STATISTICAL SELECTION · RECALL ENABLED

The
Second-Best
Gate

Learn the field from eight. Keep the best two. Then let one credible challenger end the search.

  1. 01Draw eight candidates at random.
  2. 02Rank them; retain the best and second best.
  3. 03Continue until a newcomer beats the second-best scout.
  4. 04Choose the best candidate seen so far.

Model used here: sampled candidates can be recalled, arrival order is random, ranks are distinct, and if nobody crosses the gate, the best scout is selected.

LIVE EXACT MODEL

Interactive field size

This is exact enumeration over relative ranks, not a randomized animation. “Rank percentile” is 100% for the best candidate and 0% for the worst.

EXPECTED RANK PERCENTILE——
CHANCE OF LITERAL #1——
EXPECTED CANDIDATES SEEN——
expected rank percentilechance of literal best

COMPUTATIONAL RESULT

Stopping rule

With 19 candidates, the selected candidate’s expected rank percentile is 97.02%: on average, only about 3% of the field ranks higher. At 24 candidates it is 96.15%; at 100 it is 93.45%.

As the field becomes very large, expected rank percentile approaches roughly 92.6%, while the search still ends after about 16 evaluations on average. The system’s durable advantage is this quality–effort trade: fixed scouting cost, a forgiving second-place threshold, and near-constant expected search effort as the pool grows.

ONE RUN

Role of the second-best scout

SCOUT EIGHT4267915834847351
SEARCH637988
UNSEEN9670

Gate = 84. The 88 is the first newcomer to clear it, so evaluation stops.

Selection = 91. Because recall is allowed, the best candidate seen so far—not necessarily the triggering candidate—wins.

The trade. The unseen 96 shows what early stopping buys and what it risks.

THE SECRETARY CONSTANT

Relationship to classical secretary rules

The classical secretary problem observes and rejects about the first n/e candidates, then accepts the first later candidate better than every candidate in that sample. With no recall and a goal of selecting the unique best, its optimal success probability tends to 1/e, about 36.8%.

The Second-Best Gate shares the sample-then-threshold structure, but changes three decisive terms: the scout count is fixed at eight, the threshold is the second-best scout, and earlier candidates remain recallable. Those changes make it faster and forgiving, but they also mean the classical optimality theorem does not transfer.

RULESCOUT PHASEGATERECALLPRIMARY GOAL
Second-Best Gate8 fixed#2 scoutyeshigh rank, low effort
Classical secretary≈ n/e#1 scoutnoprobability of #1

Reference: Oxford Probability and Computing notes derive the n/e threshold and asymptotic 1/e success probability.

AUDIT THE CLAIM

Simulation method

The enumeration

Let a and b be the best and second-best absolute ranks among the eight scouts. The probability of that pair is C(n−b, 6) / C(n, 8). Every later rank better than b, except a, is equally likely to be the first gate-crosser. Summing over every valid (a,b) pair gives the displayed expectations exactly.

The boundary

Relative-rank results are distribution-free for distinct exchangeable candidates. Real hiring may violate random arrival, stable scoring, independence, and recall. A different value distribution also changes score ratios even when rank percentiles remain the same.

The optimality boundary

“Optimal” requires an objective: best-candidate probability, expected rank, evaluation cost, or a utility combining them. This page validates the proposed rule; it does not claim that one rule minimizes every possible objective.

Download exact validatorDownload computed evidencePython standard library · deterministic · no fitted parameters
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