Sunday, May 21, 2017

Aggregating the Right Moments

Should we prefer to give one person half a million minutes (i.e. one year) more life, or to give a million people one minute more each?  If iterated a million times over (once for each person in the million), the latter repeated choice is clearly better for all (by half a million minutes).  Moreover, as I suggested in comments to that post, if we assume that the million choices are independent of each other in value -- that is, the value of making one such choice does not depend on how the other choices are made -- then it quickly follows that it's better to give the million tiny benefits rather than the one big benefit, even in a one-off choice situation.

However, it's worth flagging that on one very natural (but philosophically distorting) way of imagining the situation, the independence assumption will not hold.

It's natural to imagine bestowing the extra gift of life to someone on their deathbed.  But then giving them one minute of time is not actually worth 1/500k of 500k extra minutes, since a minute spent on your deathbed is likely to be pretty worthless, i.e. disproportionately bad compared to the average minute in your life.  With an extra year, by contrast, you may actually achieve something worthwhile.

Of course, if we're interested in whether small benefits to many can, in aggregate, outweigh large benefits to one, it's important that the putative "small benefit" actually be beneficial.  So, rather than imagining someone starting afresh after an extended hospital stay has already disrupted their life, it might be more philosophically fruitful to instead imagine the initial health disruption being delayed for the stipulated duration of time (a minute or a year, as the case may be).  This makes an important difference, because an extra minute in the midst of your life might be just what you needed to finish some worthwhile project (or at least extend a period of carefree enjoyment) that is genuinely valuable.

While it may be relatively unlikely for any given person that the extra random minute would make much difference, given a large enough number of beneficiaries it becomes quite likely indeed that the extra minute made a substantial difference for someone (and a less substantial, but still genuinely desirable, difference for many others).  This is in stark contrast to imagining all million people each getting an unrepresentative, comparatively unhappy extra minute to spend lying in their deathbed!

In sum: It can be hard to believe that giving one minute more to a million different people could really do more good than giving a full year of life to one person.  But this intuition may at least be partly due to imagining a distorted version of the case where the extra minute being given is a disproportionately worthless one.  If we instead imagine giving randomly representative minutes to a million individuals, and bear in mind all the moments in life when an extra minute could have real value, then I find the anti-aggregative intuition disappears entirely.  I now find it entirely credible that the extra minute each, to a large enough population, is better overall than an extra year to just one person would be.


  1. There still seems to be some lower limit here. Consider the analogous case with money -- whether it is to give one person a half a million pennies (i.e., five thousand dollars), or to give a million people one penny each. A parallel argument could be made. But the benefit-value of money (like time, albeit in a different way) is what you can do with it, and the fact of the matter is that there is just not much that can be done with a penny -- the benefit-value is not additive, but increases with concentration. Circumstances have to be very fine-tuned for an added penny to have great effects, and cannot be presumed to exist in any significant number even in a large-scale random distribution. And if so for a penny, how much more so for a penny, or for the tiny fractions of a cent only bank computers keep track of. One-thousandth of a cent each for a billion people can hardly do as much good as a few dollars given to a small group; there isn't actually anything anyone can do with a thousandth of a cent.

    What seems to be doing the real work here is the clause, "bear in mind all the moments in life when an extra minute could have real value". I don't know that there are that many that we are likely to hit if we are only dealing with a million minutes, but it does seems very plausible that there is some threshold of minutes given at which one is almost guaranteed to hit enough of the really, really, important moments to make it worthwhile. But while there may be lots of such cases with minutes, is it true with seconds? With nanoseconds? With picoseconds? Are there really any cases where a picosecond's difference is a difference with which anyone can do anything? It seems that at best we have an argument that minutes is the wrong unit for the objection -- that, actually, it's crude enough that you can still use it. But the anti-aggregation intuition is not eliminated by this, just pushed down to a lower threshold of measurement: there will still, apparently, be some point below which dilution creates insuperable problems for aggregation, and aggregation seems to become a sort of ethical homeopathy.

    1. Ha, good stuff, you hit upon the topic of my next post, so I'll reply to the money analogy in a comment over there :-)

    2. [On second thought, I'll reply here, since my move in response is actually more relevant to this post.]

      I think the key difference in the case of money is that not only is 1/1000th of a cent unusable by itself, but we're also disproportionately unlikely to have 999/1000ths of a cent. So the crucial "chunking" effect, of bringing us up to the next level of actually usable resources, is disproportionately unlikely to occur. In this way, it's a good analogy to the deathbed case, where you basically start at zero and so are assured of failing to achieve a meaningful 'chunk' when given a tiny benefit.

      I think it would be different if our bank accounts were typically so fine-grained that you could expect one in every thousand people receiving the 1/1000th of a cent would actually find their balance pushed up to the next spendable cent. If that were so, then we would find that such diffuse benefits could actually be assimilated to the more concentrated "penny" case.

      So the issue of literal unusability is a red herring here, given the 'chunking' response that's available. But it does leave open the first question you raise of whether the benefit-value of a penny is additive or "increases with concentration", as you put it. But I would think that a similar argument should apply here: while it's plausible that the value of a single penny when starting from zero (or some similarly privileged number in terms of usability) is disproportionately low, when talking about an arbitrary population it seems reasonable to distribute our credence evenly across possible penny-increments (modulo one dollar, say), such that the expected value of a marginal penny should equal the average value of a penny, rather than being in any way disproportionately low due to a lack of "concentration" on the margin. For example, a proportionate number of people should be precisely one penny short of being able to afford something useful, right?

      (You say that "Circumstances have to be very fine-tuned for an added penny to have great effects, and cannot be presumed to exist in any significant number even in a large-scale random distribution." But all I need is a proportionate number being apt to receive each level of possible benefits: many people getting small benefits, a few getting larger, and a very small probability of "great" benefits, all adding up to make the expected values in aggregate be at least those of concentrating a lesser number of pennies in the hands of fewer beneficiaries.)

    3. As you say, I need to go back and rethink how I'm handling proportionate number here.

      I'm not sure that the chunking response is enough, though. Time and money are measures of something else here; they aren't what we're actually trying to chunk together. What we are really trying to chunk together is some complex of action and experience -- the issue of importance would be whether a picosecond would ever, or rather, ever enough, make more worthwhile complexes of action and experience available. The question is whether, insofar as time measures this, it is picosecond-grained; our temporal bank accounts for action/experience (rather than just having numbers on a ledger) are not obviously that fine-grained -- we don't seem to be ordinarily able to distinguish plans to that precision, nor is it easy to experience a picosecond's difference (a blink of an eye is 3*10^9 picoseconds!). It seems you could literally get your extra picosecond and never know the difference. For the chunking response to work, we have to be able to cash out a time-measurement difference of picoseconds into an action/experience difference at least sometimes; but this depends on the actual causes of the latter.


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