“I’M really glad you’ve decided to stay over until New Year’s Day,” said
Adam, as the car nosed forward through the traffic. “I always find this time
after Christmas kind of anticlimactic, and it’ll be good to have someone around
to stop Josie and me strangling each otłó±đ°ů.”
Hugh grimaced. “As long as you don’t both decide to strangle me instead.”
“Well, even though you’re a mathematician, you’re usually more entertaining
than the TV,” piped up Josie.
Advertisement
“Gee, thanks!”
Adam swerved to avoid a bargain hunter from the sales who had staggered into
the road, barely visible beneath a mountain of shopping bags. Hugh thought for a
moment.
“There is something I’ve been working on that you might find entertaining,”
he said. “It’s called optimisation, and it’s about using maths to improve your
chances of happiness.”
“The only way I know of doing that is getting rid of those minuses from my
bank statements,” said Adam drily.
“Well, optimisation can help you do that too, but that’s not quite what I
mean,” said Hugh. “It’s all about making decisions. You must have looked back
sometimes on decisions in your life and wondered if you could have done better.
Finding those best decisions is what optimisation is all about.”
Just then a car pulled out of a parking space in front of them, and Adam
manoeuvred in. “That’s handy,” he said. “We’re just round the corner from the
wine merchants. You wait in the car for a bit, and Josie and I will go and sort
out the booze for the New Year’s party. Just keep an eye out for traffic
·É˛ą°ů»ĺ±đ˛Ô˛ő.”
“You’d better feed the meter then.”
“Włó˛â?”
“Because I can’t move the car if a warden turns up—and chances are
you’ll be back too late and get hit for £40.”
“We’ll be gone ten minutes and the warden comes round every hour or so. It’s
not worth paying a pound.”
“Actually, it is,” said Hugh. “And guess what—it’s a classic problem in
optimisation. Suppose that the warden comes round like clockwork every 60
minutes. Then if you don’t pay the meter and stay more than 60 minutes, you’re
guaranteed to get a fine, right?”
Adam nodded. “OK.”
“But,” continued Hugh, “what if you stay for less than 60 minutes? Now what
you’ve got to think about is the relative costs of paying the meter, and getting
a fine. When you park, you’ll catch the warden at some random point during his
patrol. So if you stay for, say, 10 minutes, the chances of your hitting their
round at a point where you’ll get clobbered is 10 divided by 60, or a sixth.
“Taken over zillions of 10-minute stays, that’s the average fine you can
expect to pay. It only makes sense to take the risk if this average is lower
than the cost of paying the meter. Now, in our case, the fine is ÂŁ40, and
a sixth of that is about £7. And that’s a lot more than £1.”
“Hang on a minute,” said Adam. “This is one of those typical mathematician’s
simplifications—everyone knows wardens don’t come round like
ł¦±ô´Çł¦°ě·É´Ç°ů°ě.”
“True,” said Hugh. “But William Woodside, the mathematician who worked this
out a few years ago, showed that the same basic rule holds for all plausible
assumptions about the warden’s patrol methods. It turns out it’s essentially
never worth taking the risk unless you know for a fact there are no wardens. So
pay up, OK?”
“When you two scrooges have quite finished,” said Josie, who by this time had
paid the meter anyway, “you may like to know that it’s now pouring down. Perhaps
you can suggest how we’re going to get to the shop without getting soaked?”
“Isn’t this another one of these problems?” asked Adam. “I remember reading
something that said walking through a storm is better than
running—something to do with getting your front wetter than your head as
you run faster. Or was it the other way round?”
“Unless you’re infinitely thin, you always get less wet if you run,” said
Hugh. “The time spent in the rain is always more important than the relative
soaking of your head and front. The real question is whether the difference is
worth the extra effort. I’ve done some work on this myself, and I reckon it’s
only worth running if there’s a reasonable headwind and/or it’s bucketing
»ĺ´Ç·É˛Ô.”
By now, the rain had eased into a light drizzle. Josie scrambled out of the
car and ran off into the distance. Adam and Hugh began walking resolutely after
her. “The things I do for science,” said Adam, as rain trickled down the back of
his neck.
Once the two men arrived at the wine-merchant, Josie could not restrain
herself: “See! I am drier than you two,” she said triumphantly. “Ah yes,” said
Hugh, “but was it worth the extra effort?”
“You bet—I arrived just in time to bag us the last bottle of Bollinger
¸ż75.”
“Touché,” muttered Hugh.
Half an hour later, all three were wandering round the local supermarket,
picking up snacks for the party. As Josie disappeared in search of the
delicatessen counter, Adam decided to tell Hugh his own bit of news. “I’m
wondering if it’s time that Josie and I, well, made things a bit more
±č±đ°ůłľ˛ą˛Ô±đ˛ÔłŮ.”
“What, get married you mean?”
“Er, yes, that sort of thing,” said Adam, staring determinedly at some tins
of catfood. “But, well, you know me and commitment. I’m worried that if I did
settle down, I’d be constantly thinking I’d cashed my chips in early.”
“You’re 39 now, aren’t you? And Josie is by far your best relationship to
»ĺ˛ąłŮ±đ?”
“She’s certainly the only one I’ve thought of marrying.”
“Then mathematically speaking, you’d better propose to her right now.”
“What?” spluttered Adam. “You’re not saying there’s a mathematical formula
for deciding when to get married?”
