Column Editor’s note: In this column’s first article by a pre-college student, Angelina Chen—a student at Princeton High School in New Jersey—discusses the difference between causality and correlation through the lens of a well-known problem in probability: the secretary problem. Using examples from the movie industry and high school life, she lightheartedly explores how we can optimize some types of decision making in the face of uncertainty. Through scenarios familiar to and language accessible for pre-college readers, Chen poses the question of “When?” with real-life optimal stopping problems.
The Oscars. We might know them as film industry awards in the form of little statues of golden men, or the thing that Leonardo DiCaprio wanted but was snubbed by for so long. We see memes about red carpet outfits and acceptance speeches. At the 2020 Oscars, we even saw Bong Joon Ho make two statuettes kiss like Barbie dolls. Yet we rarely consider the technical thought process of the creators of winning films, or even how some of them won in the first place.
As you may have read in ‘Oscarmetrician’ Ben Zauzmer’s (2020) article, “Oscar Seasons: The Intersection of Data and the Academy Awards,” the decision of when to do something is often vital for success. Here’s a recap: An investigation of the Oscars’ Best Picture nominees reveals a trend in which the winning movies’ release dates are close to the nomination decision date. Zauzmer draws the conclusion that studios tend to release contenders for the Academy Awards close to that date, and that they are wise to do so because it increases their chances to win an Oscar, which would then enhance the movie’s revenue. But when I read this article, I wondered if the logic was circular: Was it the studios making the decision to release Oscar contenders later in the year, or was it the fact that movies released later in the year were more successful in being nominated? Essentially, we cannot tell if this relationship is causal (one thing directly causing another to occur), but we can see that there is a relationship. Movies released later in the year have a higher probability to win an Oscar, even though we do not know what causes this.
As a moviemaker, it would be logical after seeing this trend, no matter the cause, to then begin to release movies later in the year. But what if all the best movies of the year came out in the few weeks leading up to the Oscar nomination date? Would that onslaught of masterpieces not make it harder for any movie to shine? After all, to paraphrase Disney’s Aladdin, what we are looking for is a diamond in the rough, not a diamond in a big pile of diamonds. So, would it not be better for studios to release their best efforts in the midst of a bunch of low-quality films, so that their work stands out to the critics and seems better by comparison?
This question mirrors a dilemma that I face in everyday life, away from the flashing lights of movie stars and cinemas (though high school classrooms can be just as dramatic). Say a student, Darren, worked really hard on a presentation, and now he needs to deliver it to his class for a grade. It is good, but Darren wants it to stand out so that the teacher remembers it and gives him an extra high score. In this class, presentation order is not predetermined. Instead, students can volunteer to follow any presentation. If Darren presents right after a peer’s exceptional turn, his presentation may not seem as good in the teacher’s eyes. It would make sense to wait until a bad presentation occurs, then go right after to wow the teacher (and maybe wake them up from their nap of boredom). But how long should Darren wait for a bad presentation until he presents? What if he waits too long and the worst has already passed by?
First of all, we should inspect whether the presentation order truly does make a difference in the scores students receive. In this situation, we could conduct a study similar to those with the Oscars, examining both order and score to see if there is a connection. The data is readily available, but it would still just be an observational study and would also do no more than find a relationship without confirming that it is a causal one.
We would not know for sure if students who present after a bad presentation are receiving better scores because of the order, or simply because students with good presentations are trying to use this strategy. To solve this conundrum, we can design a more focused and detailed human behavior experiment to investigate the relationship. For example, we could record presentations of varying qualities, and ask a group of teachers to grade videos with different orderings of the presentations. Only under such a controlled experiment would we then be able to discover or confirm a causal relationship. Once a relationship is discovered, a smart student like Darren can then optimize the timing of their presentation.
If this first part is confirmed, we can then move on to the next question: How long should Darren wait? Darren does not know the quality of the presentations before they occur, so he cannot know if there will be worse presentations later on that would be better for him to follow. This type of problem is called an Optimal Stopping Problem, also called the Secretary Problem (to hire the best secretary) or the Marriage Problem (to decide when to stop dating and settle down). Let us imagine a variation of this problem happening to a girl named Sarah during the fast approaching prom season.
