The Classic Problem: When You Only Get One Shot
Imagine you’re a department chair tasked with hiring the best possible secretary from a pool of 100 applicants. The rules are harsh: you interview candidates in random order, must decide immediately after each interview whether to hire them, and once rejected, a candidate is gone forever. You can only hire one person. How do you maximize your chances of hiring the absolute best candidate?
This puzzle, known as the Secretary Problem (or the Marriage Problem, the Dowry Problem, or more formally, the Optimal Stopping Problem), has captivated mathematicians since the 1950s. Despite its seemingly impossible constraints, it yields a remarkably elegant solution.
The 37% Rule
The optimal strategy is to reject the first 37% of candidates outright, no matter how impressive they seem. Think of this as your “calibration phase.” You’re learning what the talent pool looks like, establishing a benchmark for quality. Then, hire the very next candidate who surpasses everyone you’ve seen so far.
More precisely, you should review exactly n/e candidates (where e ≈ 2.718, Euler’s number), which works out to about 36.8% of your pool. This strategy gives you approximately a 1/e (37%) chance of selecting the absolute best candidate, surprisingly high given that you’re choosing from 100 people under such restrictive conditions.
The mathematical beauty lies in its universality: whether you have 100 candidates or 1,000, the same proportion holds. Review 37%, then commit to the next best thing you see.
The Harsh Logic of Optimal Stopping
This solution embodies a fundamental tension in decision-making under uncertainty. Wait too long, and the best option may have already passed. Commit too early, and you haven’t gathered enough information to make an informed choice. The 37% rule represents the mathematical sweet spot between these extremes.
But here’s what makes the classical secretary problem so anxiety-inducing: it assumes complete finality. Once you pass on an opportunity, it’s gone forever. No second chances, no returns, no regrets.
Plot Twist: What If You Could Go Back?
Real life, and particularly academic life, rarely works with such unforgiving rules. What happens to the mathematics when we relax the “no recall” assumption?
The Secretary Problem with Recall
Consider a modified version where you can call back previously interviewed candidates, though they might not still be available. Perhaps they’ve taken other offers, or their interest has waned. Mathematicians model this in several ways:
Version 1: Probabilistic Recall
Each rejected candidate has a probability p of being available if you try to recall them later. If p = 0.5, there’s a 50-50 chance they’re still interested. If p = 1, you have perfect recall, anyone you’ve seen remains available.
Version 2: Decaying Availability
The probability of successfully recalling a candidate decreases with time. Someone you interviewed yesterday might have a 90% chance of being available, but someone from last month might only have a 10% chance.
Version 3: Limited Recalls
You can attempt to recall previously seen candidates, but only a limited number of times, say, three callbacks total.
The Revolutionary Result
With perfect recall (p = 1), the optimal strategy changes dramatically:
- Instead of stopping at 37%, you might review 50%, 60%, or even more of the candidate pool
- You set a much higher initial threshold for acceptance
- You gradually lower your standards as you proceed
- The probability of getting the best candidate can approach 60% or higher
Even with imperfect recall (p < 1), the strategy shifts toward more exploration. The key insight: option value. Every candidate you see but don’t immediately hire becomes a fallback option, however uncertain. This accumulated option value justifies extended search.
The “Shortlist” Variant
Another fascinating variant allows you to maintain a “shortlist” of up to k candidates, making your final decision after seeing everyone. With a shortlist of just 3-5 candidates, your probability of selecting the best can exceed 90%.
This mirrors real hiring: you interview everyone, identify finalists, then make deliberate comparisons. The harsh temporality of the classic problem dissolves.
The Academic Career Translation
Now here’s where it gets interesting. The classical secretary problem, with its irreversible decisions and lost opportunities, feels intuitively applicable to academic careers. Should you accept this postdoc? Commit to this research area? Take this tenure-track position?
But academic careers actually resemble the recall versions far more than the classic problem. Let’s see why.
Journal Submissions: The Ultimate Recall Game
When submitting a mathematics paper, the classical secretary problem would suggest: try journals in descending order of prestige, and accept the first one that says yes. Once rejected from the Annals, never return.
But that’s not how it works:
- Papers improve through rejection: Reviews from top journals, even rejections, provide valuable feedback
- Editors rotate: The editor who rejected you in 2024 might be gone by 2026
- Reputation accumulates: Your next paper’s acceptance makes editors reconsider your previous work
- Topics become fashionable: A result ignored today might be celebrated tomorrow when its relevance becomes clear
The Strategic Implication: Start ambitiously. Submit to the Annals, Inventiones, or JAMS first. Use rejections to improve the paper. You can always “recall” lower-tier journals later, and they’ll often be more receptive to papers that clearly went through rigorous review elsewhere, even if rejected.
Research Topics: Building an Inventory, Not a Sequence
The classical problem suggests you should explore research areas for your first few years (your 37%), then commit completely to whatever seemed most promising.
