Modeling Monogamous Commitment

A Super Simple Model of Monogamous Commitment and Choice

This is something I came up with a long, long time ago to help build my intuitions regarding how to “know” when to commit in a relationship. It seemed people were always wringing their hands over whether their current partner is Right For Them, or whether A Better Partner Is Just Around The Corner. So I thought, why not try to explicitly quantify the dilemma and build a simple model (with very restrictive assumptions) that lets you know whether to commit to your current partner or keep playing the field? This is mostly meant as a tongue-in-cheek exercise and absolutely should not be taken seriously. If you’re actually interested in this stuff, I’m sure the human behavioral ecology folks have looked into it and developed much more sophisticated approaches and solutions.

Anyway, the idea is this: say you want to maximize expected quality time with your prospective partner(s), ahem, that’s a product, (Quality x Time). You can sample randomly from a distribution of partners, and if we wave our hands around and solemnly invoke the central limit theorem and the principle of maximum entropy, we can say that this distribution represents Partner “Quality”/Compatibility and is normal with, let’s say, a mean of 5 and a standard deviation of 1*. You’re a serial monogamist, so you keep dating partner after partner because what if the next one’s better and you commit to a suboptimal, incompatible partner (the horror!)? But you’ve also studied, and in studying you have learned that man is mortal, and so you don’t want to keep dating serially indefinitely because eventually you’re going to die and you’ll never have actually spent any time with a committed partner. You also decide that the Quality Time you derive from casual dating is inferior to that you experience in a committed relationship, to the tune of, oh, 80% (so that you get 20% less quality time dating casually vs committing to an equicompatible partner; but see endnote 5). Ultimately what you’re really interested in is spending as much time with the best partner possible**. Your strategy, then, is to date casually for a while, get the lay of the land, etc., and then after reaching a certain age commit to the first partner afterwards that’s better than the BEST partner that has come before. What age of “beginning commitment sampling”, then, maximizes Quality Time (on average)?

Let’s say that as a serial monogamist you begin dating (or at least evaluating prospective partners) at puberty, here taken as 12. You die when you turn 80, and you date partners for 30 days*** during the initial sampling, lay-of-the-land period before saying, no, sorry, I’d like to try my luck with the next person, actually probably not even then because I’m sort of a weirdly robotic asshole and my initial sampling period isn’t up yet and I’m in a stupid thought experiment/simulation. Somehow, you can also perfectly intuit each partner’s “quality/compatibility score” (but see below), though you don’t actually know what the underlying distribution is (or else you could just solve for the cumulative density and figure out how great a catch your current date is; maybe a future exercise will look at this as a question of inferring that underlying distribution and determining how much better your current partner is than all the rest).

Anyway, here’s some commented R code (damn formatting gets lost in the shuffle; sorry if it’s hard to read or if there are any mistakes!):

# Maximize Quality*Time With Partner
ageAtFirstSample <- 12 #when we first start evaluating potential matches
ageAtDeath <- 80 #when we die
evalDaysPerPerson <- 30 #how many days we sample each partner for
ageSeq <- seq(from = ageAtFirstSample, to = ageAtDeath, by = 1) #range of ages across which we can begin committing
noCommitmentReductionFactor <-.8 #how good casual dating is compared to a committed relationship
samples <- 1e3 #how many simulations from each age do we want to run?
EQT <- vector(length = length(ageSeq))
for(j in 1:length(ageSeq)){ #make a for-loop took look at age at which you might begin committing
 jEQT <- vector(length = samples) #initialize an empty vector to hold data from our simulations
 for(i in 1:samples){
 if (i %% 50 == 0) {print(c(j, i))} #print a counter to tell us where in the sims we are
 search <- rnorm((ageSeq[j] - ageAtFirstSample) * 365 / evalDaysPerPerson, mean = 5, sd = 1) #simulate partners during initial sampling
 life <- rnorm((ageAtDeath - ageSeq[j]) * 365 / evalDaysPerPerson, mean = 5, sd = 1) #simulate partners during the rest of our lives
 theOne <- which(life > max(search)) #find which partners during the rest of our lives are better than the best partner from initial sampling
 if(![1])) #if there aren't any better partners during the rest of our lives, we keep dating forever
 { jEQT[i] <- (length(life) - theOne[1]) * evalDaysPerPerson * life[theOne[1]] + #Quality Time with our chosen partner
 (sum(search) + sum(life[1:(theOne[1]-1)])) * evalDaysPerPerson * noCommitmentReductionFactor} else #Quality Time with our casual dates
 {jEQT[i] <- (sum(search) + sum(life)) * evalDaysPerPerson * noCommitmentReductionFactor} #Quality Time with our lifetime of dating if we never commit
 EQT[j] <- sum(jEQT)/samples #find the average quality time across simulations for each age of begin commitment
plot(ageSeq, EQT)

From this, we get the following graph:

