Secretary Problem and 1/e Optimal Stopping

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Optimal Stopping Times.

Your Task
You interview $n$ candidates sequentially for a position. You can only rank a candidate relative to previous candidates, and you must accept or reject immediately after the interview. Rejection is permanent. Your goal is to maximize the probability of selecting the absolute best candidate. Prove that the optimal policy skips the first $r^*-1$ candidates and then accepts the first candidate who is better than all previous candidates, where $\lim_{n\to\infty} \frac{r^*}{n} = \frac{1}{e} \approx 0.368$, achieving a success probability of $\frac{1}{e}$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.