Relationship Cycles

[Apologies in advance to anyone who finds this post sexist. I'm just having a bit of fun here.]

Recently, I casually mentioned to a group of guy friends that I thought many guys would be happiest with a 3-day-a-week girlfriend. Everyone present agreed. Some also expressed the suspicion that most women want a 7-day-a-week boyfriend. I bought this at first, until I realized women couldn’t possibly like us that much. Naturally, I have another theory.

I figure that everyone, men and women alike, derives diminishing marginal utility (MU) from relationship time. But the MU diminishes more quickly for men than for women. The figure below illustrates the difference.

For men, the MU of relationship time is given by MU(RM). It crosses the MU of alternative activities, MU(A), at 3 days per week. For women, the MU of relationship time is given by MU(RW), and it crosses MU(A) at 5 days per week. Thus, women’s optimal number of relationship days exceeds that for men, but it’s still less than 7. (Slightly different assumptions could generate essentially the same conclusion. For instance, men and women could have the same MU of relationships, but men could have a higher MU of alternative activities, thus generating the same qualitative result – a lower optimal number of relationship days per week.)

But what explains the perception that women want 7-day-a-week boyfriends? Now we come to my theory of relationship cycles. When two parties to a relationship have different preferences about timing, even a small divergence can create the illusion of a much larger one. Say that Harvey and Matilda have spent three days straight together. At this point, Harvey requests some alone time. If Matilda agrees (despite her desire to continue another couple of days), both parties get to reset their relationship clocks. Once Harvey and Matilda reconvene, say, four days later, Harvey’s and Matilda’s relationship-time MU’s have both risen back to their initial levels. But in another three days, Harvey will be asking for alone time again. Matilda never has to ask for alone time, because Harvey is always satiated sooner. Thus, the pattern of interactions creates the appearance of one-sidedness: Harvey keeps asking for alone time, while Matilda always want more together time.

The pattern would emerge even for a smaller difference; e.g., if Matilda’s optimum were 4 days per week, Harvey would still always want alone time earlier. And if the pause in between meetings is shorter, the effect gets stronger: since Harvey hasn’t had enough time to “decompress,” his MU of relationship time hasn’t risen all the way back to its start value, and so he reaches the saturation point even sooner.

The phenomenon of relationship cycles occurs in other interactions as well. Take phone call frequency. Say Ted would like to talk on the phone every two days, whereas Sheila would like to talk every day. You might think Sheila would call Ted about two-thirds of the time – but in fact, she will call him every time. If they talk on Monday, Ted plans to call on Wednesday; but then Sheila calls him Tuesday. His clock reset, Ted plans to call on Thursday. And then Sheila calls on Wednesday. Eventually, Sheila decides Ted doesn’t care about her, because he never calls.

Or consider sex frequency. Terry would like sex every other day, whereas Pat would like sex about twice a week. If they were separated for a week, Pat might initiate sex. But when they are in regular contact, Terry will always initiate before Pat does. Each time they have sex, both sex clocks get reset, and Terry’s fuse burns faster. At some point, Terry begins to doubt Pat’s affection, because Pat never initiates sex.

At this point, it seems natural to discuss the need for communication and compromise in a healthy relationship. But I’ll leave that to the experts.

UPDATE: I described the basic theory. Alex Tabarrok provides the empirics. And Eric Rasmusen extends the theory with a little Bayesian updating.