Narrow your answer space

Mark Rosewater, a designer for Magic: The Gathering, writes a lot about how “restrictions breed creativity.” The explanation he gives is simple: when someone is building a house, the more tools they have, the better off they are. But when someone is looking for something, the more space they have to explore, the worse off they are. This applies to answer space: the more narrowly defined your problem is, the easier it is to search an answers; you’ll find both more answers and better answers by looking in a well-chosen smaller space. Oftentimes the hardest problems to find good answers for are the ones with the widest scope.1

Most problems require some sort of creative thinking to overcome, and perhaps the greatest gains from this method come from applying it to your life goals. Imagine someone with a simple goal:2 they want to improve themselves. That’s admirable, but sort of bland and massively broad. It would be helpful to have a way to work from a bland, broad goal to a better goal- but what’s better, in this context? We know that restrictions help, but what sort of restrictions help the most?

Choosing Restrictions

In Getting Things Done, David Allen argues that to-do lists should only include ‘tasks.’ That is, only write down clearly identified next actions towards achieve specific goals. “Call Adam” isn’t a task, but “Call Adam about hotel reservations for the conference” is. This serves to reduce mental load (once you’ve written down the second, you can remove the task entirely from your mind, while you still need to keep a lot in memory for the first), to reduce the need to plan while doing, and to reduce the ugh field associated with getting started. A good place for a goal to be, then, is a place where looking at the goal causes you to imagine the next task, even if you lost the to-do list where you had written down the next task and then purposefully forgotten it. So, actionable is a restriction that helps (even if the action is “wait for X,” it’s a good idea to know what X is, so you can look out for it!).

But at the same time, it helps when our goals are a sentence or a paragraph long, rather than a list of every subgoal and task. They should be kept simple, for the sake of both communication and flexibility. Finally, to encapsulate what’s useful about restricting creativity in general, it should be specific. There are many actionable goals which present too many possible actions, and so we choose at random or do nothing at all.

When talking about plans and goals, David Allen uses a plane’s eye view analogy: goals are 0, 10k, 20k, 30k, 40k, or 50k feet above the ground. I prefer a math analogy- goals are worked out to 0th, 1st, 2nd, 3rd, and 4th order (and even further if necessary).3 Allen’s analogy and mine work in opposite directions, and it’s worthwhile to point out why. Allen’s primary focus is (unsurprisingly) getting things done, and that happens at the task level. Traveling upwards is done to zoom out and obtain information, not to do work while in the clouds. A good visual analogy for my approach is a tree’s root burrowing into the ground. At each spot, the root has a choice of where to go, and the point is to be there and soak up nutrients. The root also isn’t traveling but extending- it still exists everywhere it was before. Allen is happy with a satellite photo, but I need a pipeline.

Refining a Goal

When we take a 0th order goal, like “I want to improve,” there are a pretty large number of ways we could make it more specific, and a staggering (literally) number of potential actions we could take to work towards that goal. We think about our options, and settle on “I want to be cleverer.” We still want to improve- we’ve just outlined a way to do so. But we’ve also discarded most of the ways we could improve! This is a valuable thing because it narrows our answer space. We could also say it constrains our expectations and our efforts; so we upgrade that goal to 1st order.

Allen’s analogy is robust because it has a strong anchor: the next task to do is at ground level. There is no strong anchor for a 0th order goal, just a heuristic about how to rank goals. We could have started off with “I want to be cleverer” as a 0th order goal, and the rest of this example would work out exactly the same- except with slightly different numbers. So don’t focus on the numbers as much as the relationships between them and the changes in answer space.

A 1st order goal, while specific, is generally still not actionable. Here is where it’s important to keep refining the goal instead of being seduced into working. The first thing you can think of to make yourself cleverer is probably not the best thing you can do to make yourself cleverer. Again, our root pushes into the ground, seeking out nutrients, and we select an aspect of cleverness to focus on: “I want to make better decisions.” The thoughts produced by this 2nd order formulation are starting to become fertile, but we can do better.

Stop when Satisfied

A clarifying question—“what do we mean by better?”—gives us a 3rd order goal: “I want to have a solid idea of how good a decision is.” We could keep refining the goal endlessly, but at some point we have to stop planning and start producing. When is a good time to do this, given that you can only compare the present and past, not present and future? We don’t have the assurance that this is a well-behaved problem where each additional step will change our answer space less than the previous step did, so we need to be cleverer about this choice than normal. One approach is the scale of resources involved- if you haven’t gotten to the point where you could reasonably expect success with the resources you have to throw at the problem, keep drilling down until you’ve reached that point. If you’re trying to decide on what your life’s work should be, drill down until you’ve got a problem you can do significant work on in 20 years; don’t stop when you hit a goal that would take fifty people fifty years.

Another way is to look at how the goals intersect with each other- it seems like our root has curled in a different way moving from “better decisions” to “understand decision quality” than in its previous extensions- it seems like if we don’t understand the problem of measuring a decision’s quality, any other improvements we make can’t succeed at “make better decisions” because we can’t tell if the new decisions are higher quality than the old decisions! Beforehand, we were selecting from independent specializations. Now we’re looking at a necessary subgoal instead of a related goal. Looking at this another way, we’ve been choosing more and more specific terminal values and have come across our first instrumental value.4

That suggests we’ve got enough to stop deciding what goal to pursue and start actually pursuing it. When you stop your goal-selection because of a fork like this, it’s a good idea to look at what other goals are on the same order: while you should work on 3rd order problems before 4th order problems, problems of the same order are roughly equally important, and you may find you want to work on a different one or you can work on multiple of them in parallel.

