WILL I EVER MEET THE PERFECT PARTNER?
The chances and choices behind coupling up
It is well known that fewer people get married these days. Fifty years ago, getting married was expected of you, while remaining single carried a stigma, particularly for women. The term ‘spinster’, which has almost vanished from use, always seemed to carry far more negative connotations than its male equivalent ‘bachelor’. The pressure was on to get married and have children as quickly as possible, and often people married their first love, and then stuck it out for better or for worse.
Those social pressures have reduced enormously, and people have begun to think much more of marriage as a lifestyle choice. Men in particular seem much more reluctant to enter marriage. As Bridget Jones puts it, some men seem to be commitment-phobic. The same is true for some women, too, but for the sake of simplifying what follows in this chapter we will look at it mainly from the male point of view. For ‘he’ substitute ‘she’ where you think it is appropriate, and of course exactly the same arguments will apply for choosing a mate of the same sex.
Why is there this lack of commitment to a permanent relationship? One of the many reasons is that the modern male believes that he has plenty of time to make a choice. Many men set out looking for the ‘perfect partner’, whatever that might be, and put off engagement because they ask themselves, ‘What if the next is even better?’
Making the best choice: The 37% Rule
Selecting a mate is an example of a serial decision. In other words, you aren’t usually presented with all of your options at the same time, but instead they come along one after the other, and you have no idea who or what will come along next.
In fact there are similarities between finding a partner and other more mundane decisions, such as renting a flat, finding a parking space or accepting a job offer. In each case, the options are presented to you in turn, and, once you have rejected an option, you often don’t have a chance to go back to it later. If you are driving down a one-way street, you can’t turn back to take the parking spot you just ignored, and when the housing market is hot, then unless you say yes to the place you just saw, it is likely to be snapped up by somebody else. All are examples of serial decisions.
When is it right to stick with what you have? To examine this very real problem, it helps to have a slightly artificial case study, just to make the analysis simple. In this case, Jim is ideal. He is 39 years old, and is determined to be engaged by the age of 40. To remove some of the randomness from meeting possible partners, Jim has joined a dating club, which guarantees to set him up with ten dates per year. We’ll add the rather less plausible situation that every one of the dates he meets is so desperate to get hitched that all he has to do is pop the question. Jim therefore knows that one of the ten dates that he will meet will become his wife. But which one?
It seems a little callous to rank possible mates in order, but unfortunately this is necessary if the analysis is to be taken any further. (And it has to be said that men and women are not averse to rating potential partners in this way.) One of the ten dates will be the best choice of partner and one is the worst, but the order in which Jim gets to meet them will be completely random.
Jim meets his first date, and she seems OK. But is she the best one? She might be, but the chance of her being the best of all the dates he is going to meet is only 1 in 10. So it seems a reasonable decision not to commit to her, but to use her as the benchmark with which to compare later dates (we’re avoiding any judgment on Jim’s moral character here).
If he’s a real ditherer, Jim could continue being noncommittal by saying no until the final of the ten comes along, at which point he has no choice – he has to go with her if he is to become engaged as he promised. If he opts for this dithering, noncommittal strategy, he still has only a l-in-10 chance of picking the best. There must be a better strategy.
Indeed there is. One way to improve his chances is to say no to the first date, whom we will call Kate, but say yes to the first date he meets who scores higher than Kate. Using this strategy, nine times out of ten he will end up with a better mate than Kate. However, if Kate happened to be the best date, this strategy fails.
Choosing the first date who beats Kate actually increases his chances of ending up with the best possible partner to 20 per cent, or 1 in 5. The calculation of this is quite tricky, since it is necessary to add together the chance of the best date arriving second, or third (with the second scoring less than Kate), or fourth (second and third both scoring less than Kate) and so on up to tenth.
What if Jim uses more than just Kate to be his benchmark? The longer he holds out, the better a cross-section he has of all the possible dates, but the more chance there is that the best partner will already have been rejected.
In fact, there is mathematically a best solution for Jim in the above circumstances. This is to meet three dates, and then to propose to the first one who improves on all of them. This way Jim’s chances of ending up with the best available partner increase to about 1 in 3. You can simulate Jim’s experiences by playing the blind-date game described in the box.
Despite its gross over-simplification of real life, this exercise makes a good case for what many people do anyway, which is to experience a few partners before making a firm commitment. As the number of potential partners increases, the mathematical answer tends towards an extraordinarily precise ratio – if there are N partners available, you should commit after meeting N divided by ‘e’ partners, where ‘e’ is roughly 2.718, and is the number at the heart of exponential growth. When N is a larger number, this means you should settle down after meeting about 37 per cent of your potential partners.
