Pinpointing Utility


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Following Morality is Awesome. Related: Logical Pinpointing, VNM.

The eternal question, with a quantitative edge: A wizard has turned you into a whale, how awesome is this?

"10.3 Awesomes"

Meditate on this: What does that mean? Does that mean it's desirable? What does that tell us about how awesome it is to be turned into a whale? Explain. Take a crack at it for real. What does it mean for something to be labeled as a certain amount of "awesome" or "good" or "utility"?

What is This Utility Stuff?

Most of agree that the VNM axioms are reasonable, and that they imply that we should be maximizing this stuff called "expected utility". We know that expectation is just a weighted average, but what's this "utility" stuff?

Well, to start with, it's a logical concept, which means we need to pin it down with the axioms that define it. For the moment, I'm going to conflate utility and expected utility for simplicity's sake. Bear with me. Here are the conditions that are necessary and sufficient to be talking about utility:

  1. Utility can be represented as a single real number.
  2. Each outcome has a utility.
  3. The utility of a probability distribution over outcomes is the expected utility.
  4. The action that results in the highest utility is preferred.
  5. No other operations are defined.

I hope that wasn't too esoteric. The rest of this post will be explaining the implications of those statements. Let's see how they apply to the awesomeness of being turned into a whale:

  1. "10.3 Awesomes" is a real number.
  2. We are talking about the outcome where "A wizard has turned you into a whale".
  3. There are no other outcomes to aggregate with, but that's OK.
  4. There are no actions under consideration, but that's OK.
  5. Oh. Not even taking the value?

Note 5 especially. You can probably look at the number without causing trouble, but if you try to treat it as meaningful for something other than condition 3 and 4, even accidentally, that's a type error.

Unfortunately, you do not have a finicky compiler that will halt and warn you if you break the rules. Instead, your error will be silently ignored, and you will go on, blissfully unaware that the invariants in your decision system no longer pinpoint VNM utility. (Uh oh.)

Unshielded Utilities, and Cautions for Utility-Users

Let's imagine that utilities are radioactive; If we are careful with out containment procedures, we can safely combine and compare them, but if we interact with an unshielded utility, it's over, we've committed a type error.

To even get a utility to manifest itself in this plane, we have to do a little ritual. We have to take the ratio between two utility differences. For example, if we want to get a number for the utility of being turned into a whale for a day, we might take the difference between that scenario and what we would otherwise expect to do, and then take the ratio between that difference and the difference between a normal day and a day where we also get a tasty sandwich. (Make sure you take the absolute value of your unit, or you will reverse your utility function, which is a bad idea.)

So the form that the utility of being a whale manifests as might be "500 tasty sandwiches better than a normal day". We have chosen "a normal day" for our datum, and "tasty sandwiches" for our units. Of course we could have just as easily chosen something else, like "being turned into a whale" as our datum, and "orgasms" for our units. Then it would be "0 orgasms better than being turned into a whale", and a normal day would be "-400 orgasms from the whale-day".

You say: "But you shouldn't define your utility like that, because then you are experiencing huge disutility in the normal case."

Wrong, and radiation poisoning, and type error. You tried to "experience" a utility, which is not in the defined operations. Also, you looked directly at the value of an unshielded utility (also known as numerology).

We summoned the utilities into the real numbers, but they are still utilities, and we still can only compare and aggregate them. The summoning only gives us a number that we can numerically do those operations on, which is why we did it. This is the same situation as time, position, velocity, etc, where we have to select units and datums to get actual quantities that mathematically behave like their ideal counterparts.

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 an equivalence class of utility functions with numbers coming out, which don't agree on the actual numbers, but can be made to agree with a linear transform, rather than making me think of a utility function as a space I can measure distances in. I'm an engineer, not a mathematician, so I find it much more intuitive and less confusing to think of it in terms of units and datums, even though it's basically the same thing. 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.

So what does it mean that being a whale for a day is 400 orgasms better than a normal day? Does it mean I would prefer 400 orgasms to a day as a whale? Nope. Orgasms don't add up like that; I'd probably be quite tired of it by 15. (remember that "orgasms" were defined as the difference between a day without an orgasm and a day with one, not as the utility of a marginal orgasm in general.) What it means is that I'd be indifferent between a normal day with a 1/400 chance of being a whale, and a normal day with guaranteed extra orgasm.

