# The Secretary Problem

It isn’t what you think.

The “secretary problem” in this case is the name of a very famous math conundrum originally conceptualized around the best method for hiring a secretary. It has also been called the “sultan’s dowry”, the “marriage problem” or simply the “optimal choice” problem. For the sake of political correctness and to be slightly more up to date, I will just call it…the “speed dating” problem.

Here is my version. Imagine you are engaged in a speed dating exercise with 50 potential matches. You meet each sequentially in random order and immediately after each individual conversation you must decide whether that person is your preferred match. Once you select a person the exercise ends and that is your final choice. Should you move past a candidate to the next suitor that candidate is no longer available to be selected. You can’t go back. So with every disqualification you reduce the remaining pool of potential mates and you cannot know whether there is someone better waiting or whether you are passing up the best available match.

How do you increase the probability of making the best possible selection?

Mathematicians have answered precisely that question with something called “optimal stopping” theory. Optimal stopping in this scenario requires automatically rejecting an initial number of potential dates and then selecting the very next candidate who is better than all the previous ones. Of course, how many potential dates should we initially pass over? Too small a group and we don’t see enough candidates to make a really good comparison, but too many and we may miss all the really hot dates and be left with a small number of only poor suitors at the end. So what is the right number? Mathematicians can easily calculate the stopping point which gives you the very best probabilities. In very large pools, for instance, the optimal stopping point arrives after you have seen approximately 37% of the total pool. Once there, your best odds result from selecting the very next candidate who is better than all the previous ones you have evaluated.

But here is the interesting bit….we don’t do it.

When behavioral scientists study how we actually make selections they have shown that people tend to decide things much too quickly. In a way it is not surprising. Science may coach us to invest more time in thinking, but that notion remains unrewarded in popular culture. Traditionally in business, sports, politics and elsewhere the heroic decision maker is one who chooses with speed and certainty. The more difficult the problem the more we crave the simple quick solution. The New Yorker’s Borowitz Report captured the sense with a recent headline parodying public frustration with America’s cautious response to complex global crises: “Growing Pressure on Obama to Do Something Stupid”.

The general perception is that our consideration aptitude, that willingness and ability to thoughtfully consider alternatives, even those alien to our own biases and instincts, is diminishing. I am constantly aware of the negative impact of my own confirmation bias as I scour the web, read articles and evaluate factual evidence. But what if, as humans, we compound the bias problem by consciously or unconsciously choosing not to invest the time and effort to be truly thoughtful? Like rearing children, nurturing good judgment takes both quality time and quantity time.

In the early 17th century, the German astronomer and mathematician, Johannes Kepler, confronted a variation of The Secretary Problem when he undertook the process of deciding on his second wife. His first wife, Barbara, had died of illness and Kepler chose to pursue the selection of a new wife with the same mathematical scrutiny that he had successfully used to plot planetary motion. Kepler selected 11 ladies and, over a two year period examined their suitability in succession.

Alas, even Kepler was not immune to process flaws. After selecting Candidate Five, Susanna Reuttinger, he was talked out of the engagement by friends and family who convinced him that Candidate Four was truly the better choice. Happily, it turns out, Four was, by that time, no longer available for marriage so Kepler returned his affections back to Susanna who, he said, won him over with love, loyalty economy and diligence. By all accounts it was a wonderfully happy marriage.

It is a good reminder. Too often our culture heralds the quick choices made by soft heads with hard hearts, those who think sloppily and cannot tolerate contradiction. Give me the Keplers, hard headed and soft hearted citizens, persons whose thoughts are disciplined, who give ample time to consideration, and who make space for their souls to be spoken to.

They will usually find their way to a good decision.

Application:

For those who better understand math, here is an interesting review of the history of The Secretary Problem: Thomas Ferguson’s 1969 article, “Who Solved the Secretary Problem?” in Statistical Science:

http://projecteuclid.org/euclid.ss/1177012493