How to kill the magic of human relationships with mathematical precision

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My friends are pretty cool. Your friends are probably pretty cool too.

Sometimes I feel like mine are the best in the world and you probably feel the same way about yours.

You probably stop there, move on and continue to enjoy your life. The tragically rational among us stop and realize that’s mathematically impossible or at least statistically improbable.

How can you not be overwhelmed by the number of people you don’t know? Or even the people you do know but have not developed a relationship and past history with?

But then again if you spend all your time meeting new people you never develop meaningful relationships with anyone. Clearly there is some appropriate balance between developing relationships you already have and expanding your social horizons. But how do you find that balance?

Not knowing where the correct balance, we can turn to a mathematical model. To make our lives easier, let’s examine a particular type of relationship: Marriage. The rules are you get to pick one person from the set of people you’ve met (no mail order brides) and can not divorce.

Today’s question: What is the appropriate age to get married (or more precisely pick a partner from the set of people that you already know)? Until what age do you continue to search and when do you … for lack of a better word … settle with the best you have?

I put the rest under the fold because it involves a fair but of math and even some integrals .

For those who don’t care about math, the answer is 50 years. You can stop reading now. For the geeks, onward we go…..

– Define S to be the set of all people who meet your criteria as eligible for marriage. Perhaps they have to be of a certain religion, in a certain age range, of a certain intelligence, a certain ethnicity or sex . The only important thing is that this set is smaller than the number of people in the world.

– Define R(A) as the rate at which you meet people at any given age A. This will vary over the course of your life and is thus a function of A.

– Define P(A,S) the % chance at any age A that a person you meet is in S. This will also vary over the course of your life (e.g. for most chances are higher a random person you meet is within S freshman year of college rather than within a corporate context). Also note, P(A,S) increases monotonically with the size of S. The larger the set of eligible partners, the larger the probability that someone random is in that set.

– Define M(A) as the % of people within S who are married (and thus removed from consideration) at any given age A. M(A) is dependent on the type of S, but not particularly strongly on the size of S.

The important part: In this simplified model, you want to choose your husband/wife out of S at the age A in which you know the maximum number of people within S who are still single. In other words, you want to maximize your options.

The number of single people you know within S at a given age A is:

eqtn-1

The optimal age, t, to get married is then

eqtn-2

Doing out the calculus and simplifying:

eqtn-3

This is the answer. Now, let’s actually try to make this useful by making some simplifying assumptions. I want to use the following exponential function to describe M(A), the % of people married at any age, A:

eqtn-61

If we choose A = 1 and k ~ .02, then we get some reasonable values of M(A)

  • Half of people are married by age 35
  • Three-quarters are married at age 70.

The new equation boils down to this:

eqtn-41

Now this may seem like we just substituted one useless equation for another, but this actually simplifies the problem quite a bit.  We’ve now boiled down this marriage equation to one simple parameter, k, which represents the how early people get married. Lower k means fewer people people married early.

But we need to make another simplifying assumption. Admittedly, this shells the accuracy of the exercise, but will allow us to easily arrive at a final answer.

Suppose you meet the same number of people and the chance they are in S does not vary over time.

R(t) = constant = number of people you meet per year
P(A,S) = constant = chance that a someone you meet is in S

In this special case, those terms drop out and we are left with:

eqtn-51

Taking our k = 0.2 value, we arrive at an optimal marriage age of 50 years. If you think half your friends get married by age 25 (k = .027), we get an optimal marriage age of 36.

And it’s mathemagic. Hold out, people. You’ve got time.

 

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9 Responses to “How to kill the magic of human relationships with mathematical precision”

  1. doc mac Says:

    Sounds great as long as there’s no relationship between marriage and children. The average age of menopause in the U.S is 50; fertility declines starting in the early 40s.

    I’ve always been curious about the people who marry in middle age. What’s that relationship like? At that point more than half of a person’s life is behind them–you’ve each already built your own lives, in lieu of constructing a shared one. On the other hand, its well established that young (ie immature) marriages fail at a higher rate than those of older people.
    Can you maximize something that’s a function of maturity vs remaining years of life or fertility or something?

  2. phil Says:

    Doc,

    Good points all. This model is missing a few important factors — the only issue is including them requires some sort of notion of “utility” which is pretty much unworkable in anything but an abstract sense.

