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:
The optimal age, t, to get married is then
Doing out the calculus and simplifying:
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:
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:
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:
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.