“Why not?” said Hugh. “Years ago, a mathematician named Dennis Lindley looked
at making optimal choices from options that pop up at random, and where there’s
no going back once you’ve rejected each one. If you think about it, that’s just
like picking a partner for life.”
“Well, the women in my life have definitely turned up at random, and there’s
absolutely no chance of going back to any of them,” said Adam.
“Yes, anyhow,” interrupted Hugh, “What Lindley did was to make precise what
you’re fretting about—not cashing your chips in too early as you so
delicately put it—but on the other hand, not leaving it too late. Clearly,
there’s going to be some optimal age at which one should at least start thinking
about settling down.
“It won’t be 16—that’s the legal minimum. And realistically, no one’s
going to be too interested in a clapped-out old Adam past, say, 60 either. Now
suppose that potential partners turn up at random during those 40-odd years,
some better than others. Lindley proved that if we want the best chance of
making the optimal choice, we should not even think of choosing until we have
met some minimum number, but after then we should keep looking until we meet
someone who beats all the others thus far. And if she likes you too, you’re
˛ő´Ç°ůłŮ±đ»ĺ.”
“Well, that’s all very clever, but how do you know how long to wait before
starting to look seriously?” said Adam.
“Lindley showed that the waiting time is given by the total available search
time—44 years in our case—divided by “e”, the base of natural
logarithms. That gives about, what, 16 years. And since you start searching at
16, that means you should marry the best person you meet after the age of 32.
So, like I said,” said Hugh, with a broad grin, “I’d get down that aisle pronto
it I were you.”
“Josie’ll think I’ve gone crazy when she hears this,” said Adam. “Speaking of
which, where is she?”
Both of them looked around. They’d spent so long proving that Josie was the
mathematically optimal partner for Hugh that it seemed they’d lost her. “I’ll
check this way, you go down there,” said Adam, pointing towards the checkouts as
he began to run towards the delicatessen.
“Hang on—we’d better be careful. One of us could find her and then
spend the rest of the day looking for the otłó±đ°ů.”
“What do you suggest, then?”
“Searching optimally, what else? Lyn Thomas at Edinburgh University sorted
this one out a few years back. What we should do is decide to meet back here
again in, say, five minutes. If neither of us has found her, we should go and
look for another four minutes, and then meet up again. And we keep repeating
that, gradually decreasing the amount of time we spend apart until we find
łó±đ°ů.”
“What’s so smart about that?”
“It’s the best way of dividing our time between looking for Josie, and
staying in touch with each other. Obviously, the chances of finding her improve
the longer we spend looking for her, and so the priority increasingly becomes
making sure we don’t lose each otłó±đ°ů.”
Just as they were about to put theory to the test, Josie appeared, looking
decidedly irritated. “Come on—let’s get this lot through the checkout,”
she said, pushing her trolley at Adam. He decided to tell her about
mathematically optimal partners some other time.
Standing in the queue did nothing to improve Josie’s mood. On either side the
queues were moving forward at a great rate, and the one they had chosen seemed
rooted to the spot. “We’re going to be last, as usual,” she complained.
“Well, almost 70 per cent of the time one or other of our neighbouring queues
will beat ours,”said Hugh. “You see, all the checkouts are as likely to suffer
random delays as every other, so the chances that our queue will beat both our
neighbouring queues is just one in three. But look on the bright side—two
times out of three, we won’t finish last, eitłó±đ°ů.”
At that moment, the cashier announced that the barcode reader had broken
down. “Er, of course that still means that one time out of three, we will,”
added Hugh. Josie scowled at him, and carried on unloading the trolley.
Back at the house, Josie had recovered her Christmas spirit. While the other
two unloaded the car, she disappeared into the kitchen, returning with tea and a
huge, gooey chocolate cake which she put on the table. Adam cut himself a vast
slice.
“Ah, now there’s a great example of what I’ve been talking about,” said
Hugh.
“What, me being a greedy pig?”
“Well, that too, but what I meant was that was a chance to optimise the
division of the cake. When you cut the cake, it was pretty obvious which bit you
were going to have. But we could both have had our fair share if one of us had
cut, but the other had chosen which bit to have.”
“Oh, sorry Hugh. OK, that way we’re forced to be fair in cutting the cake,
because if we try to cut ourselves a big slice, the other person will choose
ľ±łŮ.”
“Exactly. Now imagine the cake is some vast new mining area found by a
multinational company in the coastal waters of some developing country.
Obviously, they’re obliged to share some of it with the country—but if
you’re a company full of fat cats, you’ll want the best bit, right?”
“Right—but you could ask the company to divide up the find, and let the
country choose which bit it wants.”
“And that’s exactly what the 1994 Convention on the Law of the Sea requires.
Everyone gets their fair share. Smart, huh? Anyhow, enough of all this. Made any
New Year’s resolutions yet?”
“Absolutely. I’m going to try to have an Optimal New Year, starting right
now. Here, finish this bit of cake for me Hugh—I feel sick.”
- An optimal decision rule for the parking meter problem
by William Woodside, Mathematical żěè¶ĚĘÓƵ, vol 15, p 36, (1990) - Finding your kids when they are lost
by Lyn Thomas, Journal of the Operational Research Society,
vol 43, p 637 (1992) - Making decisions
by Dennis Lindley, 2nd edition, Wiley - Fair Division
by Steven Brams and Alan Taylor, CUP, 1996