But mathematical research allows remarkable flexibility:
- Dormant projects reawaken: That representation theory problem you abandoned in year 3 might suddenly become solvable when you learn new techniques in year 7
- Skills transfer: Time spent learning algebraic geometry is never truly “wasted” even if you switch to analysis as mathematical maturity accumulates
- Problems wait for you: Unlike job candidates, mathematical problems don’t accept other offers. The Riemann Hypothesis will still be there tomorrow
- Multiple threads: You can pursue several research programs simultaneously, unlike hiring exactly one secretary
The Strategic Implication: Keep detailed notes on every research direction you explore. What seemed like a dead end might become viable with new techniques, collaborators, or theoretical developments. Your “failed” explorations are call options on future breakthroughs.
The Job Market: Memory and Reputation
A pure secretary problem approach to the academic job market would be terrifying: you get one shot at each institution, and timing is everything.
Reality is far more forgiving:
- Institutions remember strong candidates: That department that couldn’t hire you as an assistant professor might invite you as a tenured associate
- Networks persist: Committee members move institutions, carrying memories of impressive candidates
- Reputation is cumulative: Each market attempt builds visibility, even unsuccessful ones
- Multiple games simultaneously: Unlike hiring one secretary, you might field offers from multiple institutions
The Strategic Implication: Go on the market when you’re ready to be competitive, but don’t catastrophize about timing. Strong candidates who go “too early” often get invited back. The market has memory.
Collaborations: The Portfolio Model
The secretary problem assumes you’re choosing one option forever. But mathematical collaborations shatter this framework entirely:
- Dormant ties reactivate: That collaborator from your postdoc might resurface years later with the perfect complementary expertise
- Multiple simultaneous partnerships: You might have 3-5 active collaborations at once
- Low cost to experimentation: Trying a collaboration and having it not work out has minimal downside
- Network effects: Good collaborations lead to more opportunities
The Strategic Implication: Say yes to more collaborative opportunities early in your career. Unlike the secretary problem, there’s little penalty for exploration and enormous option value in maintaining broad networks.
The Meta-Lesson: Why Academic Anxiety Might Be Miscalibrated
Many mathematicians, especially early-career researchers, operate under secretary problem anxiety: the fear that passing on an opportunity means losing it forever, that there’s one optimal path that must be identified and committed to at the right moment.
But academic careers, particularly in mathematics, operate under “recall rules”:
- Ideas don’t expire: Unlike job candidates, mathematical ideas wait patiently to be explored
- Relationships persist: Academic networks maintain long memories
- Reputation accumulates: Past work continues to pay dividends
- Skills compound: Everything you learn remains in your toolkit
- Multiple attempts allowed: You can retry journals, reapply for grants, return to problems
This has profound implications for career strategy:
Explore More Broadly Than Your Instincts Suggest
The 37% rule assumes exploration has a steep opportunity cost, while you’re looking, options disappear. But in mathematics, exploration builds option value. That semester you spent learning category theory? It might pay off five years later in unexpected ways. Those three papers you submitted to ambitious journals and got rejected? They taught you the standards of top venues and improved your work.
Set Higher Standards Initially
With recall options, you can afford to be ambitious. Submit to the best journals first. Apply for the most prestigious postdocs. Propose the most innovative research ideas. The downside is smaller than you think, and the information value is enormous.
Document Everything
In a world with recall, memory becomes crucial. Keep notes on:
- Abandoned research directions (with clear documentation of why you stopped)
- All professional contacts, even casual conference conversations
- Referee reports, even negative ones
- Ideas that seemed promising but weren’t quite ready
Treat Rejections as Information, Not Failures
Every rejection in mathematics carries information: about standards, about current fashion, about your work’s presentation. Unlike the secretary problem where rejection means permanent loss, academic rejections often strengthen future attempts.
The Paradox of Choice, Resolved
The classical secretary problem creates anxiety because it combines high stakes with irreversibility. You must optimize under harsh constraints, knowing that mistakes are permanent.
But mathematical careers offer something remarkable: the ability to revisit, reconsider, and recombine. Your research path is less like hiring a secretary and more like tending a garden, ideas planted early might bloom years later, collaborations can go dormant and revive, and skills compound in unexpected ways.
This doesn’t mean every decision is reversible or that strategy doesn’t matter. Tenure clocks tick, opportunities do sometimes expire, and timing can be crucial. But recognizing that your career operates under “recall rules” rather than “secretary problem rules” should be liberating:
- That “failed” research project might become feasible with new techniques
- The journal that rejected you might accept your next paper
- The collaboration that didn’t work out taught you valuable lessons
- The job market will likely give you multiple chances
The optimal stopping point in the secretary problem is 37% because you can never go back. In academic mathematics, you can often go back, which means the optimal strategy is to explore more, aim higher, and worry less about perfect timing.
Your mathematical career is more forgiving than the secretary problem suggests. Act accordingly.