When Barbara Mueller died of Typhus in 1611¹, Johannes Kepler did what any urbane, technologically sophisticated person does to find a new mate.
He joined OKCupid.
Well, not really. Unfortunately for poor Johannes, neither Tinder nor OkC nor Coffee meets Bagel were available options for eligible bachelors of the day. What was he supposed to do?
Aided in the candidate selection process by a number of friends, he shortlisted 11 women and over the course of two years proceeded to examine them with a care worthy of one the pillars of modern science, until he arrived at his final selection. More on that later.
Unbeknownst to Kepler, his tribulations did not go unnoticed and his problem was given renewed consideration by mathematicians in the 20th century. First called the marriage problem², it later went by other names such as the the secretary problem, the fussy suitor, or the sultan’s dowry problem (depending on the scenario used as the set up). These all share the problem of determining when to cease sampling through a fixed number of n options in such a way that one minimizes cost while maximizing the probability of a successful outcome. Problems of this type are generally called Optimal Stopping Problems.
While the antecedents to probability theory go back a number of centuries, it is really in the 20th century that optimal stopping problems develop a firm axiomatic basis. The problem in its simplest form presupposes the following:
- There is a single desired outcome.
- There are N possible outcomes and N is known.
- The outcomes can be ranked without ambiguity.
- Possible outcomes are selected at random
- Outcomes are selected or rejected solely by their ranking relative to the outcomes already seen.
- Decisions are final.
Given these, how could Kepler the eligible bachelor proceed? Since optimal stopping problems and their solutions were formalized long after Kepler’s death, we can’t really blame him for not trying this approach. How might Kepler have gone about securing a wife?
Here’s an example. Below we have 16 possible candidates for a given position, and each row represents different ways we might sample those candidates. In the first row, we stop after seeing the first 2. In this case, the sample appears too small. Intuitively we can say that there is likely insufficient information to make a valid conclusion. Likewise for the second row, we could probably guess that we looked at too many, 13 of 16 possible candidates. But is there an optimal number beyond which further searching would be superfluous?
It turns out there is. Mathematician Thomas Bruss³ derived a result showing that the optimal probability of winning is always 1/e. Euler’s number or e, is an irrational number that is a mathematical constant given by the following formula:
According to Bruss, the optimal stopping point comes after rejecting the first n*(1/e) candidates. Subsequently we continue to interview candidates until we find the first one better than the best of the ones we reject. That’s the one we keep.
And Kepler? What did he do? Remember that he had 11 choices. Not having Bruss’s formula at the ready, he simply proceeded the old fashioned way and took notes about his process. Had he utilized 1/e, he would have skipped over the first four and of the rest, choosen the first one who was better than those. In Kepler’s case? Two years later in 1613, he married Susanna Reuttinger with whom he had several more children, and to whom he remained married until his death in 1630. His biographers say they were quite happily married. Susanna was the fifth of the 11 women.
 Not to be confused with another mathematical problem called the Stable Marriage problem. See : https://en.wikipedia.org/wiki/Stable_marriage_problem
 Bruss, Thomas; 2000. Sum the Odds to One and Stop. projecteuclid.org/download/pdf_1/euclid.aop/1019160340