# You’re one in a (quarter of a) million: the mathematics of finding love

Okay, enough on the physical aspects of attraction (this isn’t Love Island – but if you’re interested go and read my previous article). Let’s take a look a bit deeper at those aspects of attraction that are beneath the skin.

There are a number of traits that science has proven to be commonly attractive for us humans, with regards to heterosexual relationships (unfortunately, there’s a huge lack of data on homosexual and other types of relationships). These include: a good sense of humour, owning a pet, socialising in groups, and even a good credit score [1]. So, even though we all have our own personal preferences, we all generally look for similar things (just like with physical attraction). Okay, great, now we have our checklist! But now, how do we find someone that ticks all of those boxes?

In times gone by (and by that I mean anything before the Noughties), the norm was to go out, meet someone either on purpose or at random and start chatting to them to see if they tick off any of those boxes on your mental checklist. (I’ve said “mental checklist” because it might be a bit strange if you had taken a physical checklist and were ticking it off as your potential partner speaks to you.) That’s all well and good, but what are the chances of actually meeting someone that ticks ALL of your boxes?

“You’re one in… er roughly 285,000”

Back in 2010, one lonesome teaching fellow at the University of Warwick called Peter Backus decided to take a well-known equation called the Drake equation to estimate the likelihood of him finding a ‘suitable’ girlfriend [2].

The Drake equation was actually originally developed as a way to estimate the number of highly evolved civilisations (essentially aliens) that might exist in our galaxy – The Milky Way. It goes like this…

The equation is generally specified as:
$$G = R \times f_p \times n_e \times f_l \times f_i \times f_c \times L$$
where
$G =$ the number of civilisations capable of interstellar communication,
$R =$ the rate of formation of stars capable of supporting life (stars like our Sun),
$n_e =$ the average number of planets similar to Earth per planetary system,
$f_l =$ the fraction of life-supporting planets where intelligent life develops,
$f_c =$ the fraction of planets with intelligent life that are capable of interstellar communication (those which have electromagnetic technology like radio or TV),
$L =$ the length of time such communicating civilisations survive.

The Drake equation [3]

Backus thought to himself: “Hey! The minuscule chances of finding aliens seems pretty similar to my chances of finding a suitable girlfriend…” and decided to see if he could apply the Drake equation in this context. He first re-defined the parameters as follows, with the numbers or estimates shown in brackets:

$G =$ the number of potential girlfriends

$N^*$ = the population of the UK as of 2007 (60,975,000)

$f_W =$ the fraction of people in the UK who are women (0.51)

$f_L =$ the fraction of women in the UK who live in London (0.13)

$f_A =$ the fraction of women in London who are age-appropriate (0.20)

$f_U =$ the fraction of women in London with a university education (0.26)

$f_B =$ the fraction of university-educated, age-appropriate women who I find physically attractive (0.05)

Then, given these parameters, Backus stated that:

$$G = N^* \times f_W \times f_L \times f_A \times f_U \times f_B =$$

$$60,975,000 \times 0.51 \times 0.13 \times 0.20 \times 0.26 \times 0.05 = 10,510$$

So, there are just over ten thousand people in the UK that satisfy Backus’ girlfriend criteria! In real terms, this means that, on a given night out in London, there is a 1 in 1000 chance of Peter meeting a woman who is “girlfriend-worthy”. That’s pretty good odds, right?

Well, actually, we’re forgetting quite a few important factors still… To name a few: the woman in question would need to find Peter attractive too, she would need to be single, and maybe most importantly, they would need to get along. Roughly estimating these factors as 0.05, 0.5 and 0.1 respectively, those 10,510 women suddenly become just 26 women. For context, this would reduce Peter’s chances of meeting a woman who is “girlfriend-worthy” on a given night out in London to just 1 in 285,000. Pretty slim. Maybe some of Backus’ criteria are a little picky, but it really does make you wonder how anyone would find a partner just by chance. That’s where online dating comes in…

A mathematical match: OkCupid

This “go-out-and-leave-it-to-luck” method of finding your partner has multiple downfalls: it’s slow, you have to actually leave the house and, most of all, it involves physically meeting a pretty much stranger. Luckily, following on from the widespread adoption of the Internet, a handful of people began to wonder if and how we could use the World Wide Web to solve some of the dating difficulties encountered by adults.

Among these people was a group of four young mathematics graduates from Harvard University; Chris Coyne, Christian Rudder, Sam Yagan and Max Krohn [4]. They collectively founded the company OkCupid in 2004, as an “improved” version of some of the earliest online dating websites like Match and eHarmony [5].

