Tuesday, August 2, 2011

Relationship advice for economists: Don't do it with models.

Economics has plenty to say about relationships, both in terms of individual behavior and how it aggregates. Today I want to frame the individual's problem as a peculiar sort of search problem.

For today, take as given that each individual has a bunch of characteristics, which make different people better or worse matches for each other. For instance, you may prefer someone who likes football as much as you, or someone who shares a lot of your beliefs.

The objective, loosely, is to spend as long as possible with as good a match as possible. But there is incomplete information. You don't know anyone else's vector of characteristics, and you have to meet and get to know them to find out. Some characteristics are uncovered quickly (e.g. physical appearance), others take weeks, months, or years to discover ("just how much can I trust this person?").

So you set about to find a good match, repeatedly sampling from a pool of candidates. You're dating someone, all the while learning more about them and their compatibility with you. On any day you can either opt to continue the relationship another day, or reject your partner and move onto the next possibility. In this simple example, the expected value of rejecting/starting anew is constant, so as usual the problem gives rise to a rejection threshold: if you should ever learn enough bad information that the expected value of continuing the relationship drops below the expected value of moving on, you move on.

Now here's a twist. All through your life, you continue to meet people and learn things about them, whether in a relationship or not. But if you are in a relationship, there's a cap on how much you're allowed to learn about anyone else. Conversely, there's a cap (not necessarily the same) on how much you are able to learn about someone else (in terms of relationship compatibility) without being in a relationship with them.

Together, these say that to find out whether someone will be a good enough match for you, you will eventually have to enter into a relationship, and once in a relationship, your outside search behavior is critically limited. Also, we are now imagining a model where the expected value of moving on is not constant, but rather depends on your current best candidate outside of the relationship, which is determined by what you've learned about them. (You can think of this as the best candidate who is willing to be in a relationship with you).

Excuse me while I make up some terminology. Say that a relationship is t-stable if in period t, neither party prefers to match with their best candidate outside of the relationship. That is, a relationship continues to the next period precisely if it is t-stable. Furthermore, say that a relationship is t-superstable if it would be t-stable for any possible best outside candidate. That is, given each partner's period-t expected value of the relationship, there's no possible outside option that could entice them away from their current relationship.

(The t is there to remind us that this really is a dynamic system that unfolds over time; superstability is a function of the information you currently know.  You could have a really great first date and be superstable in the first period of the relationship, but lose it later on as more characteristics become known.  And you may never know whether your relationship will last into old age; people change in ways they can't entirely predict. Perhaps it takes being 50 to know that you'll be compatible as 50-year-olds.   Keep it in mind, but for the rest of the post I'm just going to drop the t.)

And let me be clear that superstability does not mean you're matched with your "soul mate," your best possible match in the universe.  It means that out-of-relationship signals are simply not informative enough to compete with all the great things you know for sure about the person you're with.

To be sure, whether we should expect to find superstable matches depends on the particular social rules that govern the sort of search people are allowed/able to do inside and outside of what society has defined as a "relationship."  But I think the answer is generally yes.  Compatibility depends on so many important things that are impossible to observe outside of a relationship; attractiveness and surface personality and so forth are ultimately not a large enough part of the quality of a match. After enough searching, you may find yourself in a particular relationship and discover that it's a really great match, way better than you expected beforehand, way better than you could ever expect beforehand, from anyone. And then you will be done.

I know this is a long post, but I don't like to split these things up into disjoint pieces.  Read on if you want to see what else falls out of the model, now that we have it.

Observations, questions, extensions

Which sorts of characteristics are really important for match quality?  I haven't said anything about how characteristics get converted into match quality, but presumably the function is not simple.  It takes the two vectors of characteristics and combines them in a potentially complicated way.  In reality, perhaps we observe the match quality directly, but not the function that produced it from the characteristics.  We can theorize about the relationship between characteristics and quality, but how good should we expect these theories to be?

If we know that certain characteristics lead to higher match quality, we can put a lot of effort into surrounding ourselves with people who have those characteristics.  We can apply to schools or jobs that attract like-minded people, and let the process do a lot of screening for us.  On the other hand, if we don't actually have a good idea of where match quality comes from, then our attempts to screen for preliminary characteristics are largely a waste of effort.  If we mostly need to be in a relationship to find out the quality of the relationship, then we should try to have more relationships, not "better" ones.


