Later on, we will prove rigorously that the two optimal entities converge to the same value of roughly 0
You may find more optimism in the fact that as we increase the range of our dating life with N, the optimal probability of finding Mr/Mrs
There are some interesting observations here: as we increase the number of candidates N that we consider, not only does the optimal probability decreases and see to converge, so does the optimal ratio M/N. 37.
You may wonder: “Hang on a minute, won’t I achieve the highest probability of finding the best person at a very small value of N?” That’s partially right. Based on the simulation, at N = 3, we can achieve the probability of success of up to 66% by simply choosing the third person every time. So does that mean we should always aim to date at most 3 people and settle on the third?
Well, you could. The problem is that this strategy will only maximize the chance of finding the best among these 3 people, which, for some cases, is enough. But most of us probably want to consider a wider range of option than the first 3 viable options that enter our life. This is essentially the same reason why we are encouraged to go on multiple dates when we are young: to find out the type of people we attract and are attracted to, to gain some good understanding of dating and living with a partner, and to learn more about ourselves along the process.
Perfect does not decay to zero. As long as we stick to our strategy, we can prove a threshold exists below which the optimal probability cannot fall. Our next task is to prove the optimality of our strategy and find that minimum threshold.
Let O_best be the arrival order of the best candidate (Mr/Mrs. Perfect, The One, X, the candidate whose rank is 1, etc.) We do not know when this person will arrive in our life, but we know for sure that out of the next, pre-determined N people we will see, X will arrive at order O_best = i.
Let S(n,k) be the event of success in choosing X among N candidates with our strategy for M = k, that is, exploring and categorically rejecting the first k-1 candidates, then settling with the first person whose rank is better than all you have seen so far. We can see that:
Why is it the case? It is obvious that if X is among the first k-1 people who enter our life, then no matter who we choose afterward, we cannot possibly pick X (as we include X in those who we categorically reject). Otherwise, in the second case, we notice that our strategy can only succeed if one of the first k-1 people is the best among the first i-1 people.
I don’t want to bore you with more Maths but basically, as n gets very large, we can write our expression for P(S(n,k)) as a Riemann sum and simplify as follows:
Does it mean you should swipe left on the first 37 attractive profiles on Tinder before or put the 37 guys who slide into your DMs on ‘seen’?
The final step is to find the value of x that maximizes this expression. Here comes some high school calculus:
The model provides the optimal solution assuming that you set strict dating rules for yourself: you have to set a specific number of candidates N, you have to come up with a ranking system that guarantees no tie (The idea of ranking people does not sit well with many), and once you reject somebody, you never consider them viable dating option again.