    Other factors that make the ideal age younger include:
    – Fertiility / ability to have babies (as you suggested)
    – If marriage is a good thing, more marriage (being married for longer) is a better thing
    – Volatility around expected age of death of you and a potential spouse. This is fundamentally asymmetrical because the death of one partner (early) means the end of the marriage. But unexpected long life in one partner does not mean any additional marriage time.

    I’m sure there are plenty of others…

  3. Molly Says:

    Great point about babymaking, Doc, but your numbers are off. A woman’s fertility starts to decline in her late 20s, not early 40s: http://www.sciam.com/article.cfm?id=study-shows-fertility-dec. That’s why they start whining about rings and biological clocks.

  4. doc mac Says:

    I may have been a little optimistic, Molly, but I think that article is misleading. Quoth scientific american, “A woman’s fertility begins to decline in her late 20s, but her overall chances of becoming pregnant do not start to slide so soon”.
    The article is pointing out that two horny teenagers can get pregnant in 2 months if they time things right, ladies over 35 should give themselves a few more months. Its still very reasonable to have children in your 30s, and not impossible (though difficult and sometimes expensive) in your 40s.

    The bad news first, about those eggs, from the American College of Gynecologists (ACOG): “The maximum complement of oocytes is 6–7 million and exists at 20 weeks of gestation in the female fetus. The number of oocytes decreases to approximately 1–2 million oocytes at birth, 300,000–500,000 at puberty; 25,000 at age 37 years”–in some sense, female fertility is ‘declining’ even prior to birth. And yes, 25,000 eggs isn’t SO many. ACOG suggests that if you’re >35 and don’t get pregnant within 6 months of trying, talk to a gyn to see if you need help. But don’t throw in the towel! Babies are still possible.

  5. doc mac Says:

    I may have been a little optimistic, Molly, but I think that article is misleading. Quoth scientific american, “A woman’s fertility begins to decline in her late 20s, but her overall chances of becoming pregnant do not start to slide so soon”.
    The article is pointing out that two horny teenagers can get pregnant in 2 months if they time things right, ladies over 35 should give themselves a few more months. Its still very reasonable to have children in your 30s, and not impossible (though difficult and sometimes expensive) in your 40s.

    The bad news first, about those eggs, from the American College of Gynecologists (ACOG): “The maximum complement of oocytes is 6–7 million and exists at 20 weeks of gestation in the female fetus. The number of oocytes decreases to approximately 1–2 million oocytes at birth, 300,000–500,000 at puberty; 25,000 at age 37 years”–in some sense, female fertility is ‘declining’ even prior to birth. And yes, 25,000 eggs isn’t SO many. ACOG suggests that if you’re >35 and don’t get pregnant within 6 months of trying, talk to a gyn to see if you need help. But don’t throw in the towel! The good news: Babies are still possible.

  6. Molly Says:

    Completely true, Doc. But there’s more to think about: the risk of having a child with Down syndrome is 1 in 1,300 for a 25-year-old woman; at age 35, the risk increases to 1/365. http://www.aafp.org/afp/20000815/825.html

    Anyway, I’m revealing my own prejudices about when people should have babies, and how silly people are to think that they can have babies in their 40s without any trouble. As you can tell, it’s a bit of a pet peeve.

  7. phil Says:

    This post has been hijacked.

  8. Molly Says:

    And since I’ve already hijacked it, let me throw in this nugget for baby lovers everywhere: http://www.youtube.com/watch?v=Y6rE0EakhG8.

  9. doc mac Says:

    You’re right, Molly, that the risk of Down’s rises with age. And that’s definitely something to think long and hard about. If I’m not mistaken, however, the maternal age group with absolute highest number of Down’s children is still women in their 20s–because they are having so many more babies than everybody else.

    Couple of questions, for everybody, just to play devil’s advocate:

    If one is approaching 35-40 without kids, should one go ahead and have them even if the job/marriage/retirement fund isn’t in place yet?

    Is it responsible to try having kids as a 40-something (or even older!), given the demands of children on aging parents, or vice-versa?

    Are men aware that the risk of other kinds of genetic disorders also increases with increasing paternal age? Do they care?

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