Like other dating websites, OkCupid uses algorithms to decide whether two people should go on a date and, to provide data for the algorithms, asks users a bunch of questions. These can be anything from “How often do you go to the gym?” to “Do you like alone time?”

With some questions, to find a good match, OkCupid should match like-with-like. For example, two people who go to the gym every day are more likely to be a good match than one person who goes to the gym every day and another who never goes to the gym. But, with other questions, matching answers on a like-with-like basis could lead to a disastrous match. For this reason, OkCupid not only lets the user answer the questions themselves but also to choose how they’d like other users to answer and how important others’ answer to this particular question is.

The founders of OkCupid decided to create the following importance scale:

• Irrelevant = 0
• A little important = 1
• Somewhat important = 10
• Very important = 50
• Mandatory = 250

When matching two people, the algorithm then seeks to answer the following questions:

1. How did Person A score on Person B’s scale?
2. How did Person B score on Person A’s scale? [6]

To illustrate this, let’s give an example using the two questions above:

Person A and Person B are asked a common set of questions (just two for simplicity) that have multiple-choice style answers. Here are the questions and their answers…

Person A

Person B

Okay, so now we have our data. Next, we can work out each person’s scale…

For Question 1, Person A can score a maximum of 50 (“Very important”) on Person B’s scale. Person B can score a maximum of 10 (“Somewhat important”) on Person A’s scale.

For Question 4, Person A can score a maximum of 10 (“Somewhat important”) on Person B’s scale. Person B can score a maximum of 1 (“A little important”) on Person A’s scale.

So Person A could score a possible total of 60 points on Person B’s scale and Person B could score a possible total of 11 points on Person A’s scale.

Now we can go about finding how each person scores on the other’s scale…

Person A scores 50/50 for Question 1 and 0/10 on Question 4 on Person B’s scale. So Person A scores 50/60 (83.3%) on Person B’s scale. Person B scores 0/10 on Question 1 and 1/1 on Question 4. So Person B scores 1/11 (9.09%) on Person A’s scale.

The algorithm can now tell us how compatible Person A is with Person B from each person’s perspective, but we need to turn this into one singular compatibility rating. OkCupid does this by using the geometric mean, which is suitable for data sets with a large range, and lots of data points representing very different properties.

The geometric mean of n variables can be calculated as follows:

The geometric mean of $n$ variables can be calculated as follows:

$$\sqrt[n]{x_1 \cdot x_2 \cdot \cdots \cdot x_n}$$

Using this on the scores of Person A and Person B, we get:

$$\sqrt[2]{\text{Person A’s score} \cdot \text{Person B’s score}}$$

So in our example, Person A and Person B would get a compatibility percentage of:

$$\sqrt{83.3\% \times 9.09\%} = 27.52% \text{(to 2 d.p.)}$$

This method obviously isn’t failproof; OkCupid users might be put off by being asked to answer a large set of questions, the numerical figures applied to the importance scale might not represent it accurately, and, of course, users might not answer truthfully. But, relatively speaking, many people are likely to have much more dating success using this method rather than just going out and hoping to find someone suitable. At the very least, your chances of finding a suitable partner on any given day should be more than 1 in 285,000. Maybe Peter Backus should have given OkCupid a go.

Finding ‘the One’

Great, now we’ve (sort of) mastered how to find a partner to date – hurrah! But what about something more long-term? What about finding the right person to marry? Surely mathematics can’t help with that…

Well, Dr. Hannah Fry certainly thinks it could! Dr. Fry is a British mathematician with a passion for studying human behavior and how mathematics can apply to them.

Back in 2015, she gave a TED talk titled The Mathematics of Love, which was based on her recently released book The Mathematics of Love: Patterns, Proofs and the Search for the Ultimate Equation. In this mini-lecture, Dr. Fry explains her top tips for finding love, and in particular, for finding “the One” – the person you should settle down with. She suggests that by using optimal stopping theory, we can decide when to settle down to give us the highest chance of long-lasting happiness with a partner.

In particular, Dr. Fry’s suggestion is:

You should reject the first 37% of the total number of people you date, and after this point, choose the next person who comes along who is better than everyone else you’ve previously dated.

