How can it be that despite the best intentions of many couples, such a significant proportion will not endure?  As one always should, we can turn to mathematics for possible answers.  In fact, José-Manuel Rey of the Department of Economic Analysis at the Universidad Complutense in Madrid has done just that, by proposing a mathematical model to explain the dynamics of long term relationships.  His model is based on the following assumptions: first, that the individuals in the couple have similar traits (this is spelled out more precisely in Rey’s paper, but basically this is so that he can consider one utility function for the couple rather than separate utility functions for each individual), and second, he assumes a so-called law of thermodynamics for sentimental relationships.  This law can be stated as follows: “There is tendency for the initial feeling for one another to fade away. This kind of inertia must be counteracted by conscious practices.”  In other words, the natural deterioration in a relationship must be counteracted by continual effort, such as a romantic evening, a heartfelt conversation, or listening to some slow jams by R. Kelly.

Given these assumptions, Rey constructs a model to try and explain the paradox (which he terms the failure paradox) in the fact that people who believe they will be together forever will, with a high degree of probability, end up separating.  And sure enough, his model offers us an explanation.  His work is perhaps best summarized by the following picture:

The graph above is of feeling (x axis) versus effort (y axis) – in other words, how satisfied is the couple in their relationship compared to how much effort they are willing to put into it?  Rey’s model asserts that there is a minimum happiness (denoted x_min above) below which a relationship cannot survive.  In other words, if your happiness crosses the red vertical line, your relationship is doomed.  The couple’s initial happiness is given by the value x₀ – note that this is greater than the happiness of the equilibrium point, reflecting the common observation that happiness within a couple frequently decreases from its initial state (you can think of this as the honeymoon period, if you like).  This isn’t to say that couples in long term relationships are unhappy, for indeed the happiness at the equilibrium point is still above x_min; all this is saying is that happiness is lower at the equilibrium stage than it was initially.

What about the effort involved?  Heuristically speaking, the amount of effort you put into a relationship should increase your own happiness, up to a point, but then should begin to negatively affect your happiness.  Driving your partner to the airport every once in a while will probably make you feel good about yourself, but driving your partner to the airport every weekend will probably start to wear on you.  On the graph above, c* represents the turning point from when the amount of effort you put in positively affects your happiness to when it negatively affects your happiness.

Here is where an important feature of the model comes into play: the amount of effort at the equilibrium point is greater than c*!  In other words, in order to maintain equilibrium in your relationship, you need to be putting in more effort than you would optimally choose to.  It is this s0-called “effort gap” that Rey identifies as being the cause of so many problems.  Note that if the effort gap is large enough, then maintaining the appropriate level of effort may drive happiness below the minimum required value, and even if this is not the case, maintaining the appropriate level of effort will still require some loss of the couple’s happiness, since the effort required will always be greater than c*.  To put it more bluntly, relationships require sacrifice.

I’d encourage you to keep this in mind as you navigate the unpredictable seas of love.  When it comes to the effort involved in keeping your relationship afloat, mathematics warns you against complacency.  When in doubt, it’s always best to go the extra mile.  And if you need help wooing your special someone, you can always call on R. Kelly to help set the mood.

Aziz Ansari does a mighty fine impression of Kells.

Some time ago, I wrote an article on the optimal way to select a mate, assuming you know how many eligible partners exist, and that once you’ve dated someone, you can’t go back and date them again (sorry, Drew Barrymore and that dude from the Apple commercials).  This is less romantically known as the secretary problem.  Let me briefly recall the problem and its solution: suppose you have n candidates, from which you want to pick the best one.  This applies to a variety of situations, from hiring a secretary to finding a girlfriend to apartment hunting.  In either case, the outcome is the same: you should look at roughly the first n/e of them (yes, that e), and then select the first one after those n/e which is better than all that you have seen so far.  While this strategy won’t guarantee you get the best choice, it will give you the best choice around 37% of the time.

The major problem with this model is that in many situations, the value of n is unknown.  There are ways to circumvent this problem, which I will not discuss here.  Instead, in the context of finding a mate, I offer the following method to calculate the number of partners you could reasonably expect to find in your area.  This method recently gained some attention when Peter Backus, a Ph.D. candidate from the University of Warwick wrote a paper titled “Why I Don’t Have a Girlfriend.”

The basic technique involves modifying the Drake Equation, an equation used to estimate the number of potential extraterrestrial civilizations in our galaxy.  For those who have never been introduced to this equation, it asserts the following:

N = R · f_p · n_e · f_l · f_i · f_c · L.

