THERE is a universal language that helps connect people across the globe and
bring solace to the lonely. Treat it well, for it can offer you the key to
happiness, mending broken hearts and soothing troubled breasts. What is this
mysterious panacea? Yes, it鈥檚 鈥� maths.
Mathematicians have been puzzling over the business of pairing up for nearly
forty years. They claim it helps for understanding physics, economics and even
the future of robot technologies. And thanks to all these efforts, there are now
a few simple rules which could help us all get some sweet lovin鈥� tonight. Follow
these and everyone you know鈥攊ncluding your very loveable self鈥攃ould
soon be lucky in love.
It鈥檚 an age-old problem: there are plenty of available people out there, so
why can鈥檛 you find your soulmate? While the mathematical nitty-gritty of the
problem is easily stated, the solution is less straightforward. It involves
weighing up people鈥檚 preferences and finding an optimal solution where the most
people are the happiest. Exhausting as it may be, you can work through the
problem, and at the end of it everybody will have found a partner that they鈥檙e
satisfied with. Won鈥檛 they?
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Well, not necessarily. The trouble with love started, of course in the
Sixties. In 1962, University of California researchers David Gale, based at
Berkeley, and Lloyd Shapley in Los Angeles took it upon themselves to find out
if global coupling was possible. They proposed a simple algorithm, and set about
solving what is now known as the 鈥渟table marriage problem鈥�.
A computer generates two sets of people, 100 men and 100 women, say. Each
person composes a wish list by randomly ranking the others in order of
desirability. No one has any intrinsic value: beauty is truly in the eye of the
beholder. There鈥檚 no reason why Man 7 should be ranked higher on average than
Man 26, for instance. He is as likely to be ranked top by one woman as bottom by
the next.
It cuts both ways: each of the men assigns a similarly random rank to each of
the women. Then the computerised dating frenzy begins. The first man proposes to
the woman at the top of his list (that鈥檚 just the convention; it would work just
the same if the women took the initiative). She accepts: she鈥檚 had no other
offers, so why would she spurn her chance of happiness? Man 1 is happy, and for
the moment his woman is content too. Then Man 2 proposes to the woman at the top
of his list. Unless she happens to be Man 1鈥檚 new fianc茅e, and happier
with her original mate, this woman accepts too. So it goes on until the last man
has made his proposal.
Any time a woman receives a proposal from someone higher up her wish list
than her current partner, she accepts the offer, and a new engagement forms. Her
disgruntled ex is then free to try his luck elsewhere. Eventually a stable state
forms in which it is impossible to find a man and a woman who would both rather
be married to each other than stay with their current partner.
In this simple situation, a man and a woman humbly attempt to do the best
they can according to their own personal standards. And this restrained society
is rewarded for its demure behaviour. Gale and Shapley proved that you can
always match everyone up. There might be some heaving of bosoms and a few
furtive, longing glances, but at least everybody has settled with a partner
they鈥檙e pretty happy with.
Depending upon people鈥檚 randomly assigned preferences, there may be more than
one way to match everyone up and keep things stable. But some solutions are
better than others. Add up the rank of everyone鈥檚 partner, and you get a measure
of happiness: the lower the score, the happier the society.
The best outcome would occur if, by chance, everyone got their number-one
choice. Researchers have written algorithms for optimising happiness, but
because the whole business revolves around people looking to maximise their
personal benefit, selfishness and conflicting interests tend to stop the best
global solution ever emerging. Yi-Cheng Zhang of Fribourg University in
Switzerland has even proved, strangely enough, that the happiest matching is an
unstable one: the best stable solution leaves society nearly 20 per cent less
happy than this optimum.
But the world has changed since Gale and Shapley鈥檚 day. This is the thrusting
new millennium, and Guido Caldarelli, a physicist based at the University of
Rome, has brought the stable marriage problem up to date. He thinks he might
have found a vital clue in the quest for contented love. Working with Andrea
Capocci of Fribourg University, he made a small but pertinent modification to
the model that produced a large change in the results. They introduced the
concept of beauty, and watched global happiness sink through the floor.
