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Category Statistics & Probability

Quick Scenarios:

Dating Pool Size Total potential partners you expect to meet (2–10,000)

Exploration Phase %

Optimal is ~37% (1/e). Adjust to see effects.

Mutual Interest Rate (Optional)

Probability a candidate you choose also chooses you

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About

The optimal stopping problem in dating contexts applies the Secretary Problem algorithm to romantic partner selection. Given a pool of n potential partners encountered sequentially, the mathematical optimum involves rejecting the first 36.8% (precisely 1e ≈ 0.3679) unconditionally while establishing a quality benchmark. After this exploration phase, you commit to the first candidate who exceeds all previously observed options. This strategy maximizes the probability of selecting the single best partner at approximately 37% - counterintuitively optimal when alternatives include random selection (1n probability) or exhaustive comparison (impossible with sequential, non-recallable encounters).

The model assumes candidates arrive in random order, each can be ranked against previous candidates, rejected candidates cannot be recalled, and exactly one selection must occur. Real-world deviations include non-random encounter sequences, imperfect ranking ability, and the possibility of mutual rejection. The calculator provides probability distributions across different stopping points, expected rank outcomes, and sensitivity analysis for parameter variations.

Formulas

The optimal stopping rule derives from maximizing the probability of selecting the absolute best candidate from a sequential pool. For a pool of n candidates with rejection threshold r, the probability of selecting the best candidate equals:

P(best) = rn n∑k=r+1 1k − 1

To find optimal r, we take the derivative with respect to r and set equal to zero. In the continuous limit as n → ∞, substituting x = rn:

P(x) = −x ln(x)

Maximizing yields dPdx = −ln(x) − 1 = 0, giving x^* = 1e. The optimal rejection count thus equals:

r^* = ⌊ne⌋ = ⌊0.3679n⌋

where r^* = optimal rejection threshold (floor function), n = total candidate pool size, e = Euler's number (2.71828...). The expected rank of the selected candidate under optimal strategy follows E[Rank] ≈ e − 1 + ln(n) for large n.

Reference Data

Pool Size (n)	Optimal Reject Count (r)	Selection Phase Starts	P(Best) Exact	Expected Rank	Worst-Case Rank
5	2	Candidate 3	43.33%	1.52	5
10	4	Candidate 5	39.87%	1.88	10
15	6	Candidate 7	38.95%	2.12	15
20	7	Candidate 8	38.42%	2.31	20
25	9	Candidate 10	38.06%	2.47	25
30	11	Candidate 12	37.81%	2.60	30
40	15	Candidate 16	37.49%	2.82	40
50	18	Candidate 19	37.30%	3.00	50
75	28	Candidate 29	37.05%	3.28	75
100	37	Candidate 38	36.91%	3.49	100
150	55	Candidate 56	36.83%	3.78	150
200	74	Candidate 75	36.80%	3.98	200
500	184	Candidate 185	36.79%	4.61	500
1000	368	Candidate 369	36.79%	5.10	1000
n → ∞	ne	36.79% of pool	1e ≈ 36.79%	O(ln n)	n

Frequently Asked Questions

The convergence to 1e emerges from the continuous limit of the discrete optimization. As pool size n increases, the harmonic sum approximation n∑k=r1k → ln(n) − ln(r) transforms the probability function into −xln(x), whose maximum occurs at x = e⁻¹. This is a fundamental constant in optimal stopping theory, independent of specific distribution assumptions.

The classical Secretary Problem assumes unilateral selection power. To model mutual selection, modify the success probability by multiplying with reciprocal acceptance rate P_accept. If candidates accept with probability p, optimal rejection phase shifts to approximately 1ep of the pool. For 50% acceptance rate, explore roughly 74% before committing. The calculator's advanced mode includes this adjustment.

Under strict optimal stopping, you must select the final candidate regardless of quality. This represents the model's primary risk: approximately 37% chance of ending with the last candidate, who may be suboptimal. The expected rank of e − 1 + ln(n) accounts for this tail risk. Practical applications may allow rejection of the final candidate (no selection), which requires modified threshold calculations.

The algorithm generalizes to any sequential selection problem satisfying: candidates arrive one at a time in random order, immediate accept/reject decisions required, no recall of rejected options, and cardinal ranking capability. Apartment hunting with 20 viewings suggests rejecting the first 7 to establish baseline, then committing to the first apartment exceeding all previous. Job offer timing differs slightly due to exploding offers and negotiation possibilities.

The 36.79% probability assumes perfect ranking ability and truly random arrival order. Human cognitive biases (recency effects, contrast effects, satisficing) reduce effective success rates. Empirical studies show actual performance ranges from 20% to 31% in laboratory settings. The model provides an upper bound on achievable performance given the structural constraints of sequential, non-recallable selection.

Partial recall fundamentally changes optimal strategy. If rejected candidates can be recalled with probability q, the exploration phase shrinks. At q = 1 (perfect recall), optimal strategy becomes simply ranking all candidates then selecting the best. For intermediate q, numerical methods determine thresholds. Dating apps with match persistence approximate partial recall scenarios with q ≈ 0.3 to 0.5.