Friday, September 10, 2004

Optimal Haircuts

How short should your barber or hairdresser cut your hair? I suspect a lot of people want a haircut that looks “just right” immediately after cutting. At least, that’s what the lady who cuts my hair (and who does a great job, by the way) always wants to give me. Yet that is obviously a suboptimal haircutting strategy. I always have to ask her to cut it shorter than feels right to her.

The farther your hair length is from the “right” length, in either direction, the greater is your disutility from bad hair days. Too short and too long are both troublesome. Let’s suppose the problem is symmetrical, so that (for instance) hair one inch too long and hair one inch too short are equally undesirable. For simplicity, let’s say your hair grows one-half inch per week, and you get one unit of disutility for each inch of difference from the right length. If you’re on a 5-week haircut cycle, and you start off with the right length, your total disutility is:
0 + 0.5 + 1 + 1.5 + 2 = 5
whereas if you got your hair cut an inch too short, your disutility would be:
1 + 0.5 + 0 + 0.5 + 1 = 3
Clearly, you’re better off asking for the shorter haircut, since that minimizes your disutility.

But it’s unrealistic to assume every inch of difference corresponds to the same amount of disutility. I’d rather have two weeks of hair one-half inch too long than one week of hair a whole inch too long, because a whole inch is more than twice as annoying as a half inch. If your marginal disutility is proportional to (say) the square of inches, the comparison above becomes even more stark. Starting with the right length, you get disutility of:
0 + 0.25 + 1 + 2.25 + 4 = 7.5
whereas if you started an inch too short, your disutility would be only:
1 + 0.25 + 0 + 0.25 + 1 = 2.5
In such a case, having an optimal haircutting strategy matters a great deal more.

Now consider the hastened haircut effect. If there’s some threshold disutility above which you simply cannot stand it any longer, you’ll break from your planned cycle and get an early haircut. Say your threshold is 2 units of disutility, and you start with the right length. Then, using the numbers above, you’ll cut your cycle short and have a haircut after four weeks instead of five.

A cynical person might suggest that your barber or hairdresser, in sending you out of the shop with the “perfect do,” deliberately tries to short-circuit your cycle to squeeze more haircuts out of you. But I’m not a cynic on this one. I suspect hairdressers just don’t trust you to attribute your great hair two weeks from now to the great cut they gave you today.


Anonymous said...

That was a hirsutulous and pulcritudinous post if ever there was one. Since I sport a Mohawk hairdo, the mathematics of disutility becomes more complicated due the the non-uniformity of starting hair lengths. A spikey neon green punk hairdo is even more complicated mathematically and will require the use of random number theory. And what about a bad dye job? We must factor in the disutility of grey hair roots growing in prematurely. I suppose baldness would result in a high constant disutility unless you fancy the Yule Brenner look. Well, if you are bald and hate the look you can alway wear a toupee (no hair cut needed just occasional washing and combing). Of course, you can always wear a baseball cap or wrap your head in a turban. If you are on a budget, you can wrap your head in toilet paper but be prepared to be called some very unflattering names. You can blame the toilet paper idea on your previous post.


Blar said...

Bootleg Trackback: The Perfect Haircut from Blargh Blog.
Glen Whitman at Agoraphilia continues his efforts to use practical mathematics to unveil underlying truths about our day-to-day lives. This time, he argues based on economic reasoning that it is best to get a haircut that is a bit shorter than your ideal hair length. I am concerned, however, that his model leaves out an important fact.... [Read More]

Al Brown said...

Of course, this will all be irrelevant when a machine is created that can learn your haircut and re-apply it at any time.

You'd want the original haircut to be perfect in that case, so that you can perfect your head any time you want.

Don't know what all of the hairdressers will do at that point. Fortunately, that's still another ten or fifteen away.

Glen Whitman said...

Joshua -- I addressed your first point already, by considering disutility as the *square* of inches.

As for the measurement of utility, you're correct, of course. But using cardinal (measurable) utility is just a convenient rhetorical device, since everything I said here could be rendered in the form of ordinal (ranking only) utility. It would be a lot more complicated, since I'd have to introduce some other values against which hair satisfaction could be traded off, but it could be done.

Anonymous said...

Some of us are better off in this area than others. I am blessed with Male Pattern Baldness, which results in a "ring" or "doughnut" hair pattern, from ear to ear, wrapping behind the head, but sparse, if not naked on top of my head.

Being a geek, I really only have three choices of hair style: Shaved, Clipped and Pony-Tail. Since I was born after the 60's, and my hair is currently short, the Pony-tail is out. While it would be financially cheap to grow it out, I would suffer much ridicule and emotional distress during that time.

So, I'm left with Shaved and Clipped. Shaved, as a previous commentor suggested, has problems of its own. You have a continual expense of buying razors (forgo the shaving cream for soap you already use in the shower!). Not only that, but it takes time to shave carefully and if you cut your scalp, you bleed something fierce. You could be a stud and buy a straight razor for a fee, but that's just scary and you better keep that thing sharp and out of the way of your kids.

The Clipped option is the way to go. You buy a pair of clippers and use one (or NO!) attachment. Your marginal disutility is cut in half because your hair only grows in one direction :)

12 years ago, I bought a pair of Oster clippers for $14.95. They've never had to be fixed. I keep them maintained with the original brush and oil they came with. They use negligible energy (I clip about once every two weeks). What hair I have is neat and tidy, easy to maintain and wash. I don't need extra hair products like shampoo and conditioner. I don't need special shaving products like razors, lotion, styptic(sp?) sticks, etc.

I recommend everyone do it, even if you're not balding. It's cheap, fashionable and ... optimal!

Mr. Nosuch said...

Sadly, the idea that hair length contributes to utility strikes me as false, or at least in most cases.

What a hair stylist does isn't simply setting the length of your hair to some optimal value. What they are doing is reducing visible entropy. A good hair cut has low entropy. As the hair grows, the entropy increases.

A perfect example of this for a man: when you get your hair cut, the stylist will frequently shave the upper part of the neck to provide a clean line. Since in most cases, the stylist will shave down to the skin, there is no way to "overshoot" or shave more.

This, and other things, show that a haircut frequently loses "utility" simply as a time function for many people. Most grooming functions have utility because they improve our attractiveness which is related to signaling low personal entropy (because low personal entropy translates to a high level of health and/or wealth and/or free time, all desirable).

Unknown said...
This comment has been removed by the author.
Anonymous said...

Good theory of perfect haircut length! I love it! May be it'll be interesting to read about perfect hairstyle !

Yahya said...

Mr Nosuch makes good points about "reducing visible entropy" and the utility of doing so, in terms of controlling other people's judgment of one's social status, wealth etc. There is, however, a hidden paradox in the arena of "visible entropy", which we should perhaps rather label "apparent entropy". That paradox is this: The appearance of not caring about appearances often signals greater personal freedom. We see it in the young man whose job requires him to wear suit and tie; however, his tie is always artfully loosened just enough to show he's not a stuffed shirt. Or the IT dude who seems to flout convention by wearing t-shirt, jeans and sneakers to the same office where everyone else dresses more formally. Or, to return to our onions, um, haircuts, my haridresser saying to me: "What you really want me to do is to make your hair a little more messy"; to which I could only assent, vigorously. The last thing I'd want is to look too neat and tidy - because it might look like I had to!

I remember my wife telling me, years ago, to stop tucking my t-shirt into my slacks; to her mind, "ultra-neat" translated to "uptight". Still haven't figured out how to wear the t-shirt out and still show off that groovy leather-and-silver jeans belt ...