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Linearity - a useful assumption in evaluating careers and causes

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Introduction

When analysing the good done by different paths, we’ve often found it useful to assume that the value of your resources are linear - i.e. donating $2mn is roughly twice as good as $1mn, persuading two people to support a cause is roughly twice as good as persuading one person, and so on. For more in-depth examples, see our upcoming analysis of the value of becoming a politician or this analysis of the ethics of consumption.

This assumption, however, faces a number of objections. In this post, Paul Christiano, a Research Associate at 80,000 Hours, responds to these objections to linearity, arguing that it’s normally a reasonable approximation to make.

What do I mean by “linearity”?

More precisely, the assumption is:

The value of a resource is very likely to be linear when considering changes that are a small fraction of the current supply of that resource; is very likely to be diminishing through most of the range; and is likely to be increasing only as you come to control the majority of that resource, and even then only in some cases.

In the abstract it’s not a very objectionable sounding claim, but below I go over a few common objections in particular cases.

Note that “current supply” means resources that would be used in the pursuit of similar goals. When it seems like the current supply is negligibly small, I think we are probably drawing the boundaries wrong: don’t consider money being spent on a very narrow cause, consider money being sensibly spent on improving the world, etc. In the most extreme case, where the relevant supply of resources really is tiny, then this number will still be driven up by incidentally relevant behavior by people with completely different goals.

Of course I don’t think this is an ironclad law, but in practice I rarely believe objections people make against local linearity. That said, there is lots of room for me to revise my views here.

First, I should say that linearity seems to be the right prior presumption. If we do something twice, a priori we should suppose that the second time we do it will have the same (expected) effect as the first time we do it. So I see my role here (at least with respect to linearity) as defending the prior presumption from various objections that might be raised.

Examples

Some examples that have come up recently:

  • Convincing one additional senator of something is roughly 1/2 as good as convincing two additional senators, and more than 1/k times as good as convincing k senators

  • $1M of donations is roughly 1/2 as good as $2M of donations, and more than 1/k times as good as $k M of donations

  • One additional do-gooder working on a problem probably does roughly half as much additional good as two additional do-gooders working on that problem, etc.

Why not?

Some arguments against this view (which are intended to be examples of a broader class):

  1. There are critical thresholds, such as having a majority or supermajority in a legislature; the value of crossing that threshold is much larger than the value of increments below the threshold.

  2. It is easy to imagine our current best-guess intervention running out of funding between $1M and $2M, so that the second million does much less good than the first.

  3. With a lot of funding, one can start to engage in qualitatively different projects, and spend more time (absolutely) doing research.

  4. There is sometimes a mismatch between nearly balanced resources (e.g. money and personnel) and expanding one to catch up with the other has a big effect which is hard to match.

Why?

I think that the counterarguments above apply only very rarely in practice. While it is easy to see simple cases in which these arguments would apply, those situations tend to be highly brittle. As we add uncertainty and realistic complexities, local linearity becomes a better approximation.

  1. Unlikely to be rapidly increasing or decreasing returns across the whole range. When we are considering changes which aren’t too large relative to the current stock of fungible resources (for example, < 10%), it is very hard for there to be a strong enough general trend towards increasing or diminishing returns (unless the returns are very concave or convex). For example, suppose I am considering the value of supplying $2M to a project that currently has $10M, and it seems to me that the second million is roughly half as good as the first million. If this is the general situation, then the tenth additional million would be 1/1000 as valuable, which is implausibly extreme. So most claimed instances of non-linearity rest on the existence of particular important thresholds or critical regions, rather than on a general trend.

  2. Uncertainty about status quo. Even if there are special thresholds, we are often very uncertain about what the real level of a resource is (e.g. I don’t know how much support something currently has amongst politicians, and I don’t know quite what kind of support will end up being relevant, and I don’t know how things will change over the next 10 years, and I don’t know what other similar opportunities other people will follow). Averaging over many possible situations will tend to wash out the effect of special thresholds, though it can preserve an overall trend of increasing or diminishing returns.

  3. Uncertainty about where the critical points are. If there is some critical threshold but we don’t know where it is, then the effect of that threshold gets smoothed out. across a wider range. Given that there aren’t rapidly increasing or decreasing returns across the whole range, we have to be confident to identify a particular point as one where there are rapidly increasing or decreasing returns.

  4. Many opportunities. It is easy to see increasing returns with respect to a particular opportunity, e.g. with respect to a particular charitable project or passing a particular bill, or with respect to a particular funding shortfall right now. But typically it is difficult to foresee exactly what opportunities will be most important, and the nature of future opportunities (and hence what the important thresholds are, whether thresholds that need to be reached or thresholds beyond which the marginal value drops off) is uncertain. So arguments for increasing or rapidly decreasing returns that rest on the nature of particular foreseeable opportunities tend to seem quite weak to me.

  5. Dubious detailed stories. Most sources of data about the value of resources in particular cases, even particularly solid sources like reports from organizations that could use money or capital, or the opinions of legislators, seem less robust than the general pattern of linearity. It is very easy to do the accounting wrong—to neglect indirect effects, to neglect the benefits from resources redirected to other organizations or other times, or (relatedly) to overestimate the importance of the particular problem we are working on at this instant.

  6. Trades and bargains. In some cases we have access to a bargain that lets us turn a bit of influence into a small share of a lot of influence. For example, with money, we can pursue very risky enterprises, or even put half of our money on red at the roulette table. Though less straightforward with other resources, it seems that similar trades are often possible, whether explicitly or implicitly. When a resource can be used in many different ways, it becomes increasingly unlikely that all of them exhibit the property of increasing returns, and the options with diminishing returns are particularly likely to be the best options when you have only a small amount of a resource. On top of that, given the ability to exchange resources freely, increasing returns open up the possibility for positive-sum exchanges (e.g. if there are increasing returns to political influence we strongly expect people to form coalitions or engage in tit-for-tat). This further decreases the plausibility of increasing returns unless there is some reason that such exchanges can’t occur.

Conclusion

Ultimately an assumption of linearity must be justified in each particular case. However, the general arguments in favor seem strong enough that is a safe “working assumption” while beginning to think about an area. I would be hesitant to accept a story describing substantially non-linear returns unless it was backed by a very compelling case, and in complicated situations I am happy to continue using linear returns as an approximation until I understand the domain quite well.


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