In a paper submitted to Europhysics Letters, Caldarelli and Capocci
take into account the looks of their participants by adding a 鈥淰ogue
factor鈥� to Gale and Shapley鈥檚 random function. They give each person an
intrinsic beauty which is then multiplied by a weighting factor U. This
determines how much influence beauty has in society. When U equals 0,
beauty plays no part, and each person is ranked by potential mates in the random
way as before. In that case, every one of the 1000 men in Caldarelli鈥檚 game
landed a partner ranked better than 70 out of the 1000 women on his list. But
when U is even slightly greater than zero, beauty can outweigh the
random factor, and the intrinsically beautiful people rise to the top of
everyone鈥檚 wish lists.
The result of this makes distressing reading for the plain and ordinary. With
beauty on the scene, you鈥檙e now much less likely to be matched with your
number-one choice, unless you happen to be one of the beautiful people yourself.
With every man vying for the attention of the loveliest women, an averagely
attractive man can do as badly as getting lumbered with Ms 900. In fact, he鈥檚
about as likely to get her as he is to get Ms 200. It鈥檚 enough to make a
physicist鈥攁nd almost everybody else鈥攇ive up hope. 鈥淓ven if the more
beautiful players have a larger satisfaction by far, the general dissatisfaction
in the system increases,鈥� bemoans Caldarelli. He draws a simple, melancholy
conclusion: 鈥淲hen the concept of 鈥榤ost beautiful鈥� in the world tends to be the
same for everyone it becomes more and more difficult to make more people
丑补辫辫测.鈥�
But does this tell us the recipe for making love in the real world? Thanks to
the homogenisation due to TV, cinema and magazines like Vogue, many of
us鈥攑articularly in the West鈥攁re bombarded with images of 鈥渂eautiful鈥�
men and women, unsubtly dictating our ideals. Have the mass media given us a
standardised concept of beauty? And has beauty become ever more important in our
choice of partner?
Hard questions
鈥淭hese are hard questions to answer. None of it is cut and dried,鈥� says Merl
Storr, a sociologist at the University of East London. Caldarelli says he
doesn鈥檛 want to get drawn into debates about the implications of his results.
That鈥檚 probably wise: physicists do well to keep out of sociological
waters. In fact, even sociologists are keeping their distance. 鈥淎 lot of the
time sociologists are not even asking those kinds of questions any more. There鈥檚
a real trend towards evolutionary explanations of our perceptions of beauty,鈥�
Storr says.
So, let鈥檚 turn to a professor of Darwinian aesthetics. According to Dev Singh
of the University of Texas, Austin, we can鈥檛 help but be motivated by beauty, at
least as an initial draw towards a partner. 鈥淏eauty has a direct link to the
quality of your reproductive success,鈥� he says. It鈥檚 a subtle drive that we鈥檙e
hardly aware of. 鈥淵ou don鈥檛 go up to a woman and ask if she would like to have
your babies,鈥� he points out. 鈥淏ut it鈥檚 like eating sugar: you don鈥檛 say, `Ooh
good, I鈥檝e got lots of calories here.鈥� It just tastes sweet.鈥�
However, there is hope. Our craving for beauty is tempered by many other
social and cultural factors. 鈥淓volution is designing your fantasy and your
desires, but not your real behaviour,鈥� says Singh. 鈥淵our preferences are shaped
by evolution. Your choices, on the other hand, may not reflect your ultimate
preference.鈥� A 60-year-old man who doesn鈥檛 have a realistic chance of a
19-year-old mate is also quite likely to accept that youth and beauty aren鈥檛
everything. Pragmatism often wins and we can compromise our evolutionary lust
for beauty.
So maybe we know how to make ourselves happier already. But we can at least
use Caldarelli鈥檚 result as a guide to the first lesson in love. Although there
is no empirical way to tell whether people are now more unhappy than they have
ever been, let鈥檚 err on the side of caution.
Lesson in love 1: Forget what some call beauty. Enjoy the random
factor鈥攃all it 鈥減ersonality鈥� if you like. Be an individual, with personal,
eclectic tastes.
Having said that, the old Gale-Shapley scenario does have its own peculiar
drawback. Without a universal concept of beauty, the proposing sex is always
better off than those who receive the proposals. In 1000 stable solutions of 512
couples, the average rank of partner for the proposer is around 8, while for the
receiver it鈥檚 80.
With beauty in the equation, only the really ugly receivers are worse off
than the proposers, and even then not by so much. The player ranked 200 in the
beauty stakes gets their 50th choice on average, player 300 gets their 100th
choice, and player 400 gets number 175 on their list, regardless of whether they
are proposer or receiver. Only at the tail end, beyond 500, do differences start
to occur. The proposer gets a partner around the 250 mark, while a receiver gets
350 or so. So things are a bit more equal, but it鈥檚 small consolation, as almost
everybody is far worse off.