One of many flaws in this cunning plan, of course, is that you have no idea beforehand how many potential partners you will meet. Still, if you estimate that you might expect to meet forty potential mates in a lifetime, then after meeting fourteen of them it is time to look to settle down.
How commitment-phobic are you? Play the blind-date game.
Pick out ten cards from a pack that are numbered ace to ten, with ace scoring low. These cards represent your blind dates, and the aim of the game is to end up with the highest-scoring card. Shuffle the cards, and deal them out face down:
Starting at the left, you can choose how many of the cards you want to date, but not get serious with. This is you, ‘playing the field’ without making a commitment. The lowest playing-the-field (or PF) number is zero – in other words, you’ll take the first card that comes along. The highest PF number is nine, which means your commitment is at a minimum but you are forced to commit to the last card you turn up, the tenth. Once you have chosen your PF number, turn over that many cards, and note the highest score. This is your benchmark. Now start turning over your potential partners. The first card that is higher than the benchmark is your partner. If none scores higher, the tenth card is your partner regardless of its value.
On average, PF numbers of 0 or 9 give you the worst out comes. The best results come from choosing a PF number of 3. This will give the best available partner about one time in three.
The perfect match?
The strategy just described gives you a good chance of meeting the best possible mate of the ones you are destined to meet. This isn’t quite the same as meeting the perfect partner. For example, if your main interest in life is travelling to remote countries, you’re going to be disappointed if none of the people you meet has a desire to travel any further than Clacton-on-Sea.
To find somebody truly compatible requires a more focused search. This is where dating agencies can come into their own. By asking lonely hearts to fill in a questionnaire, they can then use the data to match up possible couples based on their interests and preferences. Statisticians like to use distance measures to find the degree of match between two sets of data. The principle is that the smaller the overall gap between the two sets of data, the better matched the two items will be.
As an example, consider Annie, who is seeking a partner, and fills in a short questionnaire. For questions with yes/no answers, she puts a 1 for yes and a 0 for no. Here are her answers, together with those of two possible mates, Ken and Josh:
The final two columns show the distance between Annie and Ken and between Annie and Josh for each factor. Annie/Ken have a total distance of 3, while Annie/Josh have a distance of 0. This makes Annie and Josh a perfect match according to this simple test.
Not surprisingly, however, this simplistic approach is flawed. What happens if the questionnaire includes questions with numerical answers that are not in the range 0 to 1, such as age?
Added to the previous result, Annie’s distance from Ken is now 2 years + 3 points from the other questions, making 5, while her distance from Josh is 6 years + 0 points, making 6. Since Ken has the lower distance, he is now deemed the more compatible by this scoring system.
There is something wrong with this. Annie is completely compatible with Josh on preferences, but this match is wiped out by their age difference, which is on a completely different scale. A better distance measure would make sure that each category was scaled to give it a similar degree of variability. The fact that Josh is six years older than Annie may be worth, perhaps, two points of difference, rather than the six shown above. Even yes/no questions may not all be on the same scale. If Annie’s main passion in life is cats, then it might be more appropriate to score your liking for cats on a scale of 0 to 5, so that it outweighs other less critical factors.
There is another potential problem with questionnaires, too. Here are some more of their answers:
Here Annie and Ken score a difference of 4 (they only match on garage music), while Annie and Josh score 2, so once again Annie and Josh are the more compatible. However, four of the five questions are closely related to each other. An interest in clubbing will tend to mean an interest in the various types of music heard in clubs, too. Since Annie and Josh both like clubbing, this is bound to have distorted the scoring in Josh’s favour.
It would be so much better if there was a statistic that could reduce the distorting effect of questions that were closely related to each other, such as the clubbing and music ones above. Sure enough, a statistic exists that does this and adjusts the scaling to remove anomalies like the age differences earlier. It is called the Mahalanobis distance, based on the principles that have just been discussed but calculated using an intimidating set of vectors and matrices that will do nothing but confuse matters if reproduced here.
Mahalanobis is widely used in the field of statistical matching. For example, companies wanting to know when to advertise their products on TV will benefit from information that shows them which type of programme is watched by the sort of people who buy their product. A bit of data fusion, courtesy of Mahalanobis, will give them a good indication.
