My grandest ‘real-life’ use of probability theory until recently was to estimate my odds in Risk and poker. Then, while skimming a math-themed book, I singled out a principle that will shape how I will spend $6,000 and about 200 hours of my life in the upcoming year. As far as the math goes, it doesn’t matter whether I’m referring to getting a girlfriend, a pet, an apartment, or a job. But at the moment I am actually picking a piano teacher. The teacher-picking theorem I have in mind is based on an oversimplified scenario that has an optimal strategy. Trying to apply it has been interesting. But before I reflect on my story, let’s glance at what the math says.
Start by imaging a series of candidates, among which we want to choose the very best. We have to assess candidates one at a time, and we have to decide whether to accept or reject them immediately after inspecting them. The candidates come to us in random order, so that the first one we assess is just as likely as the last one to be the best. The optimal strategy in this setup is to inspect about 37% (or exactly 1/e) of the candidates and then accept the next one that is better than the first 37%. This approach is optimal (i.e., mostly likely to snag the best option) because 37% of candidates is just enough to estimate which ones are exceptional without rejecting too exceptional ones.
“Sequential decision making,” as this scenario is sometimes called, first came to my attention in a psychology study. When healthy people do computerized sequential decision tasks—sometimes simulating job interviews or shopping—they tend to jump the gun. However, this study in Germany found that people suffering from depression tend to wait longer and more closely approximate the best strategy. While this line of research has interesting implications for the causes and origins of depression, my main personal reaction was a sense that I probably don’t sample enough options in my life decisions. A prime example heads off my piano story: I picked a piano teacher a two months ago by emailing a music professor I did not know and taking his recommendation. Out of 30 available teachers listed in a directory for my area, I ended up ‘sampling’ only 1/30 ( 3.3%) of them. That’s all the more suboptimal because with piano teachers I can choose earlier candidates, which justifies sampling a larger fraction of teachers than in “37% rule” I’ve described. My only defense is that the recommending professor seemed to be familiar with the candidates, so he probably ruled out some of the least compatible candidates.
Fast forward to two weeks ago; I was now looking for a new piano teacher. Emboldened by my pet theorem, I emailed nine teachers (about 30% of my options) based on what little information I found online. It was not hard to pick four of the more interesting teachers to meet in person. I have now met three out of those four and I feel glad with my approach, but I am still making sense of the challenges involved. The main challenge is making fair comparisons. I started my search by de facto rejecting all the teachers I did not email based on little or no information; then I rejected some email respondents based on unreliable cues in their messages; and, in the final step, I made snap judgments from meetings that were subject to confounds like mood, time of day, shared expectations, and who knows what else. My two conclusions here are that (1) quality-assessments are imprecise, especially when they involve judging a match between people and (2) “counting” how many candidates have been “assessed” is subjective unless they all receive similar attention under similar conditions.
My next big hitch is about social emotions, not strategy. Specifically, my meetings so far were all at some point comically awkward, because in each case I did not want to admit how widely I had casted my net. I vaguely mentioned “considering my options” and, at most, I acknowledged meeting one other teacher. Although I don’t feel ashamed about my attempts to find a good match, I don’t want to upset teachers who might frown on my approach. After all, no one likes being compared to others or facing rejection. Thus I tried to treat each teacher as my top choice without making false promises. Looking to the future, I also wonder if more awkward situations will develop. Will any teachers gossip disapprovingly of me? Will I bump into “rejected” teachers at future recitals? These questions suggest that trying out multiple teachers has had a minor “emotional cost” for me. I imagine this type of cost could be much heavier in other choice processes like child adoption.
My final complaint with my piano teacher experiment is that it has been resource-intensive. The time and effort spent emailing, driving to and meeting people has felt subjectively like “too much.” The root of this complaint is that sequentially finding the best piano teacher in my area is not my only goal in life. I have other uses for my time, energy, and gasoline, like finding a part-time job, catching up with friends, and sleeping. Therefore I constantly have to weigh the value of finding a slightly better teacher against improving some other aspect of my life. At some point, my search is no longer worth it (economists, read: diminishing returns or increasing opportunity costs). The inevitable trade-offs we all face could partly explain, from both evolutionary and practical perspectives, why we tend not to sample quite enough alternatives to make the ‘optimal’ choice. Satoshi Kanawaza makes a related point about dating in densely populated cities like New York—at some point there are so many eligible bachelors around that meeting or speed-dating 37% of them is infeasible. We usually settle for “good enough,” and most times we have to.
Murky math and puppy paradoxes
These reflections should make clear that an idealized model of sequential decision making cannot replace mental assets such as good intuition, resourcefulness, and common sense. In spite of the apparent certainty of the “37% rule,” its most useful lesson for daily choices is vague: get a good sense of the candidate pool. In my search for piano teachers, the easiest way to scope out the field has been to contact a lot of teachers directly. But other ways to do that include asking experts or reviewers and drawing on relevant past experiences.
Unfortunately, knowing our options can be just as counterproductive as it is helpful for some highly subjective decisions. As Sheena Iyengar and Barry Schwartz are fond of pointing out, we humans are susceptible to “choice overload.” When we see too many retirement plans or job offers, we take irrational shortcuts and sometimes we feel less satisfied with whatever pick we make. For some of us (depending on culture, personality, and exact circumstances) the most adorable puppy possible is one of the first we see. If beauty is in the eye of the beholder, then the beholder is liable to get fatigued and jaded by alternatives.
It’s no surprise that marrying a mathematical theorem like the 37% rule with the complexity of human decision-making requires a laundry list of caveats. But even with the caveats, mathematicians and other logical sorts are on hand to help us if we ever get the urge to approximate “rational” thinking. The rest of the time, our unconscious brains can run a decent autopilot for us—and thank goodness for that.
(Disclaimer: I disavow myself of any responsibility, moral or legal, for terrible choices of piano teachers or puppies that result from your reading this post).