With a maximum value at age 14! (also, pay attention to the axes in all of these plots — the bottom left corner is not the origin. The horizontal axis is the age at which our simulated people start being ready for life-long commitment, and the vertical axis represents Expected Quality Time) So this very simple model says to get ready for commitment after only a few years of playing the field! This doesn’t mean that 14-year-olds are getting married, though, because they keep sampling prospective partners until they find one better than all the ones that have come before. Actual age at commitment (in the instances when commitment does occur and our simulated people don’t die as casual daters) looks like this (with line y=x plotted):Rplot01
Suppose though that as we age we find more and more compatible partners because we engage in self-improvement and also because we grow wiser with age and better know what we want. Say this Age Wisdom Bonus is linear and manifests by having an 80 year old sample from distributions with means of, oh 2 points greater than a 12 year old (equivalent to 2 standard deviations — so the mean of an 80 year old’s sampling distribution is equivalent in compatibility to the ~95th percentile of a 12 year old’s sampling distribution). Rerunning the code from before, we get the following graph:
So now it’s “better” to hold off until you’re around 18 — 6 or 7 years in — before Being Ready To Commit to the best partner you’ve ever met.

Alternatively, we can model this as uncertainty in our evaluation of each partner, with our ability to discern our own values and partner compatibility improving as we age. Say then that this introduces error in our perception of partner compatibility (upon which we make the choice to commit), but Quality Time is determined by their actual compatibility value. This error at 12 could be sampling from a normal distribution with mean 0 and sd 0.5, which decreases linearly to mean 0 and sd 0 by the time we hit 80. Running these simulations, we get the following plot:Rplot06

Which looks a lot like the plot in the first set of simulations, probably because the error that we use has expectation 0 (since we’re sampling from a normal distribution with mean 0), which across 1000 simulations largely cancels out?

We might also be interested in asking some of the things of the model, such as how good our chosen partner at each age really is? Using the cumulative distribution function, we get the following:Rplot09
Whoever we end up committing to with the “commit to someone better than all who have come before” strategy, then, is pretty darn good.

So I guess to close, what have we learned? Not much, since this entire exercise was pretty goofy. I suppose, though, that if the above, overly simplistic model adequately represents your own dating experiences, and you’re in your 20s or something and wondering whether you should commit to your current partner — a person better than anyone you’ve met before**** — then you can maximize expected “Quality Time” by committing to them and not continuing to Play the Field in the hopes that a better match might come along. At least, well, maybe.*****

*(more realistically, we’d sample from a multivariate normal representing what we value in a mate. The distances from the origin to each draw are roughly normally distributed with a long right tail, and I think lognormally or gamma distributed maybe? Anyway, running basic simulations with higher and higher dimensionality and applying the n-dimensional Pythagorean theorem seems to give us something that looks more and more normal, and, ooh, you can think of our univariate Quality distribution as Partner Compatibility or something. Also, the numbers here are completely arbitrary and you can sample from whatever normal distribution you’d like and it shouldn’t qualitatively change the results much, but people loooove their 10 point scales and this roughly corresponds to one)

**(hence, maximizing expected (Quality)(Time). Even though quality/compatibility is distributed normally, how much you benefit you reap out of progressively more compatible partners scales linearly with their Quality. So even though a partner of compatibility “8” or higher is found once in every 741 prospective partners, and a partner of compatibility “9” or higher is found once in every 31574 prospective partners, a partner of compatibility “9”, all else being equal, will only bring you 12.5% more (Quality)(Time) than a partner of compatibility “8”. I think this is a decently realistic but mostly a simplifying assumption. Say, for instance, that you strongly value scientific competence in a mate. Nobel prize winners are way rarer than members of the National Academy of Sciences, but not necessarily “greater” in a manner proportional to their rarity)

***(sure, as a serial monogamist you might date each partner for much more than 30 days, or for much less, who knows. Realistically, you’re also “evaluating” more than whoever you’re currently dating – your friends, acquaintances, and even your entire peer group are up for consideration, so you’re not sampling so much in series as in parallel. Also, maybe your partners don’t want to keep casually dating after 30 days, so that’s when you have The Talk and commit or part ways? In any case, 30 seems reasonable enough for my purposes here, and have I mentioned simplifying assumptions yet?)

****(in the future, I might return to this and relax this assumption and allow commitment to partners in the top Xth percentile and figure out how picky simple model folk should be, as well implement rejection/courting, non-uniform dating periods, break-ups, difficulties finding partners when you age, exiting the “honeymoon period”, etc. for greater realism. And also plot everything as intervals and not just averages to show simulation variance)

*****I also played around with the noCommitmentReductionFactor, which is the ratio of benefits of a non-committed relationship to the benefits of a committed one. Setting this to 1 did not, as I first anticipated, make committing at any age give equal expected Quality Time, since when you keep casually dating you go through a lot of incompatible partners, but if you commit to a really good one from the start you’re with them until you die. Setting it sufficiently high above 1 did lead to the conclusion that you should never commit, since that implies you like non-committed relationships much more than committed ones. The point at which you should never commit was around 1.55, meaning you value Quality Time with a new partner 55% more than you value time with an equivalent compatibility committed one (due to New Relationship Energy or something). The graph here looks like:Rplot07.png
Meanwhile, setting the ratio to something like 3 gives us:Rplot08


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