I got married today, to the particular fellow mentioned in my Turning 30 post. In a sort of 'inverse cat tax'[1] for a sappy announcement, here's a mathematical model of whether you should expect to like your partner more than yourself. I don't mean this in a moralistic way ('thou shalt love thy parents'), tho that might be another post for another time, or necessarily in a utilitarian way ("I would rather they get this ice cream than me"), but as a matter of raw respect ("I think they're a better person than I am, according to my values").

For simplicity's sake, let's consider everyone as having a 'stat' vector and a 'preferences' vector with the same dimensionality, and giving a candidate partner a 'score' based on the dot product of those two vectors.  We'll assume that all of the stats are universally good (no one ever prefers an uglier partner over a prettier one, tho they might not care about physical attractiveness much). The preferences vector we'll normalize to have unit magnitude (so it's just an angle in N-dimensional space, basically, defined as a point on the positive sector of the N-spherical shell). For reasons, I'll run simulations with the stat vector as 6 dimensions with 3d6 per stat, leading to a discrete distribution a bit like a truncated normal, with no correlation between the stats.[2]

Let's start by considering the heterosexual version of the stable marriage problem, in which people partner up using the well-known Gale-Shapley algorithm, and a simulation with 1,000 each of randomly sampled men and women. Mating is highly assortative; a correlation between total stats of 85.7%, with 83.3% of people have an average total stat difference of less than 6 (the dimensionality).

The interesting result is that 91% of people like their partner at least as much as they like themselves, with an average net satisfaction of 1.24. Note that we haven't baked in any correlation between one's stats and one's preferences, and so this result is, in some sense, not very surprising. The preferences are exerting pressure on the partner (thru who you can stably match with) and not exerting pressure on the self, and so you should expect that pressure to result in higher other-satisfaction.[3]

So let's add an adjustment to the preferences with a scalable parameter corr, so that now it's the (renormalized) sum of the stat vector (times corr) and the previous preference vector (times abs(1 - corr)). As we smoothly vary this from 0 to 1, the average net satisfaction decreases to -0.2 (liking themselves more than their partner) and the fraction happier decreases to 21%. 

While the change in net satisfaction is relatively smooth, the change in fraction happier looks much more sigmoidal, with the main drop between corr = 0.4 and corr = 0.8. The main change here is in self-satisfaction, which increases by about 5 points while other-satisfaction increases by only about 3 points.

You can also imagine situations where people specifically want their complements, rather than their mirror. Negative correlations between your stats and your preferences seem unlikely; a more appropriate model seems to be something like relationship satisfaction being a function of the minimum stat between the two partners (or the minimum plus half the maximum, or so on).

The 'marriage' situation with full bisexuality is typically called the stable roommate problem, solved with a similar algorithm. I'll leave it as an exercise for the reader how that impacts the results.[4]

 

Anyway, my sense is that when people talk about their 'better half', they're mostly being serious, and this is something that can easily be symmetric.

  1. ^

    On Imgur, it's common for cat owners to end posts that collect images of use for some other reason with a picture of their cat, referred to as the 'cat tax'.

  2. ^

    Of course in the real world, everything is correlated; not only is there g for intelligence, but GFP for personality, and wealth causes many material factors to be correlated, and so on. You could try to rationalize this by splitting out the 'natural' variables (like intelligence and wealth) into corrected variables (like intelligence and intelligence-adjusted wealth), but then it seems odd to have a uniformly random preference vector (as intelligence in the intelligence-adjusted model is more important than in the non-intelligence-adjusted model, given that some wealth-preference has now been moved over to intelligence). I currently don't expect that taking this into account will affect the analysis much (tho doing the analysis with univariate Gaussian stats leads to some odd effects with self-satisfaction, which I'm avoiding here to keep things simple).

  3. ^

    Correlation between stats and self-satisfaction, of course, is high (0.69), because we insisted that the preferences all be positive, and so people with higher stats will like themselves more accordingly.

  4. ^

    Naively, I would expect that everyone is more satisfied with their relationships (as they can sample from a wider pool). I think it's likely more assortative in terms of total stats, but it's a little unclear what will happen with the similarity (as corr increases) and what will happen to the crossover point of average net satisfaction (but I'd guess the 0 point is a bit to the right, with the 'increased satisfaction' effect swamping the 'when you try for people similar to you, you can get closer' effect).

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5 comments, sorted by Click to highlight new comments since: Today at 8:27 PM

Congratulations on getting married!

In the Gale-Shapley algorithm, there is an asymmetry between the two genders. One gender (typically the male) is proposing while the other is choosing. The resulting matching is the optimal stable matching for each member of the proposing gender, so I would think it makes a huge difference for your expected level of satisfaction if you belong to the proposing gender or the choosing gender. 

Are your statistics about both genders or just one of them? In either case, I would love to see separate statistics for the two genders. 

Thanks!

I would think it makes a huge difference for your expected level of satisfaction if you belong to the proposing gender or the choosing gender. 

For the simulations I ran, it was a pretty small difference, with the proposing gender typically (but not always) having an edge. Rerunning it now (with corr=0), I get an average other-satisfaction of 23.89 for proposers and 23.77 for reviewers.

Congratulations! I enjoyed reading this, it makes me slightly more optimistic towards towards dating.

Great post! Intuitively, it feels like allowing for negative preferences (i.e. disliking something that other people like, like chattiness or introversion) would actually increase equitableness and maybe allow for increased other-satisfaction. (Depending on what distribution we draw people from)

Rerunning the sim with 'non-negative preferences', I get an average self-satisfaction of 21.44 and average other-satisfaction of 23.82; doing it with a random preference direction I get an average self-satisfaction of 0.53 and average other-satisfaction of 5.43 (with 95% preferring their partner). So with preferences that are less correlated between people, the matching has more ability to drive up other-satisfaction, which makes sense.

[I was expecting the average self-satisfaction to be 0, so I'm not quite sure why it's so high; maybe the randomness of the simulations is high enough that's a reasonable result? Running it some more times I get 0.28, -0.37, 0.10, -0.13, and -0.04 for self-satisfaction averages, which seems consistent with "you had an abnormally pleased group that time, but the average is 0".]

(I'm noticing I divided by net numbers by 2, which means they're "per-relationship" numbers instead of "per-person" numbers, and that was probably a mistake; oops!)