What's love got to do with it?

Forget conventional notions of beauty. Michael Brooks has a mathematical way for you to find your soulmate

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's ... 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' tonight. Follow these and everyone you know--including your very loveable self--could soon be lucky in love.

It's an age-old problem: there are plenty of available people out there, so why can't 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's 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're satisfied with. Won't they?

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 "stable marriage problem".

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's 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's just the convention; it would work just the same if the women took the initiative). She accepts: she's 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's new fiancée, 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're pretty happy with.

Depending upon people's 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's 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's 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 "Vogue factor" to Gale and Shapley's 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's 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's wish lists.

The result of this makes distressing reading for the plain and ordinary. With beauty on the scene, you're 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's about as likely to get her as he is to get Ms 200. It's enough to make a physicist--and almost everybody else--give up hope. "Even if the more beautiful players have a larger satisfaction by far, the general dissatisfaction in the system increases," bemoans Caldarelli. He draws a simple, melancholy conclusion: "When the concept of 'most beautiful' in the world tends to be the same for everyone it becomes more and more difficult to make more people happy."

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--particularly in the West--are bombarded with images of "beautiful" 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

"These are hard questions to answer. None of it is cut and dried," says Merl Storr, a sociologist at the University of East London. Caldarelli says he doesn't want to get drawn into debates about the implications of his results. That's probably wise: physicists do well to keep out of socio- logical waters. In fact, even sociologists are keeping their distance. "A lot of the time sociologists are not even asking those kinds of questions any more. There's a real trend towards evolutionary explanations of our perceptions of beauty," Storr says.

So, let's turn to a professor of Darwinian aesthetics. According to Dev Singh of the University of Texas, Austin, we can't help but be motivated by beauty, at least as an initial draw towards a partner. "Beauty has a direct link to the quality of your reproductive success," he says. It's a subtle drive that we're hardly aware of. "You don't go up to a woman and ask if she would like to have your babies," he points out. "But it's like eating sugar: you don't say, 'Ooh good, I've got lots of calories here.' It just tastes sweet."

However, there is hope. Our craving for beauty is tempered by many other social and cultural factors. "Evolution is designing your fantasy and your desires, but not your real behaviour," says Singh. "Your preferences are shaped by evolution. Your choices, on the other hand, may not reflect your ultimate preference." A 60-year-old man who doesn't have a realistic chance of a 19-year-old mate is also quite likely to accept that youth and beauty aren't 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's 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's err on the side of caution.

Lesson in love 1: Forget what some call beauty. Enjoy the random factor--call it "personality" 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's 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's 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 "weakly stable". 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--"can't decide" and "too ugly"--and we're all in serious trouble. As soon as someone says they wouldn't go out with you if you were the last person on Earth, and someone else can't choose between Naomi and Claudia, we all hit a brick wall. In mathematical terms, Manlove says, the problem becomes "NP-complete".

"Membership of this class of problems is the most convincing way of proving to someone that a problem is likely to be intractable," 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's security codes. It's 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't 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 "Man, woman and dog". It doesn't work: it's always NP-complete. "That's not too surprising," says Manlove. "If we have three parties the complexity increases radically."

Lesson in love 4: A ménage à 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's not a desire to sort out society's 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 "The Marriage Problem and the Fate of Bachelors" (Physica A, vol 252, p 178). "I had been alone a long time when I wrote this," 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's qualities (or lack of them), but also on the way society is structured:

Lesson in love 5: The maths can always show it's somebody else's fault.

"I was thinking, 'Is it me, or is it the situation?'" Nieuwenhuizen recalls. "But 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." It's not medicine, or even time: it's maths that heals a broken heart.

From New Scientist magazine, 28 October 2000.