The mathematics of marriage, or, less happily ever after

An article I recently ran across in PLoS One (Rey J-M, 2010 A Mathematical Model of Sentimental Dynamics Accounting for Marital Dissolution. PLoS ONE 5: e9881 doi:10.1371/journal.pone.0009881) sets out to provide a mathematical framework for understanding why many marriages fail.  I’d say that qualifies as a topic of general interest, though many might doubt the ability of mathematics to offer insight.  As so often with mathematical treatments of complex problems, one of the key benefits of creating the model is that it forces you to make explicit your assumptions.

Rey’s basic assumptions are these:

1. The initial level of mutual affection, x(0), has a natural tendency to decline at a constant rate, r. Rey calls this, pessimistically or realistically depending on your point of view, the second law of thermodynamics for relationships.

2. Efforts made by a couple (c) can increase the level of mutual affection (x) with a certain efficiency (a)

3.  Effort has a cost, D.  The cost is not monotonic; some efforts are pleasurable. The cost of effort is at a minimum (and negative) at a particular level of effort, c*, and goes up from there.

4.  There is a utility function, U (the happiness you derive from your relationship), that increases with the level of mutual affection but eventually plateaus.

From assumptions 1 and 2 he develops the following equation:

dx/dt (t) = –rx(t)+ac(t).

(all terms are positive);

and using this and assumptions 3 and 4 he goes on to ask what is the optimal level of effort, c, to maximize a couple’s overall satisfaction with life, i.e. what is the trade-off between the happiness derived from the relationship and the cost required to sustain the relationship.   This leads to a system of differential equations in x and c, and so it is possible to use a rather standard kind of mathematical analysis to ask — is there a steady state?  If so, is it a stable one?  And the answers are yes, and no.

The steady state (the point in the coordinate space of c (effort) and x (affection) where dx/dt and dc/dt are both zero) for the set of equations defined by these assumptions is saddle-shaped and thus unstable.  In two of the four possible directions, if you move a little away from the steady state you slide back down towards it; in the other two directions, even the smallest perturbation takes you away from the steady state.  And remember, the “second law” states that x always decreases, if left to itself — so unless you precisely calculate the amount of effort that will exactly compensate for the tendency of x to go down, and thus balance yourself at the steady state, you’ll always be in a state of dynamic instability.  But there is a region in which, provided c(t) is consistently high enough, you can stay above a minimum level of x and maintain a functioning relationship.  Here’s a diagram of acceptable relationship space, which is Figure 3 in Rey’s paper [please ignore the fact that his mathematical notation includes little hearts, which I can’t help feeling detract from the seriousness of the analysis]:

The big result here is that the level of effort that is enough for stability is always higher than c*, the level of effort that you find pleasurable because it’s nice to please your partner.  In other words — you might find this less than revelatory — you have to work at a relationship for it to be stable.  Mathematically proven.  Although I do wonder about Rey’s assumption that c* is the same for all levels of mutual affection.  Wouldn’t you expect that c* would go up, at least slightly, with x?  Or am I a hopeless romantic?

One of the directions that takes you away from the steady state (W(u)+) is a direction of constantly increasing x and c; in this area, you are putting in more and more effort, but because this effort is effective x is increasing, which means that your happiness due to your relationship is increasing, so you can afford to put in more effort and still be reasonably happy.  The other  way to slide away from the steady state is shown by the line W(u)–.  This is the direction along which collapse happens.  Once effort declines to zero, there is nothing left to prevent the second law from taking over, and then x slides down to zero too.  (Actually, below zero — but this analysis doesn’t get into negative numbers.  Probably just as well).  Note also that effort is essential, but effort is not a panacea.  There are regions of the phase-space diagram where the couple is putting in a lot of effort but the cost of the effort is so large that they’re unhappy anyway.

Rey’s analysis of what makes a relationship sustainably happy is this: relationships will be stable if the effort gap, the difference between the effort you need for stability and c*, the level of effort you enjoy putting in, is tolerable.  If it isn’t, you will tend to reduce your level of effort below the level required for stability; this takes you out of the area where you can sustain an acceptable level of x, and into the unstable region.  Reduced effort leads to reduced affection, which leads to reduced happiness, which leads to a reduced ability to put in more effort while remaining tolerably happy.  As Mr. Micawber says in a different context: Result, misery.

So there you have it.  Marriage, or whatever committed relationship you’re allowed to engage in according to the laws of the land, is fundamentally (but not unavoidably) unstable.  Interestingly, PLoS One accepted the paper on February 14th — Valentine’s day.  Perhaps the editor responsible had relationship dynamics on his/her mind for some reason?  If so I hope they were able to determine their optimal c(t).

Rey JM (2010). A mathematical model of sentimental dynamics accounting for marital dissolution. PloS one, 5 (3) PMID: 20360987