Lesson in love 2: Unless you are fantastically good-looking, you need to take
the initiative.
Of course, with beauty to worry about, other issues rear their head too.
Ugliness, for example: it can cause a lot of trouble if some people are just too
ugly for you to consider. David Manlove of Glasgow University has worked on the
problem of unacceptable partners and found that it yields solutions that are
only 鈥渨eakly stable鈥�. That means that, while no man and woman would definitely
rather be with each other than with their current partner, there might still be
other acceptable options. Only the effort involved in breaking one relationship
and forming another stops couples endlessly splitting.
There could also be people of equal beauty. Manlove has shown that such dead
heats also make the outcome weakly stable. But the algorithm still works if you
let the computer split such ties and impose an arbitrary ranking on the two.
However, combine these two complaints鈥斺�漜an鈥檛 decide鈥� and 鈥渢oo
ugly鈥濃�攁nd we鈥檙e all in serious trouble. As soon as someone says they
wouldn鈥檛 go out with you if you were the last person on Earth, and someone else
can鈥檛 choose between Naomi and Claudia, we all hit a brick wall. In mathematical
terms, Manlove says, the problem becomes 鈥淣P-complete鈥�.
鈥淢embership of this class of problems is the most convincing way of proving
to someone that a problem is likely to be intractable,鈥� says Manlove. It means
there is no stable solution, not even a weakly stable one. Some people will be
forever unmatched and unhappy, marriages will break up and eyes will always
wander. If you could find a solution, when faced with equal rankings and
unacceptable partners, you could also solve other NP-complete problems, such as
cracking the Pentagon鈥檚 security codes. It鈥檚 not looking hopeful.
There are ways of approximating the solutions to the problem, though, by
settling for fewer stable partnerships. Manlove, together with Glasgow colleague
Rob Irving, has found he can guarantee that each individual has at least a
fifty-fifty chance of finding a stable matching. Comforted? Thought not. Which
leads us to:
Lesson in love 3: Be clear about where everybody stands, and don鈥檛 rule
anyone out.
A quick word of warning for the more daring reader. Researchers of the stable
marriage problem have modelled three-way matching, a situation they refer to as
鈥淢an, woman and dog鈥�. It doesn鈥檛 work: it鈥檚 always NP-complete. 鈥淭hat鈥檚 not too
surprising,鈥� says Manlove. 鈥淚f we have three parties the complexity increases
谤补诲颈肠补濒濒测.鈥�
Lesson in love 4: A m茅nage 脿 trois can ruin things for
everyone. Please refrain.
So what exactly motivates the average physicist, mathematician or computer
scientist to have a stab at the stable marriage problem? Well, it鈥檚 not a desire
to sort out society鈥檚 marital issues. In the messy world of human attachments,
even the most complex mathematical models would struggle to account for the
diversity of pressures and preferences we encounter in our search for a
soulmate.
Instead, Manlove and Irving are experts in using the stable marriage problem
to match medical students with residency vacancies in hospitals. Zhang applies
it to economics, examining the marriage of supply with demand. Others use the
maths to find the best candidate to fill job vacancies. Or with roaming robots
working on a space station, it can help determine the best way to distribute
battery recharging stations. Theo Nieuwenhuizen of the University of Amsterdam
has used the stable marriage problem in his studies of solid-state physics,
measuring the energy costs of pairing particles.
But physicists have feelings too. In 1998 Nieuwenhuizen wrote a paper
entitled 鈥淭he Marriage Problem and the Fate of Bachelors鈥� (Physica A,
vol 252, p 178). 鈥淚 had been alone a long time when I wrote this,鈥� he says. In
the paper he shows that the main equation imposes a cost on society whenever two
people form a couple rather than remaining as individuals. This, he says, proves
that remaining single is not just dependent on the bachelor鈥檚 qualities (or lack
of them), but also on the way society is structured:
Lesson in love 5: The maths can always show it鈥檚 somebody else鈥檚 fault.
鈥淚 was thinking, 鈥業s it me, or is it the situation?'鈥� Nieuwenhuizen recalls.
鈥淏ut I showed that the problem of remaining single depends on an exponent
related to society, not to me. That made me feel a lot better.鈥� It鈥檚 not
medicine, or even time: it鈥檚 maths that heals a broken heart.