# Sketchy dating after breakups

(I thought I’d put this post up now, since it relates to a friend’s recent post elsewhere.)

Generally, when people end a long-term relationship, they want to take a bit of a break from dating to get their feet back on the ground. Break-ups can be very emotionally taxing, and recovery takes some time. There are several rules of thumb as to how long one should wait; I won’t go into those, since that’s not the point of this post. What is interesting, though, is that often these rules are not well-kept. The question is, why?

For starters, let’s model a person’s payoff for entering a relationship. Let’s assume for now the person is a woman (also, let’s call her Fiona). Obviously she doesn’t want to enter one immediately after the breakup; but how much she does not want to do so depends on how much time has elapsed. More specifically, the payoff increases, eventually approaching a certain (bounded) value, at which point she is totally over her ex.

(Formally, we assign her a utility function $U(t)$, where $U(0)=0$, and $lim_{t\rightarrow\infty}U(t)=B$, where $B$ is some positive number. For example, when $B=1$, we could have a function like this:)

Fig. 1: sample graph

The guy who wants to ask her out also shares the same payoff (and we’ll call him Scotty). After all, of course he would – he’s only happy if she is, right? Thus it’s better for both of them if they wait longer to start up the relationship.

The thing is, Scotty doesn’t know if other people will have their eyes on Fiona. So, if he wants to lock her up as his only, he’s got to act quickly (by some time $\tau$). Suppose, for simplicity, he’s the first one to arrive on the scene (the same reasoning will apply even more strongly if there are others already competing with him for Fiona’s attention). Other suitors can be expected to arrive at a pretty much constant rate ($r$) if she’s still single. If he’s willing to ask her out by time $\tau$, then they definitely will by a later time ($t>\tau$), since they get an even higher payoff. Fearing this competition, Scotty will ask out Fiona at exactly the point where the gains from waiting are balanced by the losses in potential competition. (That is, $U^{\prime}(\tau)=rU(\tau)\geq0$.)

As the model is set up now, Fiona still has no reason to accept Scotty’s request. But if we introduce a cost of rejection (C) into the model, things change, even if such a cost is small. We can account for this as a natural consequence of social interactions: for example, things might be awkward between them if she turns him down. And no matter what, she cannot get more than a payoff of $B$ later. Thus, she will certainly accept as long as $U(t)\geq B-C$.[1] Though she’d have a higher payoff if he asked later, accepting this request is the best response to his move.

To close things off, I should explain why I assumed initially that the person was a woman. Since even in this age of gender equality, guys are generally the ones doing the asking, women will encounter the possibility of being asked out, even if they are not yet looking around for new opportunities to date. Hence they incur the cost, $C$. By contrast, men might not look until they can get a payoff closer to $B$, without incurring the cost $C$. This makes it more likely that this situation will come up when woman have recently broken up with their boyfriends, rather than with the men.

Obviously, both sides in this equation would rather wait longer to start something up. But it’s just too risky to do so, since they might lose out altogether. So, we end up with much sketchiness. Haaaaaaai!

[1] That is, this is a sufficient condition; she might accept an even lower payoff depending on how frequently she expects guys to ask her out later.