That is, utilities are fundamentally about how your preferences react to uncertainty. For example, You don't have to think that each marginal year of life is as valuable as the last, if you don't think you should take a gamble that will double your remaining lifespan with 60% certainty and kill you otherwise. After all, all that such a utility assignment even means is that you would take such a gamble. In the words of VNM:

We have practically defined numerical utility as being that thing for which the calculus of mathematical expectations is legitimate.

But suppose there are very good arguments that have nothing to do with uncertainty for why you should value each marginal life-year as much as the last. What then?

Well, "what then" is that we spend a few weeks in the hospital dying of radiation poisoning, because we tried to interact with an unshielded utility again (utilities are radioactive, remember? The specific error is that we tried to manipulate the utility function with something other than comparison and aggregation. Touching a utility directly is just as much an error as observing it directly.

But if the only way to define your utility function is with thought experiments about what gambles you would take, and the only use for it is deciding what gambles you would take, then isn't it doing no work as a concept?

The answer is no, but this is a good question because it gets us closer to what exactly this utility function stuff is about. The utility of utility is that defining how you would behave in one gamble puts a constraint on how you would behave in some other related gambles. As with all math, we put in some known facts, and then use the rules to derive some interesting but unknown facts.

For example, if we have decided that we would be indifferent between a tasty sandwich and a 1/500 chance of being a whale for tomorrow, and that we'd be indifferent between a tasty sandwich and a 30% chance of sun instead of the usual rain, then we should also be indifferent between a certain sunny day and a 1/150 chance of being a whale.

Monolithicness and Marginal (In)Dependence

If you are really paying attention, you may be a bit confused, because it seems to you that money or time or some other consumable resource can force you to assign utilities even if there is no uncertainty in the system. That issue is complex enough to deserve its own post, so I'd like to delay it for now.

Part of the solution is that as we defined them, utilities are monolithic. This is the implication of "each outcome has a utility". What this means is that you can't add and recombine utilities by decomposing and recombining outcomes. Being specific, you can't take a marginal whale from one outcome and staple it onto another outcome, and expect the marginal utilities to be the same. For example, maybe the other outcome has no oceans for your marginal whale.

For a bigger example, what we have said so far about the relative value of sandwiches and sunny days and whale-days does not necessarily imply that we are indifferent between a 1/250 chance of being a whale and any of the following:

  • A day with two tasty sandwiches. (Remember that a tasty sandwich was defined as a specific difference, not a marginal sandwich in general, which has no reason to have a consistent marginal value.)

  • A day with a 30% chance of sun and a certain tasty sandwich. (Maybe the tasty sandwich and the sun at the same time is horrifying for some reason. Maybe someone drilled into you as a child that "bread in the sun" was bad bad bad.)

  • etc. You get the idea. Utilities are monolithic and fundamentally associated with particular outcomes, not marginal outcome-pieces.

However, as in probability theory, where each possible outcome technically has its very own probability, in practice it is useful to talk about a concept of independence.

So for example, even though the axioms don't guarantee in general that it will ever be the case, it may work out in practice that given some conditions, like there being nothing special about bread in the sun, and my happiness not being near saturation, the utility of a marginal tasty sandwich is independent of a marginal sunny day, meaning that sun+sandwich is as much better than just sun as just a sandwich is better than baseline, ultimately meaning that I am indifferent between {50%: sunny+sandwich; 50% baseline} and {50%: sunny; 50%: sandwich}, and other such bets. (We need a better solution for rendering probability distributions in prose).

Notice that the independence of marginal utilities can depend on conditions and that independence is with respect to some other variable, not a general property. The utility of a marginal tasty sandwich is not independent of whether I am hungry, for example.

There is a lot more to this independence thing (and linearity, and risk aversion, and so on), so it deserves its own post. For now, the point is that the monolithicness thing is fundamental, but in practice we can sometimes look inside the black box and talk about independent marginal utilities.

Dimensionless Utility

I liked this quote from the comments of Morality is Awesome:

Morality needs a concept of awfulness as well as awesomeness. In the depths of hell, good things are not an option and therefore not a consideration, but there are still choices to be made.

