I’ve previously discussed some mathematical approaches to dating. Specifically, we have seen how choosing a partner can be modeled as a type of secretary problem, and, if you like, you can estimate the number of candidates you should consider by using a modified Drake’s equation. However, as you know, building a lasting relationship is about more than choosing the right partner; maintaining a happy relationship takes work. And even though most people go into a relationship believing they will not end up as a statistic, the unfortunate reality is that nearly half of all marriages in this country will end in divorce.

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 x_{0} – 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.

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