Mathematics of Love

In this weekend’s Wall Street Journal, there is an article entitled “In Love, Formula Suggests Only Fools Rush In”. The article looks into the marriage problem, first analyzed in Scientific American in 1960. The idea is essentially a decision about whether you should stay with the current individual you are courting, or dump that individual for someone else who may be a better option. The risk in staying with the individual you are currently courting is that there is someone “better” out there. However, the risk in moving on to another option is that you may be giving up your best option. What to do?
Why not have a little fun with this problem? Let’s say you have three options for a mate. Call them Option 1 (Ideal Mate), Option 2 (Acceptable Mate), Option 3 (Worst Choice). Suppose your selection strategy in choosing a mate is to choose the first person you meet. If you meet the individuals in random order, what is the probability you choose your ideal mate? To answer this question, you could simply lay out the possibilities.
Option 1, Option 2, Option 3
Option 1, Option 3, Option 2
Option 2, Option 1, Option 3
Option 2, Option 3, Option 1
Option 3, Option 1, Option 2
Option 3, Option 2, Option 1

Notice, the probability of selecting your ideal mate first is 1/3. Can the odds of choosing the best mate improve with a new selection strategy? Let’s try a different strategy.
Suppose you cannot choose the first person. The first person becomes the baseline for choosing your mate. If the next person you meet is an improvement on the previous person, you go with that option. Here are the outcomes for that scenario. In the first possibility you end up with Option 3 because Option 2 is not better than Option 1, so you move on and end up with the Worst Choice.
Option 1, Option 2, Option 3
Option 1, Option 3, Option 2
Option 2, Option 1, Option 3
Option 2, Option 3, Option 1
Option 3, Option 1, Option 2
Option 3, Option 2, Option 1
Notice you end up choosing the ideal mate 50% of the time! Not going with the first choice appears to be a good strategy. Is there an optimal strategy?
One strategy suggests rejecting the first 37% of potential mates is a decent way to go. In fact, it only takes a set of 20 potential mates using the 37% rejection rule to obtain a probability of finding the ideal mate equal to 0.38. So, if you have 20 potential mates, you should date the first 0.37(20) = 7 of them with the goal of learning the traits you desire in a lifelong mate. You are not allowed to choose any of these 7 as your lifelong-mate, however. After dating the seven test cases and learning the traits you desire, run a simple comparison test. Begin by comparing the 8th potential mate with the previous 7. If any of the previous 7 are thought to be superior to the 8th, you should move to the 9th potential mate and the process repeats. While dating the 9th potential mate, decide if any of the prior 8 are superior to the 9th. If so, move on; otherwise choose the 9th as your soul-mate. Continue until you have found your lifelong mate.
I mentioned this approach in my stats class and one of my students mentioned that this is how the online dating world works! The mathematics of love on this Valentine’s Day.

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