This approach would be great for use in the databases of dating agencies too – except for two things. The first is a tendency to distort the truth when describing oneself, which means that the distance calculation will only be finding the compatibility with the person you say you are. That might be very different from the real you.
The second is even more critical. There is an old saying that opposites attract. If there is any truth in this then the whole argument about matching by finding the shortest distance goes out of the window.
Maybe this is why most dating agencies don’t bother to use the data-matching techniques that are used by other database industries. Instead, they tend to leave it to chance and let nature take its course.
Competing for mates
Finding a suitable partner is a tough enough problem when there is no competition. The complexity increases enormously when a whole society of people are jostling to find their ideal mates. What’s the chance that people will find a mate, and what chance is there that, having found one, they will be happy?
Sociologists, counsellors and psychologists have devoted much thought to this problem, and mathematicians have also had their say.
In fact the stable-marriages problem was analysed with notorious success by two mathematicians called Gale and Shapley back in 1962. They came up with a series of instructions – known as the Gale-Shapley algorithm – for setting up couples that guarantees a high level of satisfaction, at least for one half of each couple. All that is required for the model to work is the same number of males and females, with a fundamental assumption that all are seeking a partner of the opposite sex. We will stick with tradition and assume that it is the men who propose to the women, though everything that follows could of course apply in reverse. The routine goes like this:
• Each man in turn approaches the woman who is top of his list and proposes to her.
• If she has nobody with her at the time, she says ‘Maybe’, and the man stays with her.
• If she already has a man with her, she then chooses between the two, saying ‘No’ to the one she likes less. The rejected man then approaches the next woman on his list.
This continues until every man has found a woman who does not reject him. It may sound Victorian, but this process does at least ensure that every man ends up with the best woman who is willing to have him. The men can therefore be said to have achieved the best outcome they could have hoped for. Unfortunately things are not nearly so rosy for the women, who end up in what is usually a far from ideal arrangement from their point of view.
Take the following example, with three men – Andy, Brett and Charles – being paired up, most conveniently, with Xenia, Yvonne and Zoe. The order of preference is shown for each of them:
So, for example, Andy’s order of preference is Xenia, then Yvonne and lastly Zoe.
In the first round, Andy and Charles both choose Xenia, while Brett joins Zoe. This leaves Xenia facing a choice between Andy and Charles, and she opts for Charles because he is higher in her list. Andy, now rejected, therefore opts for his second choice, Yvonne. Each person has now become part of a couple.
Andy-Yvonne
Brett-Zoe
Charles-Xenia
Brett and Charles both got their first choices, while Andy ended up with his second choice. No wonder they are smiling. But their partners don’t look nearly so happy. None of them got their first choice, and in fact Yvonne and Zoe both ended with their bottom choice.
This approach produces a different outcome if it is the women who are doing the proposing. In this case, going through exactly the same process but with the women in the box seat, the couples end up as:
Xenia-Charles
Yvonne-Brett
Zoe-Andy
Now it is the women who are happier, with Yvonne and Zoe paired with their first choice male and Xenia landing her second choice.
In fact, the Gale-Shapley algorithm will always favour the proposers. Whatever else you might think of it, one benefit of the approach is that the marriages it produces are ‘stable’. That is, however much one partner may want to split up, no other partner will prefer the would-be divorcee to the partner he or she already has.
As with so many mathematical theories, once the algorithm had been applied to marriages, people began to see applications to all sorts of other areas where partners were being brought together. For example, what about qualified doctors applying to work at hospitals, or prospective tenants looking for a house to lodge in? By following Gale-Shapley’s rules, whatever the final arrangement, one party will be as happy as it can be, although there may be some squabbling over which side gets to be the proposer, since overall they will be the winners.
There have been attempts to improve on this algorithm, to come up with solutions that are closer to the ideal for both sides. Unfortunately, real life throws up complications that limit the effectiveness of even the cleverest systems. One problem is that some people refuse to accept second best. Once their ‘ideal’ partner has turned them down, they would rather stay single than commit to anyone else. Even worse, however, preferences change over time. The ideal partner of today may be very different from the ideal partner in ten years’ time. Stable partnerships therefore become unstable.
It would seem that the ideal system for identifying the perfect partner needs to consider a number of different factors. It needs to be able to detect genuine signals from distorted ones. It needs to be able to forecast who else is going to turn up. And, most of all, it needs to have the power to forecast how people will change. Needless to say, the perfect system for finding the perfect mate hasn’t been found yet.