If you’re on the search for a new secretary, how would you choose who to hire? For June’s evening CAMI meeting, Audrey shared a classic problem with the group:
The Popular Secretarial Problem!
There are N applicants for a secretarial position. The applicants are interviewed in random order, and you must accept or reject a candidate immediately after interviewing them.
After you reject someone, there is no way to bring them back.
There is only one position available. So as soon as you accept a candidate, you’re done. What strategy should you use in order to maximize the likelihood of hiring the best candidate out of the N applicants?
Some of the assumptions we used when exploring the problem:
- There is one best candidate.
- If you could see all the candidates at one (you can’t), you would be able to rank them all from best to worst.
Seems pretty complex! Audrey began by clarifying that the goal of the meeting was not to solve the problem (a tough task, to say the least!), but to explore ideas and come up with strategies on how to develop our understanding of the problem. Another goal was to discuss the applicability of mathematical problems like these in making real-world decisions.
We were given a few minutes to think about the secretary problem individually and write down our first thoughts. We then went around the room and shared those thoughts.
We broke up into partners based on similar initial directions, and went off to explore!
After our exploration, we wrapped up by sharing what we had worked on with our partners.
Lotus, Nadia, and Maya came up with a visual representation of the problem, showing how the ranking options change with the addition of each candidate in the pool.
Sophie and Yi simplified the problem to N=3, listing out all the possible ranking orders of the candidates and looking for patterns among the options.
Eric and Katherine also looked at a simpler case of fixed N, testing out one hiring strategy in particular and looking at the outcomes.
And Benny and Kevin explored a few different strategies and compared them, finding that some strategies gave better candidates on average than others.
We ended the meeting by discussing some applications and limitations of the secretary problem. A few ideas that came up:
- In what other situations could this apply? Finding a partner? Picking a house?
- Is this a realistic situation? Do we usually have to make an immediate decision?
- Should we be looking for the best candidate or just a good candidate?
- Could we explore the problem from another angle, avoiding getting a “dud”?
This problem has been extensively studied, and the “answer” may surprise you! Feel free to search the secretary problem and explore what others have found on the topic. Do you think that this could be a useful tool for making life decisions, or are the constraints too unrealistic?
