rounders and accumulators, satisficers and maximizers

In the Dilbert blog, Scott Adams makes a distinction between two types of people, who he calls "rounders" and "accumulators":

ROUNDERS: This group rounds things off. A problem that’s a two on a scale of one to ten gets rounded to zero. If a rounder has five problems that are all about a two on a scale of one to ten, he’ll tell you he has no problems.

ACCUMULATORS: Accumulators add up all the little problems until they equal one big problem. If an accumulator has five problems that are each a two on a scale of one to ten, that feels like having one problem that’s a ten.

This reminded me of another distinction I've read about. Barry Schwartz, author of a book entitled The Paradox of Choice: Why More Is Less, draws a distinction between "satisficers" and "maximizers". Satisficers are the people who, when they want, say, a computer, will walk into a store and buy the first computer they think is "good enough"; maximizers are the ones who will spend ridiculous amounts of time worrying about which is the "best" computer. In the end they might save $100 (or get $100 more of computing power for the same price) by doing so, but it's not worth the stress it causes them. This is becoming more and more of a problem because there are so many consumer products out there. Psychologically, maximizing turns out to be quite unhealthy. This isn't exactly the same thing as the rounder/accumulator distinction, but they're similar. In both cases, there's a type of person who is more able to shrug problems off and as a result has better mental health; the difference is in exactly how that shrugging off of problems comes about.

How can we model this mathematically, though? One of the threads that runs through Schwartz's book is that in some sense satisficers really are maximizers -- they just value their time and energy more, so settling for a "good enough" thing that isn't the best possible is maximizing, for them. The question becomes -- at what point does the time one spends examining one more alternative outweigh the possibility that that one more alternative might be "the one"?

Say that the quality of some pool of objects are given by random numbers uniformly distributed from 0 to 1. (This is unrealistic, I know, but it makes the computation easier.) It can be shown that if you choose n such numbers at random, the expected value of the maximum is n/(n+1) -- if you pick a single number its expected value is 1/2, the expected value of the larger of two such numbers is 2/3, and so on. Notice that we gain quite a bit in going from one object to two -- we gain 1/6. In going from n objects to n+1 we gain 1/(n+1)(n+2), which shrinks pretty quickly.

Now let's say that the cost of "picking" a single object and determining which random number it corresponds to is k. Then if we pick out n objects and assess their quality, we expect to have one which has quality n/(n+1); but we expended kn in finding it, so overall we win by n/(n+1) - kn. We want to maximize this. It turns out this is maximized when n = 1 + 1/√k.

What is k be, numerically? Roughly, it's the reciprocal of the potential variation in the "quality" of your objects divided by the cost of researching one well enough to know what it costs. For example, say we're looking at computers which are priced at $1000, and you think they're "really" worth somewhere between $900 and $1100. Furthermore, let's say it takes you fifteen minutes to determine what you think a particular computer is "really" worth, and you value your time at $20 an hour. Then k should be on the order of $5/$200, or 1/40. You could probably argue for other values but they wouldn't differ from this by more than a factor of two or so. But if computers varied more, k would be bigger -- say their true values varied between $500 and $1500, then k would be five times as large. Or say you were buying a $20,000 car instead of a $1,000 computer, and again the "true value" could be as much as ten percent off, between $18,000 and $22,000; now k should be twenty times as large. The scaling is much more important than the actual number.

So if we're looking at more expensive things, we should look harder -- but not that much harder. If the things we're looking at are a hundred times as expensive (or if we're buying a hundred times as many of them -- say we're buying computers for a business, not for ourselves), we should spend ten times as much time looking.

The uniform distribution probably isn't the right one to use here. But that's not really the problem; the real problem is that we don't know the distribution in advance. And the job of the advertising industry is basically to convince us that the distribution isn't what we thought it was -- that there is a product that's so great that we should spend a day in line waiting for it, even if you're the mayor of the country's fifth sixth-largest city where six murders in a day is only barely front-page news.