Behavioral Model Interactive

Optimal Stopping

The 37% rule: explore first, then commit to the first option that beats your best. Mathematically optimal strategy for sequential decision-making.

🎯 The Secretary Problem

You must hire one candidate from N applicants. You interview them one at a time, and must decide immediately (accept or reject forever). How do you maximize your chance of picking the best?

The 37% Rule

1. Look: Reject the first ~37% of candidates
2. Note: Record the best you've seen
3. Leap: Pick the first one who beats that benchmark

Why 37%?

1/e ≈ 36.8%

This is the mathematically optimal explore/exploit boundary. It gives you a ~37% chance of picking the best candidate.

Better than guessing (1/N) and better than any other fixed rule.

Parameters

5 50

5 80

Reject first 7 candidates

Then pick best from remaining 13

📊 Simulation Results

38.8%

Success Rate (1000 trials)

✅ Near optimal! You're using the 37% rule.

Success Rate by Explore %

Peak around 35-40%. Both too little and too much exploration hurt performance.

The Mathematics

Optimal Threshold

n/e ≈ 0.368n

Reject the first n/e candidates, then take first that beats best seen.

Success Probability

≈ 1/e ≈ 36.8%

Probability of picking the absolute best candidate.

🏀 Betting Applications

Line Shopping

Check 37% of books, then take first that beats best seen

Player Selection

Evaluate 37% of options, then pick first above threshold

Cashout Decision

Wait until 37% of game, then cashout if value exceeds best prior

Season Betting

Observe 37% of season, then bet when pattern beats baseline

R Code Equivalent

# Optimal stopping simulation
secretary_problem <- function(n, look_pct = 1/exp(1), trials = 1000) { 
  look_n <- floor(n * look_pct)
  wins <- 0
  
  for (t in 1:trials) { 
    candidates <- runif(n)
    best_idx <- which.max(candidates)
    
    # Look phase
    look_best <- max(candidates[1:look_n])
    
    # Leap phase
    picked <- NA
    for (i in (look_n + 1):n) { 
      if (candidates[i] > look_best) { 
        picked <- i
        break
      }
    }
    if (is.na(picked)) picked <- n
    
    if (picked == best_idx) wins <- wins + 1
  }
  
  return(wins / trials)
}

# Test different look percentages
results <- sapply(seq(0.1, 0.8, by = 0.05), function(p) secretary_problem(20, p))
cat(sprintf("Optimal at ~37%%: %.1f%% success\n", secretary_problem(20) * 100))

✅ Key Takeaways

• 37% rule: explore first, then commit
• Gives ~37% chance of picking the best
• Better than guessing or any fixed strategy

• Apply to line shopping, player selection
• Too little exploration = poor benchmark
• Too much exploration = best option passed