Note that even though we’ve make the decision to stop planning and start producing, we’re probably going to run into some 4th order problems. Goals often have subgoals and instrumental values often have instrumental values based off them; the same methodology will work at every level and often represents a much faster way to search through answer space than brute force (especially since it’s typically very hard to force your brain to brute force massive problems). Oftentimes there will be a domain-specific response which is more appropriate than this method, though (or, at least, resembles this method only in the abstract).

Carve at the Joints

One thing I have barely mentioned but is of crucial importance is that you need significant knowledge to effectively narrow down the answer space you’re considering. Consider an international corporation trying to create a human resources department. Their 0th order goal might be something like “make higher profits,” their 1st order goal is “streamline corporate functions to reduce cost without significantly reducing revenue,” and their 2nd order goal might be “task an entity with managing hiring, pay, benefits, and employee relations.” Now they have a lot of 3rd order goals to choose from, and they decide “create an HR office in each department.” After all, they’ve already got their corporation partitioned that way, and having one HR department for R&D and another for Sales will mean that the hiring expertise of each HR department is much better because of specialization.

But this misses the reality of HR departments, which is that their functions are strongly tied to the nation that employees live and work in. The R&D HR office might find itself having to deal with ten different sets of tax laws, requiring ten different tax specialists. Hiring laws in one country might require one procedure, while in another country they’re totally different. The benefit of increased hiring specialization might not be unique to this plan- due to interviewing, travel costs, and legal changes, this setup probably requires one hiring officer for each department for each country, as well as a tax specialist for each country for each department. But if you split up the HR departments by country instead of by department, you would only need one tax specialist for each country, and the same number of hiring officers.

There are three things to be learned from that example: first, hold off on proposing solutions (sound familiar?). Second, don’t be afraid to go back upwards and reevaluate your choice of goals. By choosing, you discarded a lot of answer space; if you don’t find promising things in the region you looked, you should look somewhere else.

Most importantly, we learn that the solution to a problem is often another problem. The answer we picked to “I want to improve” is “I want to get cleverer,” and we can think better and faster5 if we treat that as a full answer. After all, if you reduce the answer space of “a sentence 100 letters long” to “a sentence 10 letters long,” you have reduced it by a larger factor than reducing from “a sentence 10 letters long” to a specific sentence that is 10 letters long.6

1. My artist friends tell me that their least favorite commissions are the ones where the commissioner tells them “do whatever you want!”; I don’t think I’ve seen someone in any field ever speak positively of getting that regularly (instead of as an occasional reprieve).

2. Style note: I use ‘goal’, ‘problem’, and ‘value’ interchangeably throughout this post, based on whatever seems appropriate for that sentence. I hope this isn’t too confusing- I think there’s only type errors for values, and so when you see value recast it as “a goal to obtain this value.”

3. In physics (and many other disciplines), unsoluble problems are often approximated by an infinite number of soluble problems. For example, one can calculate sin(x) with only multiplication, division, addition, and subtraction by using the Taylor Series approximation. However, by itself this is just moving around the difficulty- your new problems are individually soluble but you don’t have the time to solve an infinite number of them. This method is effective only when you can ignore later terms- that is, take the infinite amount of trash you’ve generated and manage to throw it away in a finite volume. For example, to calculate sin(1) to three parts in a thousand requires only the first three terms: 1-13/​3!+15/​5!=.841667 while sin(1) is .841471 (both rounded to 6 digits). For well-behaved approximations, the error is smaller than the next additional term- for sin(1) with 3 terms, the error is 1.96e-4 while the next term is 17/​7!=1.98e-4. My usage of “order” is inspired by this background; a first guess at a problem (like answering 1 to sin(1)) is a first order solution that’s in the right ballpark but is probably missing crucial details. A second order solution has the most obvious modification to the first order solution and is generally rather good (5/​6 only differs from sin(1) by 1%). One note here is this implies that for well-behaved problems, one needs to do all of the nth order modifications before moving to the n+1th order- if I just give you 1-17/​7!, my answer is not really any better than my 1st order answer (and if I gave you 1+15/​5!, it would be worse).

4. The usefulness of wording things this way is limited because the boundary between the two is hard to determine. “I want to make better decisions” could easily be an instrumental value to a rather different problem (“I want to be more powerful,” say) or you could interpret it as an instrumental value for the previous value (“I want to be cleverer”). So it might actually be that you’re looking to find a narrow goal ‘as terminal as the original vague goal’ that provokes instrumental subgoals.

5. Typically, when you make a computation faster you sacrifice some accuracy. This may be one of the cases where that often isn’t true, because the computation time is infinite and thus accuracy is 0 for problems you cannot fit into memory if you try to solve them in one go. But the heuristics you use to narrow answer space can easily be bad heuristics; it helps to make this process formal so you’re more likely to notice when you jumped an order without actually checking for other ways to approach the problem. Perhaps the best advice in this article is “don’t be afraid to go back and recalculate at lower orders and make sure you’re in the right part of the tree.”

6. While it’s tempting to suggest a measure like “log(possible answers)”, that breaks down in many cases (when approaching “find the real number that is pi5“, you don’t see a change if you go from “all reals” to “all reals between 35 and 45” as possible answers) and isn’t valuable in others (if I reduce the answer space from 1,000 potential answers to 100 potential answers, but the real answer is in that 100, I’ve done better than if I reduce the answer space to 10 potential answer but the real answer isn’t in that 10). The density of good solutions matters- you can only profit by throwing away parts of the answer space because their average is lower than the part you kept.

Thanks to Aharon for the prodding to turn this from a brief mention in a comment to a post of its own, and to PhilGoetz, DSimon, and XFrequentist for organizational advice.