Selecting Candidates for an interview: e reasoning

 

This conversation with OpenAI tells us how 'e' the transcendental number, helps us to choose a candidate from many that come for an interview:

Me:

Everyday problem like secretary problem seems to be interesting, if it is not too complicated!

OpenAI:

Great choice! The secretary problem is a famous example of optimal stopping theory, and it has a surprising connection to Euler’s number $e$. Here’s how it works:

The Problem Statement

Imagine you’re hiring a secretary (or choosing a house, a partner, etc.).

• You have $n$ candidates who arrive one by one in a random order.

• After each interview, you must decide immediately whether to hire that candidate.

• If you reject someone, you can’t go back.

• Your goal: maximize the chance of picking the best candidate.

The Best Strategy & How $e$ Appears

The optimal strategy turns out to be:

1. Reject the first n/e​ (≈ 37%) of candidates—just observe their quality.

2. Then, pick the next candidate who is better than all you’ve seen so far.

Mathematically, this gives you a $\frac{\mathrm{1/}}{e}\approx$ chance of picking the best person—which is the best you can do!

Why Does This Work?

• The first $\frac{n}{e}$ helps you estimate the true quality range.

• Once you reach the decision phase, you’re more likely to find the best candidate before running out of options.

This strategy applies to real-life decisions:

• Dating: Don't settle too early! Meet ~37% of potential partners before deciding.

• House hunting: Visit ~37% of the listings, then choose the best after that.

• Job offers: If you have multiple interviews, wait until ~37% are done before accepting the first great one.

Me:

Before we delve into n/e, if I were a candidate for the interview, should I come in after the 37th of 100 people congregated for interview? If every candidate thinks the same, which candidate would come in the bracket of 1 to 37?

OpenAI:

Ah! You’re flipping the problem on its head—thinking from the candidate’s perspective instead of the interviewer’s. Brilliant! 

Let’s break it down logically.

Should You Aim to Interview After 37% Have Gone?

If you were a candidate and knew the interviewer was following the $\frac{\mathrm{1/}}{e}$ strategy, then yes, you’d ideally want to show up after the first 37% have been seen because that’s when they actually start hiring.

If you come too early (1-37%), you’ll definitely be rejected since the interviewer is just "observing."
If you come too late, they might have already picked someone.

So your sweet spot is somewhere in the selection phase (after 37%).

What Happens If Everyone Knows This?

Now, suppose all 100 candidates are aware of this strategy and try to "game" the system by arriving after 37% have been interviewed.

• Then, who would show up in the first 37%?

• If nobody comes in that period, then the interviewer has no reference point for comparison.

• If everyone tries to come later, some will still end up in the first 37% randomly due to scheduling constraints.

This means that even in a system where everyone is strategic, the first 37% will still get filled by candidates who either weren’t aware of the strategy or had no choice.