation to consider in the first place, as it's the difference u(x)-u(0) you really care about, not u(x) itself...

It is a bad idea to artificially increase the apparent distance between the work done here on decision theory and the work done elsewhere. There is way too much of that already. It makes for bad press and is phyg-like.

I don't think this does increase the distance in any substantial way.

It's not a breaking change (like, say, putting functions on the right, or declaring electrons to be positive). It's not very-similar-but-slightly-different in a way that would cause confusion (like using tau instead of 2*pi, or using Delta(z) instead of Gamma(z+1)). It's not replacing any key term that someone would be searching for (like using "meager" instead of "first category", or "false hit" instead of "type I error", or "computably enumerable" instead of "recursively enumerable"). It is a direct translation, of a term that people won't be searching for and isn't even strictly necessary, in a way that's quickly transparent and nearly self-explanatory. I am honestly having trouble imagining a less obtrusive change. So I don't think this is putting any substantial distance there, let alone approaching phyg status.

I'm worried "degrees utility" could encourage the conflation of the physical quantity 'how closely an event corresponds to a set of individuals' preferences' with the metric we select for measuring that quantity. We don't say 'degrees temperature'; Fahrenheit and Celsius are specific ways of measuring temperature, whereas I gather you don't have in mind a specific utility metric.

Indeed. It's an improvement over "utils", though, which has the same problem and also suggests linearity. I'm not sure what to do to fix this problem, but I'm also not sure it's that important (it seems pretty clear that we have to measure it somehow, after all).

Indeed. It's an improvement over "utils", though, which has the same problem and also suggests linearity.

I am reluctant to accept a terminology change to something that is broken, even if the current terminology is broken as well. Accepting such incomplete solutions serves to reduce the incentive to come up with an actual workable fix to the problem and gives people the illusion that they have something that is solved.

"Degrees Utility" is not analogous to "Degrees Fahrenheit" or "Degrees Celsius". When 34 degrees Fahrenheit is compared to 54 degrees Fahrenheit it is correct (and meaningful) to say that the latter is hotter than the former. When, using your terminology, "34 degrees Utility" is compared to "54 degrees Utility" the result is not meaningful even though it sometimes should be. For example when looking at a payoff matrix for a game involving agent A and agent B the 54 degrees Utility that B gets in some outcome cannot be compared meaningfully to "34 degrees Utility" that A gets in an outcome but can be compared to the "34 degrees Utility" that B gets in a different outcome (with the result "better"). That's just sloppy expression with the illusion of rigour.

"34 DegreesUtility" would be viable but that sort of parametrised nomenclature is not sufficiently high status to reliably enforce as a standard just now.

...actually, now that I think about it some more, I agree that there is something to your line of thinking; I'm just not certain it leads to the conclusion you suggest.

The problem is that we don't have any way of talking about this that intuitively prompts how it actually works, and "degrees utility" is problematic because it suggests it accounts for all the problems. OK. However, the thing is, so does "utils". I mean, it's possible that people see that and know to tread carefully; I don't have any data here. I just feel like I've seen people try to add 1 util and 1 util often enough that I suspect that that isn't the case, and that most people do read "utils" as indicating that it is correct to treat it as an amount of stuff.

But perhaps reverting to an even worse solution would suggest to tread carefully -- namely, bare numbers. Again, this is pure speculation, I have no data; but I get the feeling that bare numbers will raise people's hackles more than "utils". Bare numbers suggest "something's been left out here; tread carefully"; using a unit suggests "yes, this is a sensible way to measure it."

So, if I'm correct about that, "utils" actually seems like the worst suggestion of the three -- compared to "degrees utility", it's more misleading, but doesn't come with an additional warning sign; compared to a bare number, it lacks the obvious warning sign, and isn't that much more misleading. (Because adding and scaling will be the most tempting meaningless things to do anyway; multiplication seems a bit more exotic...)

Again: "Utils" has all the same problems, and more. For a single agent, the comparison is meaningful.

If you prefer sticking to stick with the existing terminology despite it suggesting even worse meaningless comparisons, OK, but don't act like you are pointing out anything that isn't obvious, or that is specific to my suggestion.

“degrees util?” (or if it has to be named after a person, “degrees von Neumann” or “degrees Morgenstein” depending on whether you're closer to Hungary or Germany).

[-]ygert-10

Maybe "degrees VNM"?

Thought-provoking indeed. I agree with you that the scale we've picked for utils is arbitrary, and the zero we've picked is also arbitrary. After reading through the comments, I begin to wonder if we should go farther. We talk about utils as if this is a quantity we can actually measure, but is that true? Are we measuring anything at all? Is a numeric measure of any kind at all helpful here?

Let me propose a situation: Given a choice between beer and steak, John chooses the steak. Given a choice between steak and ice cream, John chooses the ice cream. Given a choice between ice cream and beer, John chooses the beer. Which item has the highest utility to John? There's just no way to make sense of that in terms of real-valued utils because real numbers are transitive, and utility doesn't have to be.

If utils do make sense, I ask someone to produce an actual means of measuring them, fuzzy and approximate thought it may be. I can't figure out any mathematically consistent way to do this that doesn't resolve to some other more easily measured quantity such as money or dopamine levels or quality-adjusted life years (QALYs). And if one of those is what we're measuring, then we should probably just go ahead and say so.