Let's develop that second sentence a bit more. If all your options suck, what do you do? You still have to choose. So let's imagine we are in the depths of hell and see what our theories have to say about it:

Day 78045. Satan has presented me with three options:

  1. Go on a date with Satan Himself. This will involve romantically torturing souls together, subtly steering mortals towards self-destruction, watching people get thrown into the lake of fire, and some very unsafe, very nonconsensual sex with the Adversary himself.

  2. Paperclip the universe.

  3. Satan's court wizard will turn me into a whale and release me into the lake of fire, to roast slowly for the next month, kept alive by twisted black magic.

Wat do?

They all seem pretty bad, but "pretty bad" is not a utility. We could quantify paperclipping as a couple hundred billion lives lost. Being a whale in the lake of fire would be awful, but a bounded sort of awful. A month of endless horrible torture. The "date" is having to be on the giving end of what would more or less happen anyway, and then getting savaged by Satan. Still none of these are utilities.

Coming up with actual utility numbers for these in terms of tasty sandwiches and normal days is hard; it would be like measuring the microkelvin temperatures of your physics experiment with a Fahrenheit kitchen thermometer; in principle it might work, but it isn't the best tool for the job. Instead, we'll use a different scheme this time.

Engineers (and physicists?) sometimes transform problems into a dimensionless form that removes all redundant information from the problem. For example, for a heat conduction problem, we might define an isomorphic dimensionless temperature so that real temperatures between 78 and 305 C become dimensionless temperatures between 0 and 1. Transforming a problem into dimensionless form is nearly always helpful, often in really surprising ways. We can do this with utility too.

Back to depths of hell. The date with Satan is clearly the best option, so it gets dimensionless utility 1. The paperclipper gets 0. On that scale, I'd say roasting in the lake of fire is like 0.999 or so, but that might just be scope insensitivity. We'll take it for now.

The advantages with this approach are:

  1. The numbers are more intuitive. -5e12 QALYs, -1 QALY, and -50 QALYs from a normal day, or the equivalent in tasty sandwiches, just doesn't have the same feeling of clarity as 0, 1 and .999. (For me at least. And yes I know those numbers don't quite match.)

  2. Not having to relate the problem quantities to far-away datums or drastically misappropriate units (tasty sandwiches for this problem) makes the numbers easier and more direct to come up with. Also we have to come up with less of them. The problem is self-contained.

  3. If defined right, the connection between probability and utility becomes extra-clear. For example: What chance between a Satan-date and a paperclipper would make me indifferent with a lake-of-fire-whale-month? 0.999! Unitless magic!

  4. All confusing redundant information (like negative signs) are removed, which makes it harder to accidentally do numerology or commit a type error.

  5. All redundant information is removed, which means you find many more similarities between problems. The value of this in general cannot be understated. Just look at the generalizations made about Reynolds number! "[vortex shedding] occurs for any fluid, size, and speed, provided that Re between ~40 and 10^3". What! You can just say that in general? Magic! I haven't actually done enough utility problems to know that we'll find stuff like that but I trust dimensionless form.

Anyways, it seems that going on that date is what I ought to do. So did we need a concept of awfulness? Did it matter that all the options sucked? Nope; the decision was isomorphic in every way to choosing lunch between a BLT, a turkey club, and a handful of dirt.

There are some assumptions in that lunch bit, and it's worth discussing. It seems counterintuitive or even wrong, to say that your decision-process faced with lunch should be the same as when faced with a decision in involving torture, rape, and paperclips. The latter seems somehow more important. Where does that come from? Is it right?

This may deserve a bigger discussion, but basically, if you have finite resources (thought-power, money, energy, stress) that are conserved or even related across decisions, you get coupling of "different" decisions in a way that we didn't have here. Your intuitions are calibrated for that case. Once you have decoupled the decision by coming up with the actual candidate options. The depths-of-hell decision and the lunch decision really are totally isomorphic. I'll probably address this properly later, if I discuss instrumental utility of resources.

Anyways, once you put the problem in dimensionless form, a lot of decisions that seemed very different become almost the same, and a lot of details that seemed important or confusing just disappear. Bask in the clarifying power of a good abstraction.

Utility is Personal

So far we haven't touched the issue of interpersonal utility. That's because that topic isn't actually about VNM utility! There was nothing in the axioms above about there being a utility for each {person, outcome} pair, only for each outcome.

