杏吧原创

鈥淚鈥橫 really glad you鈥檝e decided to stay over until New Year鈥檚 Day,鈥� said
Adam, as the car nosed forward through the traffic. 鈥淚 always find this time
after Christmas kind of anticlimactic, and it鈥檒l be good to have someone around
to stop Josie and me strangling each ot丑别谤.鈥�

Hugh grimaced. 鈥淎s long as you don鈥檛 both decide to strangle me instead.鈥�

鈥淲ell, even though you鈥檙e a mathematician, you鈥檙e usually more entertaining
than the TV,鈥� piped up Josie.

鈥淕ee, thanks!鈥�

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.

鈥淭here is something I鈥檝e been working on that you might find entertaining,鈥�
he said. 鈥淚t鈥檚 called optimisation, and it鈥檚 about using maths to improve your
chances of happiness.鈥�

鈥淭he only way I know of doing that is getting rid of those minuses from my
bank statements,鈥� said Adam drily.

鈥淲ell, optimisation can help you do that too, but that鈥檚 not quite what I
mean,鈥� said Hugh. 鈥淚t鈥檚 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.鈥�

Just then a car pulled out of a parking space in front of them, and Adam
manoeuvred in. 鈥淭hat鈥檚 handy,鈥� he said. 鈥淲e鈥檙e 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鈥檚 party. Just keep an eye out for traffic
飞补谤诲别苍蝉.鈥�

鈥淵ou鈥檇 better feed the meter then.鈥�

鈥淲丑测?鈥�

鈥淏ecause I can鈥檛 move the car if a warden turns up鈥攁nd chances are
you鈥檒l be back too late and get hit for 拢40.鈥�

鈥淲e鈥檒l be gone ten minutes and the warden comes round every hour or so. It鈥檚
not worth paying a pound.鈥�

鈥淎ctually, it is,鈥� said Hugh. 鈥淎nd guess what鈥攊t鈥檚 a classic problem in
optimisation. Suppose that the warden comes round like clockwork every 60
minutes. Then if you don鈥檛 pay the meter and stay more than 60 minutes, you鈥檙e
guaranteed to get a fine, right?鈥�

Adam nodded. 鈥淥K.鈥�

鈥淏ut,鈥� continued Hugh, 鈥渨hat if you stay for less than 60 minutes? Now what
you鈥檝e got to think about is the relative costs of paying the meter, and getting
a fine. When you park, you鈥檒l 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鈥檒l get clobbered is 10 divided by 60, or a sixth.

鈥淭aken over zillions of 10-minute stays, that鈥檚 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 拢40, and
a sixth of that is about 拢7. And that鈥檚 a lot more than 拢1.鈥�

鈥淗ang on a minute,鈥� said Adam. 鈥淭his is one of those typical mathematician鈥檚
simplifications鈥攅veryone knows wardens don鈥檛 come round like
肠濒辞肠办飞辞谤办.鈥�

鈥淭rue,鈥� said Hugh. 鈥淏ut 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鈥檚 patrol methods. It turns out it鈥檚 essentially
never worth taking the risk unless you know for a fact there are no wardens. So
pay up, OK?鈥�

鈥淲hen you two scrooges have quite finished,鈥� said Josie, who by this time had
paid the meter anyway, 鈥測ou may like to know that it鈥檚 now pouring down. Perhaps
you can suggest how we鈥檙e going to get to the shop without getting soaked?鈥�

鈥淚sn鈥檛 this another one of these problems?鈥� asked Adam. 鈥淚 remember reading
something that said walking through a storm is better than
running鈥攕omething to do with getting your front wetter than your head as
you run faster. Or was it the other way round?鈥�

鈥淯nless you鈥檙e infinitely thin, you always get less wet if you run,鈥� said
Hugh. 鈥淭he 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鈥檝e done some work on this myself, and I reckon it鈥檚
only worth running if there鈥檚 a reasonable headwind and/or it鈥檚 bucketing
诲辞飞苍.鈥�

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. 鈥淭he things I do for science,鈥� 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: 鈥淪ee! I am drier than you two,鈥� she said triumphantly. 鈥淎h yes,鈥� said
Hugh, 鈥渂ut was it worth the extra effort?鈥�

鈥淵ou bet鈥擨 arrived just in time to bag us the last bottle of Bollinger
缚75.鈥�

鈥淭ouch茅,鈥� 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. 鈥淚鈥檓
wondering if it鈥檚 time that Josie and I, well, made things a bit more
辫别谤尘补苍别苍迟.鈥�

鈥淲hat, get married you mean?鈥�

鈥淓r, yes, that sort of thing,鈥� said Adam, staring determinedly at some tins
of catfood. 鈥淏ut, well, you know me and commitment. I鈥檓 worried that if I did
settle down, I鈥檇 be constantly thinking I鈥檇 cashed my chips in early.鈥�

鈥淵ou鈥檙e 39 now, aren鈥檛 you? And Josie is by far your best relationship to
诲补迟别?鈥�

鈥淪he鈥檚 certainly the only one I鈥檝e thought of marrying.鈥�

鈥淭hen mathematically speaking, you鈥檇 better propose to her right now.鈥�

鈥淲hat?鈥� spluttered Adam. 鈥淵ou鈥檙e not saying there鈥檚 a mathematical formula
for deciding when to get married?鈥�

鈥淲hy not?鈥� said Hugh. 鈥淵ears ago, a mathematician named Dennis Lindley looked
at making optimal choices from options that pop up at random, and where there鈥檚
no going back once you鈥檝e rejected each one. If you think about it, that鈥檚 just
like picking a partner for life.鈥�

鈥淲ell, the women in my life have definitely turned up at random, and there鈥檚
absolutely no chance of going back to any of them,鈥� said Adam.

鈥淵es, anyhow,鈥� interrupted Hugh, 鈥淲hat Lindley did was to make precise what
you鈥檙e fretting about鈥攏ot cashing your chips in too early as you so
delicately put it鈥攂ut on the other hand, not leaving it too late. Clearly,
there鈥檚 going to be some optimal age at which one should at least start thinking
about settling down.

鈥淚t won鈥檛 be 16鈥攖hat鈥檚 the legal minimum. And realistically, no one鈥檚
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鈥檙e
蝉辞谤迟别诲.鈥�

鈥淲ell, that鈥檚 all very clever, but how do you know how long to wait before
starting to look seriously?鈥� said Adam.

鈥淟indley showed that the waiting time is given by the total available search
time鈥�44 years in our case鈥攄ivided by 鈥渆鈥�, 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,鈥� said Hugh, with a broad grin, 鈥淚鈥檇 get down that aisle pronto
it I were you.鈥�

鈥淛osie鈥檒l think I鈥檝e gone crazy when she hears this,鈥� said Adam. 鈥淪peaking of
which, where is she?鈥�

Both of them looked around. They鈥檇 spent so long proving that Josie was the
mathematically optimal partner for Hugh that it seemed they鈥檇 lost her. 鈥淚鈥檒l
check this way, you go down there,鈥� said Adam, pointing towards the checkouts as
he began to run towards the delicatessen.

鈥淗ang on鈥攚e鈥檇 better be careful. One of us could find her and then
spend the rest of the day looking for the ot丑别谤.鈥�

鈥淲hat do you suggest, then?鈥�

鈥淪earching 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
丑别谤.鈥�

鈥淲hat鈥檚 so smart about that?鈥�

鈥淚t鈥檚 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鈥檛 lose each ot丑别谤.鈥�

Just as they were about to put theory to the test, Josie appeared, looking
decidedly irritated. 鈥淐ome on鈥攍et鈥檚 get this lot through the checkout,鈥�
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鈥檚 mood. On either side the
queues were moving forward at a great rate, and the one they had chosen seemed
rooted to the spot. 鈥淲e鈥檙e going to be last, as usual,鈥� she complained.

鈥淲ell, almost 70 per cent of the time one or other of our neighbouring queues
will beat ours,鈥漵aid Hugh. 鈥淵ou 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鈥攖wo
times out of three, we won鈥檛 finish last, eit丑别谤.鈥�

At that moment, the cashier announced that the barcode reader had broken
down. 鈥淓r, of course that still means that one time out of three, we will,鈥�
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.

鈥淎h, now there鈥檚 a great example of what I鈥檝e been talking about,鈥� said
Hugh.

鈥淲hat, me being a greedy pig?鈥�

鈥淲ell, 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.鈥�

鈥淥h, sorry Hugh. OK, that way we鈥檙e forced to be fair in cutting the cake,
because if we try to cut ourselves a big slice, the other person will choose
颈迟.鈥�

鈥淓xactly. 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鈥檙e obliged to share some of it with the country鈥攂ut if
you鈥檙e a company full of fat cats, you鈥檒l want the best bit, right?鈥�

鈥淩ight鈥攂ut you could ask the company to divide up the find, and let the
country choose which bit it wants.鈥�

鈥淎nd that鈥檚 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鈥檚 resolutions yet?鈥�

鈥淎bsolutely. I鈥檓 going to try to have an Optimal New Year, starting right
now. Here, finish this bit of cake for me Hugh鈥擨 feel sick.鈥