“‘How do mathematicians find love in a world filled with logic and rationalism?’ I was recently asked.
‘Well, that’s easy’, I said. And this is what I told them.
The Problem Setting
The problem is mostly referred to as the Marriage Problem, sometimes also the Secretary Problem. Since, if anything at all, I would be looking for a man to spend my life with, let’s call it the Husband Problem.
We assume that there is a number of n guys that I could potentially date throughout my life. I know that this is a difficult assumption to make. How would I ever know how many other guys there would be down the lane if I didn’t keep rejecting guys until there was no one left?
Nonetheless, let’s assume we have a number of n candidates to choose from. The only problem here: Once I settle for someone, I have settled. That’s it. That’s the deal I’m getting and I will never know who I am missing out on.
We also assume that I cannot go back to someone I have previously rejected.
Furthermore, we assume that for any two guys, I can decide who is better (at least for me). We also assume that the candidates are uniformly distributed. Intuitively, that means that the probability of the best candidate being in a certain position is the same for all positions.
Of course, I want the best deal I can get. I want to find the guy that is the best out of the n guys that I could potentially marry. The question is: How do I know when to stop searching? What if I decide that I am not happy with someone but I realize that there might never be anyone better? What if I keep rejecting guys and in the end, I realize that I will have to live with a poor choice or no choice at all because there are no candidates left?
Well, that’s a different story. For now, we want to maximize our chances of finding the best one.