Sarah needs to find a date. Her friends are getting proposals to prom (aka promposals) left and right, but she wants the best choice to escort her onto the dance floor. Sarah is very popular and undeniably gorgeous, so she has an abundance of suitors vying for her hand. She has just ten days left before she needs to order her dress (to match her date’s outfit), but then she must decide. Only two people per day will dare to ask her to the prom, because they do not want to seem too desperate by stumbling over each other. Sarah does not know who is going to ask her until they do. She needs to make an immediate decision once they ask, since it would be rude to leave the promposer hanging. She can say yes or reject them and wait for a better candidate. After Sarah has rejected a person, she cannot go back to them because they will still be wallowing in their misery and heartbreak. When should she stop the search and say yes?
In an optimal stopping problem such as this one, we essentially want to answer the question of when we should take an action in order to get the best result. In this case, Sarah wants to know when to say yes, so that she will achieve her end goal of having the best prom date possible to round out her high school experience.
It would be a simple decision if Sarah knew who was going to ask within the ten days, and then picked the one who would make the best date. What complicates the situation is the fact that she needs to decide right after a person promposes without knowing who the rest of the candidates are and, subsequently, how well they compare. If she says yes, she may miss a better one yet to come. If she says no, it is possible that the rest of the choices are worse and she has missed her best bet.
So then, what should Sarah do?
A good way to approach this is to take a sample of a certain number of promposers, just to get a benchmark for the type and range of people who will invite her. Sarah will reject all of the people in this sample, and only use that information to find the best prom date out of that sample. Then Sarah will wait for the rest of the promposals and say yes to the first person she encounters who is better than the best choice from the sample, rejecting everyone else. If she has gotten to the end and no one is as good, then Sarah will need to beg her cousin to go with her because she will not have a date.
How many people should be in her sample? If she uses too few people, she will not have enough information to make a good decision, but if she uses too many, then she will not have many candidates left over to choose from.
According to probability theory, the ideal number of people in this sample would be about 0.37n, where n is the total number of people to be considered. Going back to the 10 days with two askers per day, Sarah’s sample size would be about 0.37x20, or 7.4, which rounds down to 7. According to this strategy, Sarah should find her maximum from the first seven candidates, and say yes to the next person who is better than this maximum.
Using this optimal strategy, the probability that the best suitor will accompany Sarah to the dance floor is actually only about 37%. Of course, this solution relies on many assumptions. It assumes that Sarah only wants the very best choice, meaning that the second-best and worst choices are the same in her eyes. It also assumes that Sarah has no idea who will prompose, and that she has no information about the quality of the candidates beforehand. Sarah would also need to know the total number of candidates before starting to calculate her sample size. Other complex mathematical calculations can show us the best strategy under different conditions, such as if Sarah was satisfied with her date simply falling in the top five, or if she would rather have any person as her date than show up with her nerdy cousin Darren. Sarah’s prom problem shows us the simplest, beginning variation on this problem.
Okay, prom is over, now back to the classroom. This basic strategy also applies to the class presentation problem. Darren should wait for 37% of his peers to present, and just sit tight. By observing how bad the worst presentation of that sample was, for the rest of the class period he should wait for another presentation even worse than the benchmark (possibly John’s, since he is still devastated by Sarah’s rejection, not knowing the real rejector was some heartless probabilist!), and then present after that one. Realistically, an actual student might also be satisfied with presenting after someone whose presentation fell in the lower one-third of the class, or even consider presenting first or last to make a more lasting impression (which would need to be verified with data).
And, who knows, if the order really does matter, then maybe the folks in Hollywood should take a page out of high schoolers’ books and release their diamonds in the midst of the rough, and not jam-packed into December.
This article is © 2020 by Angelina Chen. The article is licensed under a Creative Commons Attribution (CC BY 4.0) International license (https://creativecommons.org/licenses/by/4.0/legalcode), except where otherwise indicated with respect to particular material included in the article. The article should be attributed to the authors identified above.