According to Wikipedia, these variables represent the following quantities:

R = the average rate of star formation per year in our galaxy,
f_p = the fraction of those stars that have planets,
n_e = the average number of planets that can potentially support life per star that has planets,
f_ℓ = the fraction of the above that actually go on to develop life at some point,
f_i = the fraction of the above that actually go on to develop intelligent life,
f_c = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space,
L = the length of time such civilizations release detectable signals into space.

Given estimates for all of these parameters, one could then estimate the number of civilizations in our galaxy.  Since we don’t know the values of any of these parameters, however, this is more of a thought experiment than anything else.

Nevertheless, the idea can be easily modified to try and find the number of eligible mates in a given area.  Peter Backus’ approach is fairly specific to him, but he links to a more general approach discussed here, in which the following equation is presented:

n = P · f_t · f_o · f_c · A · R

In this case, the parameters are given by

P = the population.  This could be the population of your university, your city, or your country, depending on how ambitious you are.
f_t = the fraction of that population which you would want to mate with, in broad terms.  If you’re a straight male, this would be the fraction of females.  If you’re a gay male, it would be the fraction of males, and so on.
f_o = the fraction of the population you’d want to mate with which wants to mate with you.  For example, if you’re a straight male who wants to mate with females, this will compensate for the fact that some females will be lesbian and therefore unwilling to mate with you.
f_c = author Raymond Francis labels this the out fraction, and describes it as the answer to the question “Of the people in your target gender and orientation, how many of them are open enough about their sexuality to engage in a relationship of the sort you’re hoping for?”  If you are straight, this value is likely 1, or very close to it.  If not, things can be a little bit fuzzier.
A = the fraction of those remaining who fall within your desired age range.  This is, of course, personal to you – if you’d like a socially acceptable age range, you could follow the “half your age plus seven” rule.
R = factors for any remaining filters you wish.  Do you want your partner to have a certain level of education, or a certain income?  Do you need a non-smoker, or demand a euphonium player?  Here’s where you can fold that into the mix.

Yes, Wikipedia has a graph illustrating the half your age plus seven rule. Amazing.

With all these parameters accounted for, N will give you the number of potential mates in your area.

Let’s take this equation for a spin, shall we?  Suppose you are a straight male living in Los Angeles, and looking for a girl to date in Los Angeles.  According to Wikipedia, the estimated population of LA as of 2008 was 3,833,995.  Of those, let’s say that 51% are female, and of the females, let’s posit that 90% are straight or bisexual.  f_c should be high in this case – to be conservative, let’s put it at 95%.

To estimate the age filter, one can obtain some data from this site.  Suppose you are a 25 year old man – then, absent any personal preference, the socially acceptable age range of women for you to date is between 19.5 and 36.  According to census data, in 2000 there were 3,694,820 people in Los Angeles, and of them, 974,004 were between the ages of 20 and 34.  Additionally, there were 251,632 people between the ages of 15 and 19, and 584,036 people between the ages of 35 and 44.  If we make the assumption that ages are roughly uniformly distributed within these brackets, this gives us an additionlal 141,970 people either between the ages of 19.5 and 20, or between the ages of 35 and 36.  Combining this gives a total of 1,115,974 people between the ages of 19.5 and 36 in Los Angeles in 2000, or roughly 30% of the population.  Let’s use this for our value A.

Assuming you have no other restrictions (i.e. taking R = 1), this gives us n = 3,833,995 · .51 · .9 · .95 · .3 = 501,544.  That’s a lot of ladies out there for the taking.  Of course, taking R = 1 is probably unrealistic.  It’s unlikely you want to date women who are married, for example, and everyone has their own personal taste that will decrease the pool even further.   Once you’ve calculated your personal value for R, however, you then know how many eligible mates will be in your area.  Given that, you’ll know how large n/e is, and then you’ll know how many people you should date before you think about settling down.

Although Peter Backus has received a fair amount of buzz for the short paper he has written on this idea, he readily admits that he is not the first to think of applying Drake’s equation in this situation.  I’ve discussed mostly Raymond Francis’ approach here, but Backus has links to many other people that have discussed the idea on his website.  In particular, here’s an exchange from CBS’s The Big Bang Theory (don’t worry, there’s a laugh track so you know when things are supposed to be funny).

In summary, not only can the Drake Equation be used to consider the existence of extraterrestrial life, it can also be used to consider potential mates right here on Earth.  The next step, of course, is obvious: we must combine these two equations to calculate the number of potential extraterrestrial mates.  Undoubtedly the number will be small, but one should never underestimate the power of love.

Interstellar love is a wonder to behold.