It turns out that if you try to compare utilities between agents, you have to touch unshielded utilities, which means you get radiation poisoning and go to type-theory hell. Don't try it.

And yet, it seems like we ought to care about what others prefer, and not just our own self-interest. But it seems like that inside the utility function, in moral philosophy, not out here in decision theory.

VNM has nothing to say on the issue of utilitarianism besides the usual preference-uncertainty interaction constraints, because VNM is about the preferences of a single agent. If that single agent cares about the preferences of other agents, that goes inside the utility function.

Conversely, because VNM utility is out here, axiomized for the sovereign preferences of a single agent, we don't much expect it to show up in there, in a discussion if utilitarian preference aggregation. In fact, if we do encounter it in there, it's probably a sign of a failed abstraction.

Living with Utility

Let's go back to how much work utility does as a concept. I've spent the last few sections hammering on the work that utility does not do, so you may ask "It's nice that utility theory can constrain our bets a bit, but do I really have to define my utility function by pinning down the relative utilities of every single possible outcome?".

Sort of. You can take shortcuts. We can, for example, wonder all at once whether, for all possible worlds where such is possible, you are indifferent between saving n lives and {50%: saving 2*n; 50%: saving 0}.

If that seems reasonable and doesn't break in any case you can think of, you might keep it around as heuristic in your ad-hoc utility function. But then maybe you find a counterexample where you don't actually prefer the implications of such a rule. So you have to refine it a bit to respond to this new argument. This is OK; the math doesn't want you to do things you don't want to.

So you can save a lot of small thought experiments by doing the right big ones, like above, but the more sweeping of a generalization you make, the more probable it is that it contains an error. In fact, conceptspace is pretty huge, so trying to construct a utility function without inside information is going to take a while no matter how you approach it. Something like disassembling the algorithms that produce your intuitions would be much more efficient, but that's probably beyond science right now.

In any case, in the current term before we figure out how to formally reason the whole thing out in advance, we have to get by with some good heuristics and our current intuitions with a pinch of last minute sanity checking against the VNM rules. Ugly, but better than nothing.

The whole project is made quite a bit harder in that we are not just trying to reconstruct an explicit utility function from revealed preference; we are trying to construct a utility function for a system that doesn't even currently have consistent preferences.

At some point, either the concept of utility isn't really improving our decisions, or it will come in conflict with our intuitive preferences. In some cases it's obvious how to resolve the conflict, in others, not so much.

But if VNM contradicts our current preferences, why do we think it's a good idea at all? Surely it's not wise to be tampering with our very values?

The reason we like VNM is that we have a strong meta-intuition that our preferences ought to be internally consistent, and VNM seems to be the only way to satisfy that. But it's good to remember that this is just another intuition, to be weighed against the rest. Are we ironing out garbage inconsistencies, or losing valuable information?

At this point I'm dangerously out of my depth. As far as I can tell, the great project of moral philosophy is an adult problem, not suited for mere mortals like me. Besides, I've rambled long enough.

Conclusions

What a slog! Let's review:

  • Maximize expected utility, where utility is just an encoding of your preferences that ensures a sane reaction to uncertainty.

  • Don't try to do anything else with utilities, or demons may fly out of your nose. This especially includes looking at the sign or magnitude, and comparing between agents. I call these things "numerology" or "interacting with an unshielded utility".

  • The default for utilities is that utilities are monolithic and inseparable from the entire outcome they are associated with. It takes special structure in your utility function to be able to talk about the marginal utility of something independently of particular outcomes.

  • We have to use the difference-and-ratio ritual to summon the utilities into the real numbers. Record utilities using explicit units and datum, and use dimensionless form for your calculations, which will make many things much clearer and more robust.

  • If you use a VNM basis, you don't need a concept of awfulness, just awesomeness.

  • If you want to do philosophy about the shape of your utility function, make sure you phrase it in terms of lotteries, because that's what utility is about.

  • The desire to use VNM is just another moral intuition in the great project of moral philosophy. It is conceivable that you will have to throw it out if it causes too much trouble.

  • VNM says nothing about your utility function. Consequentialism, hedonism, utilitarianism, etc are up to you.