Look, Then Leap

Imagine a game. I show you a stack of cards, each with a different number written on it, face down. I flip them over one at a time. After each flip you must say “stop” or “keep going.” Say stop and you win whatever is on the current card — the game ends. Say keep going and that card is gone forever; you can never come back to it. Your goal: stop on the single highest number in the stack. You don’t know the range, you don’t know what’s coming, and every card you pass is burned. When should you stop?

Your gut offers two terrible strategies. One: grab an early card that “seems high” — but seems high compared to what? You’ve barely seen any. Two: hold out for something amazing — but if you’re too greedy the big number sails past early and you’re left choosing among scraps. Both fail because they ignore the real structure of the game: information arrives over time, and every choice is irreversible. This is the shape of an enormous number of real decisions, and it has a name.

That shape is optimal stopping: a class of problems where options appear in sequence, each must be accepted or rejected on the spot, and a rejected option cannot be recalled. Hiring when candidates take rival offers, house-hunting in a fast market, selling when bids trickle in, deciding when you’ve dated enough, or simply knowing when you’ve researched enough and should just choose — all of them. And the flagship example, the one that gives the whole field its punchline, is the secretary problem: interview applicants one by one, decide each on the spot, and try to hire the best.

See it in one picture

Here is the whole idea on one chart. The horizontal axis is your look fraction — how much of the sequence you spend just looking before you allow yourself to commit. The vertical axis is how often the look-then-leap rule lands the single best option. Drag the look-fraction slider from left to right and watch the success rate rise, crest, and fall. It peaks in the middle — right around 37% — not at zero (committing blind) and not near the end (waiting too long). That hump is the secretary problem, and its peak is the 37% rule.

Then hit Deal a new sequence a few times to watch the rule fire on a single run: the grey cards are the look phase you throw away to set your bar, and the rule leaps at the first later card that beats them. Sometimes it nails the best (green); sometimes the best was hiding in the opening sample and it misses (red). Run it enough and you’ll feel why ~37% wins about 37% of the time — remarkably often for a blind, one-shot game.

Now flip the Objective toggle to Maximise expected value. The curve changes: when you get credit for a very good pick, not just the single best, the optimal look fraction shrinks below 37% — because holding out for perfection isn’t worth passing up great options. That single switch, from “the best or nothing” to “as good as possible on average,” is one of the most important lessons in the whole course, and we build it up carefully in Lesson 3.

What you’ll walk away with

By the end you’ll be able to look at a decision — a hire, a flat, an offer, a partner, a research rabbit-hole — recognise whether it has the sequential, no-going-back shape, and if it does, split it deliberately into a look phase and a leap phase instead of agonising. Here’s the map:

1. The secretary problem & the 37% rule — the exact result, why the number e shows up, and small cases worked by hand so you see the mechanism.
2. Explore-then-commit — the general pattern under the rule: a sampling phase that sets a bar, and a commit phase that takes the first option to clear it.
3. When you want the best expected value — the gentler objective, why the optimal look shrinks below 37%, and why your bar should fall as options run out.
4. Variations that matter — recall (you can go back), a known distribution (reservation values), search costs, uncertain N, and rejectable offers.
5. Transfer — and where the model lies — hiring, house-hunting, the “37% then commit” dating meme and its honest caveats, and the traps: the pure rule optimises the wrong thing, assumes strict irreversibility, and ignores what you know about the odds.

How to use this course

Every lesson opens with a quick guess, teaches the idea through a concrete story and worked numbers, and checks that it stuck. Don’t skip the guesses — committing to an answer before you know is one of the most reliable ways to actually remember. You’ll drive the optimal-stopping lab from several angles and meet sorting and matching exercises along the way.

This is an expert-tier course, so it leans on a few models you’ve ideally met already. You’ll get the most from it if you’re comfortable with thinking in probabilities (reasoning about the chance that the best-so-far is the true best, and what “37% of the time” really means), the value of information (the look phase is literally buying information about the option pool before you commit), and deciding under deep uncertainty (you must act without ever seeing the whole set of options). When you’ve finished the five teaching lessons, a graded final exam pulls it all together — it’s one-way, so once you submit an answer it’s locked.

Ready? The next lesson goes straight to the heart of it — the exact secretary problem, the 37% rule, and why the mysterious number e decides when you should stop.