I glossed over behavior out of a relationship above, but the model actually doesn't rule out it being optimal to spend time out of any relationship, even if there are currently people willing to match with you. That comes from keeping the caps separate, as I said:
If you are in a relationship, there's a cap on how much you're allowed to learn about anyone else. Conversely, there's a cap (not necessarily the same, presumably higher) on how much you are able to learn about someone else without being in a relationship with them.
By becoming close friends with someone while single, perhaps you can get relatively far into a relationship while still being allowed to search over other people. Once you are in a relationship, there may be a hard bound on how much time you can spend with other people you might be interested in. But out of a relationship, it may be that you can get pretty close to multiple people, without "dating" and thereby activating the restriction on how you spend your time outside of that particular relationship. Not necessarily a bad idea. Yes, dating can irrevocably destroy a friendship and all the time that went into it. But on the flip side, the time you spend in an ultimately failed relationship is more costly per day, since it more seriously restricts your search behavior. So the more you know about someone prior to starting a relationship with them, the less you have to figure out when time is being billed at a higher rate.

Another reason to stay (or become) single that's not captured by this model is this: when you're single, you are significantly more likely to discover who else is interested in you. Here we're assuming this is just known, but clearly it's private information in real life...info which people are reluctant to give up, especially if you're spoken for.


In a match, quality along easily observable characteristics versus less observable characteristics says something about how the match is likely to have been initiated. For example, if a really high-quality person is matched with someone who's low-quality along the most visible dimensions, it is more likely that they knew each other for long enough to uncover some of the less visible dimensions before dating. If you feel your most easily-observable characteristics don't adequately represent who you really are, you may have much better luck cultivating friendships than asking people out on the fly.


And if you do ask someone out on the fly, their expected unobservable quality could even be decreasing in their attractiveness, conditional on them saying yes. Lower quality people are more willing to say yes, so if they're willing and attractive, it could mean that they are deficient in less observable categories. Similarly, if you know your friend is really high-quality, and her boyfriend seems "not good enough" when you first meet him, then you might actually conclude that he's better than expected along the deeper dimensions you can't observe in one interaction. Better, perhaps, than you'd expect a really handsome guy to be.  (But I don't know how far it makes sense to push this line of counterintuitive reasoning.  Maybe your friend is just deluded.)

Empirically, we may find that relationship length is decreasing in attractiveness of the other party, over the range of time during which a lot of hidden characteristics are discovered. Hmm. (Perhaps economists shouldn't do it with models after all).

Note that these effects are dampened along dimensions where there is widespread disagreement about "quality."  It could be interesting to think through how more and less observable characteristics break down into matters of personal taste versus widespread agreement over quality. 


Finally, a note on this search model vis-a-vis the matching literature. The Gale-Shapley Deferred Acceptance algorithm is beautiful and simple, but it assumes complete information. How might we adapt the notion of a stable matching (across all of society) to our environment?

Well, we could say a t-stable matching is a matching where every individual relationship is t-stable. But it seems to lose its edge, no? As soon as we introduce incomplete information and a cost of searching (here it takes time), the space of t-stable matchings gets very large, and there's no reason for any such "stable" matching to survive for any length of time. On the other hand, a "t-superstable matching" is much closer to the Gale-Shapley notion of "stable matching." But in general there's no reason to expect a t-superstable matching to exist, so we might want to think about conditions that would ensure it.  I suspect they would be far too restrictive to be realistic though.  Where would we take this?  On the one hand, incomplete information is nice in that it potentially supports a lot of matchings that we see in the real world, which would not survive under complete information.  But on the other hand, that also means the theory isn't going to give any sharp predictions.  And the matchings are unlikely to be stable over time, because there is constantly new information flowing, which changes people's decisions.  I suppose superstability could potentially be a decent solution concept for the subset of people who are married, but this isn't without its problems either.

Put a bunch of people in a room. Give each person one observable characteristic, and one characteristic that can only be discovered by spending a period with them. What happens?

Bayesian approaches to matching have hardly been touched.

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