# How satisfied should you expect to be with your partner?

I got married today, to the [particular fellow](https://www.lesswrong.com/users/alex-ray) mentioned in [my Turning 30 post](https://www.lesswrong.com/posts/NCojuKN5dBMtCJHA3/turning-30). In a sort of 'inverse cat tax'[^q4b9vf48jnk] 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](https://dnd.wizards.com/charactersheets), 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.[^ss8x4i1hr5]

Let's start by considering the heterosexual version of the [stable marriage problem](https://en.wikipedia.org/wiki/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).

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/776e3d36f2a93b9495866e1d361bdfa7f3040e4ab50fa8bd.png)

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.[^qbtzf2cbrj]

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%. 

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/7d39f69a146208b1ba80e439897b1872b4b953554d7cd8eb.png)

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/e54c2574b5e001e9f5d685bf3c0efd1070e3f724cb301b89.png)

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](https://en.wikipedia.org/wiki/Stable_roommates_problem), solved with a similar algorithm. I'll leave it as an exercise for the reader how that impacts the results.[^ulf2zzxt7zo]

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.

[^q4b9vf48jnk]: 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'. 

[^ss8x4i1hr5]: 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). 

[^qbtzf2cbrj]: 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. 

[^ulf2zzxt7zo]: 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).