So, what is optimal stopping theory? And, more importantly, where does the figure of 37% come from? When applied to dating, optimal stopping theory can be described by the following equation:

$$P(r) = \frac{r-1}{n} \cdot \sum_{i = r}^n \frac{1}{i – 1}$$

Here, $P(r)$ is a probability – the probability of your choice of when to stop being the optimal decision, or in this case, the probability that the person you choose to marry is indeed “the One”.

To explain the rest of the equation, $r$ is the total number of people that you reject and $n$ is the total number of people that you date in your life. I should note here that there are some important rules to optimal stopping theory in this context too:

Rule 1: No going back – that is, you can’t go and date your ex and you can’t take someone back once you’ve declined them. Pretty solid advice for dating if you ask me.

Rule 2: You can’t see into the future (you don’t know who you may date in the future). Again, pretty realistic if you ask me.

To find “the One”, what you essentially want to do is maximise the probability. However, you probably don’t know the exact number of people you will date in your life, n. What might be easier to estimate or know is the period in which you will be dating during your life (maybe between the age of 16 and 36, for example). We can then adapt the above formula to a time-based formula, and using a continuous-time model between the two ends of your dating period, we find that to maximise $P(r)$ – the probability of success – you should reject all people within the first 37% of your dating period and then choose the next person better than all those you’ve met so far [7].

It might sound a little cold and methodical, but some species of fish have already adopted this technique when it comes to choosing a mate in the wild. They reject every possible mate that they encounter in the first 37% of the mating season and then choose the next fish that comes along that is better than all the others that they met in the 37% time window.

Of course, humans are just ever so slightly different to fish, but the idea of “playing the field” early on in our dating (or mating) career and then looking for the next person better than all of those before still remains [8].

The secret to a happy marriage

So, you’ve found that perfect person – The One. Now, all you’ve got to do is keep them. It’s easier said than done when 42% of all couples in the UK get divorced. But fear not! Maths may be able to help with that too…

John Gottman is a psychologist who focuses his research on studying why marriages succeed or fail. He developed the Specific Affect Coding System, which measures how positive or negative the exchanges between married couples are.

With the help of mathematician James Murray, Gottman was able to determine that the thing that had the greatest impact on whether or not a marriage would be successful was the influence of each person on their partner. They derived a formula to predict the positivity/negativity of the verbal response of each person in the couple:

It should be pointed out here that, although the formula is written in the context of a husband and a wife, no gender biases were taken into account when deriving the formula. This means that, in theory, the formula could be applied to couples of any gender.

Gottman, Murray, and their research team found that the couples who were most at risk of divorce were the couples with high “negativity thresholds”, which seems pretty illogical. The “negativity threshold” is defined as the point at which one partner’s negative effect becomes so great that it renders the other partner unwilling to diffuse the situation with positivity and instead to respond with more negativity.

In the above graph, $T_{-}$ is the “negativity threshold”, $T_{+}$ is the “positivity threshold” (the point at which anything more positive is much more likely to see the couple draw themselves into a nice, stable conversation with lots of positive reinforcement), $I_{HW}$ is the influence the husband has on the wife and $H_t$ is the husband’s response at time $t$. Whenever the dotted line is high on the $I_{HW}$ scale, the husband is having a positive impact on his wife, but when the line dips below zero on the $I_{HW}$ scale, the wife is more likely to be negative in her next turn in the conversation [9].

So how do these thresholds have an impact on the success of marriages? Essentially, the most successful marriages are those where when things bother one partner, they speak up immediately and don’t let small things spill out of control. For those of you looking for more marriage tips, it’s also important to point out that I’m not saying you should yell at your partner or threaten to break up with them every time they do something you disagree with. A key point in the results of the study was that negative comments should be made gently and supportively rather than aggressively.

Using this idea of the “negativity threshold”, the researchers found that they could predict whether a married couple would divorce with 90% accuracy. Pretty impressive (and kind of terrifying), hey! [10]

And they all lived happily ever after

Well, there you go, now you know how to find a lovely date, decide when to settle down and even, to some extent, how to stay with that person! Maths really can help with everything (sorry, but your childhood maths teacher was right). Just maybe don’t rely on equations alone to guide your relationship – we all know how easy it is to make errors!

##### Emilia Hurrell

Emilia is a UEA Mathematics graduate, Norwich local, and incoming Consulting Development Analyst at Accenture. Maths-wise, she is particularly interested in statistics and computing, as well as their respective applications to the finance industry. Outside of spending hours of her life puzzling over numbers, she also enjoys a good ol’ pub quiz, shopping, and traveling with her mates.