Modeling The Marriage Decision
Here are some details and extensions to my article “Modeling the Marriage Decision.”
It may sound far-fetched, proceeding to select a wife on such dry, theoretical grounds, but according to the history by T.S. Ferguson (1989), there’s one notable case of this nearly happening with none other than the celebrated German astronomer and mathematician, Johannes Kepler. How appropriate. Best known for his Eponymous Laws of Planetary Motion, which would later serve as the foundation for Isaac Newton’s theory of universal gravitation, Kepler was in search of a new wife (after his first died of cholera). He set out to interview “no fewer than eleven candidates for his hand,” a task that would take him years. Kepler’s case actually provides very interesting evidence for the theorem, because he was able to review all eleven (before deciding) and then propose to the 5th, his favorite. Corroboratively, the optimal evaluation period (r) for N=11 is 4, meaning that had Kepler known and applied theorem, he would have in fact chosen the correct wife.
What is in my opinion the most brilliant extension of The Secretary Problem is the relaxation of two important parameters, N and F, where N is the number of candidates seen, and F is the distribution of their “arrival.” In a paper by F.T. Bruss in 1984, we find the solution to the Secretary Problem in continuous time and the following two changes: N is an unspecified random variable with unspecified distribution of G, defined as the number of applicants arriving in time interval [0, t], and F being the unspecified distribution for each arrival time of the applicants. In other words, in this model you don’t have to approximate N, nor do you need to know the dynamics for how and when each candidate steps up for evaluation. Most shockingly, not only is the solution the same as in the classical Secretary Problem, but we can ensure that the probability of success, 1/e, is a lower bound! You might be wondering how one utilizes such a model if she doesn’t know the extent of her progress through the “interviewing series”—how close to t she is—which is admittedly the difficulty in applying this model. Because we don’t know N with certainty, the t/e point is reached at some z* in [0,t] that satisfies F(z*)=1/e, where F is the distribution of arrival time of each candidate. A little confusing, I know, but the model is profoundly useful, especially if you have access to some statistical metrics governing F, or if you simply already know F (however knowing the distribution, F, adds such significant information as to render this strategy sub-optimal for a subset of cases). Also, thanks to The Central Limit Theorem (probably to be discussed in a later article), sufficiently complex systems may be assumed to have a normal distribution.
Finally, if you’re interested in a full review of all extensions, I recommend the P.R. Freeman Review from 1983.