In fact, in different problems we're likely to want different kinds of utility. Sometimes a problem is best understood in terms of money. In others, it's better understood in QALYs, and money may be not the measure but rather the constraint. That is, given that we have X dollars to work with, how can we maximize QALYs?

Bu if we just use abstract "utils" or "utilons" without connecting those to something we can measure in the non-hypothetical world, I'm not sure we get any useful information that applies outside of an axiomatic system that may not model reality.

Given a choice between beer and steak, John chooses the steak. Given a choice between steak and ice cream, John chooses the ice cream. Given a choice between ice cream and beer, John chooses the beer.

Does this really happen? Can money be pumped out? For example, offer John the opportunity to pay \$0.05 to upgrade his beer for a steak, then \$0.05 to upgrade that to an ice cream cone, then \$0.05 to upgrade that to a beer. Run forever. I think if you actually did this you'd find that of the three there actually is one that John would prefer to have.

[-]asr20

Yes, real human preference is routinely intransitive. That's a topic perhaps worth its own discussion.

The parent post is still relevant, though. Inasmuch as utilitarian calculations are a useful approximation to human decisionmaking, it's worth reminding people here that utilities don't have a natural magnitude scale and that there's no natural way to compare them across agents.

But utility functions are only defined up to positive affine transformation.

As far as I understand, it's worse than that, linearity is not required, any (continuous) monotonic rescaling would do, since the only thing that needs to be preserved is the ranking of outcomes.

Linearity is required... what's preserved is the ranking of lotteries over outcomes. Preserving the order of "a cookie" and "two cookies but no dollar" and "three cookies but a dollar in debt" isn't enough, you also have to preserve "40% chance of a cookie and 60% chance of two cookies but no dollar".

[-]Benya150

There may be some confusion over terms, because economists do in fact also have use for utility functions that only express an ordering of outcomes. (Incidentally, this is also true of some of the decision theory work that has appeared on LessWrong: the utility functions in our proof-based versions of UDT only express an ordering; these models don't have a notion of probabilities at all.) The OP and the parent comment are about the utility functions given by the von Neumann-Morgenstern theorem; these are left invariant by any affine rescaling and (by the uniqueness part of the theorem) are changed by any non-affine rescaling.

The OP and the parent comment are about the utility functions given by the von Neumann-Morgenstern theorem; these are left invariant by any affine rescaling and (by the uniqueness part of the theorem) are changed by any non-affine rescaling.

It's worth mentioning that all three kinds of utility functions can be constructed: ordinal scale, interval scale, and ratio scale. For an overview of ratio scale utility functions, see Peterson (2009), pp. 106-110.

Yes, to be absolutely clear, I'm talking about the sort of utility functions you get from the VNM theorem or Savage's Theorem.

It's not really clear to me what the use is for a utility function if all you have is ordering; why not just use an ordering? Seems that using a utility function then would just be needlessly restricting what sort of orderings you can have. Well, depending on what requirements you want that ordering to satisfy... after all if you have all of Savage's axioms then you do get a utility function! But that requires ordering actions, not just outcomes...

The paradigmatic economic application I recall is consumer choice theory: You have a certain amount of money, `m`, and two goods you can buy. These goods have fixed prices `p` and `q`. Your choices are pairs (`x`,`y`) saying how much of each good you buy; the "feasible set" of choices is `{(x,y) : x,y >= 0 and xp + yq <= m}`. What's your best choice in this set? We want to use calculus to solve this, so we'll express your preferences as a differentiable utility function. The reasons VNM or Savage doesn't enter into it is that actions lead to outcomes deterministically.

In UDT, we don't even start with a natural definition of "outcome"; in principle, we need to specify (1) a set of outcomes, (2) an ordering on these outcomes, and (3) a deterministic, no-input program which does some complicated computation and returns one of these outcomes. (The intuition is that the complicated computation computes everything that happens in our universe, then picks out the morally relevant parts and prints them out.) It's just simpler to skip parts (1) and (2) in the formal specification and say that the program (3) returns a number. Since the proof-based models have no notion of probability (even implicitly like in Savage's theorem), this makes the program an order-only "utility function."

Yes, but "we want to use calculus to solve this" isn't a very natural constraint on the set of orderings. :) It's a "we want to make the math easier" constraint, not a "we have reason to believe that any rational agent should act this way" constraint.

Not that it's necessarily inappropriate in the example you give -- it probably makes sense there. Just a bit surprising that UDT would restrict itself in such a way.

In the UDT case, the set of outcomes is finite (well, or at least the set of equivalence classes of outcomes under the preference relation is finite) and the utility functions don't have any particular properties, so every possible preference relation the model can treat at all can be represented by a utility function!

(I should note that this is not UDT as such we're talking about here, but one particular formal way of implementing some of the ideas of UDT.)

Oh, OK then!

No, you don't. Risk-aversion is legal.

[-][anonymous]00

Hm. I initially posted this in Discussion before quickly moving it to Main, and now it's not showing up in the "recent posts" sidebar. Do I just have to wait or is this a bug?

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