The analysis starts with a leading threshold proportion, an asymptotic estimate of where the integer cutoff should lie as the problem grows. It then addresses the part that the leading estimate leaves open: which exact integer has the highest expected payoff. The central result is a second-order rounding rule. Under the paper's stated assumptions, the optimal cutoff is confined to the two neighboring integers around the leading estimate. Away from a shrinking transition region, the selected integer is the floor of the leading term plus a correction constant, so the correction determines the rounding direction.
The last step is the hard one
The unit of analysis is not a participant group or an empirical sample. It is a family of single-threshold optimization problems, with expected payoffs calculated over the randomness in each model. The candidate thresholds are integers, and the question is to identify the one that maximizes the payoff sequence. The result is therefore about exact solutions to mathematical models, not empirical participant outcomes.
The proof works by examining discrete payoff increments, the change in expected payoff when the threshold moves from one integer to the next. It uses unimodality, meaning the payoff sequence has a single rise-and-fall shape, and looks for the point where those increments switch from positive to negative. That sign change locates the maximum. Near the leading proportion, the increments are approximated by a local affine profile, essentially a straight-line pattern. If that approximation is uniform and its error tends to zero, then any sequence of integer approximations whose distance from the leading proportion also tends to zero eventually becomes the unique optimal threshold.
A correction decides between two integers
The second-order term matters because the leading estimate alone cannot fully resolve a narrowing band near the decision boundary. In ordinary terms, the optimizer is either the integer immediately below the leading value or the next one above it. The correction constant shifts the boundary between those two choices. When the decimal part of the leading value is sufficiently far from that boundary, the rule is clear: take the floor after adding the correction. When it lies inside the shrinking error band, the available expansion may not decide the answer. The rounding law is therefore conditional on both the model assumptions and the quality of the local approximation.
The correction changes with the model
One application is a fixed-horizon framework whose payoff increments have a harmonic form. There, the leading proportion is set by the relation between the logarithm of that proportion and a model constant: minus the logarithm of the proportion equals C. The local slope is the negative reciprocal of the proportion, and the second-order correction is one-half of one minus the proportion. Those terms give an explicit constant-order adjustment to the leading cutoff, which is the part needed to settle the final integer rounding.
The same structure changes with uncertainty about employment. In that model, the offer-acceptance probability is rho, and the leading proportion is rho raised to the power one divided by one minus rho. The second-order correction is rho multiplied by one minus that leading proportion, then divided by two. The paper places this correction strictly above zero and below one-half. This makes the shift parameter-dependent rather than a universal rounding constant, while the result remains asymptotic in the number of applicants.
A second example uses power-biased random horizons. With zero power bias, the leading proportion is e raised to minus two. For any positive power-bias parameter, the leading proportion is the unique interior solution of the model's defining equation. The associated correction is given explicitly by the second-term function and the slope of the leading term, evaluated at that proportion. Across all nonnegative values of the bias parameter, the correction is positive and smaller than one-half. These calculations show how the same rounding principle can accommodate a different horizon distribution while retaining a model-specific leading cutoff.
From approximation to an exact cutoff
The analysis also gives the rounding rule an arithmetic route. For irrational leading proportions, it applies to large continued-fraction convergents; for rational proportions, it uses exact multiples. These results connect a limiting proportion to sequences of problem sizes for which the approximation is accurate enough to identify the optimizer. In a classical factorial-horizon application, the corresponding derangement number is identified as the unique threshold. The point is a way to turn rational approximation into an exact cutoff within the stated mathematical models.
A mathematical result with a narrow brief
The result concerns single-threshold models and depends on unimodality, a uniform local expansion, and an approximation error that vanishes. Most conclusions are asymptotic, so they apply only once the relevant problem parameters or approximation sequences are sufficiently large. The leading expansion also leaves a shrinking transition region, and cases involving an integer-valued correction may need sharper remainders or higher-order terms. No empirical dataset or participant sample is part of the analysis.
The contribution is clear but conditional. The leading asymptotic calculation narrows the answer to two neighboring integers, while the second-order constant decides between them wherever the error margin is small enough. For the secretary-type payoff sequences covered by the analysis, that provides a systematic bridge from an approximate proportion to an exact integer threshold. The conclusion remains within the stated model class and assumptions.
Paper data and sources
Original title: Asymptotic Rounding Laws for Optimal Thresholds in Secretary Problems
Authors: Raúl Sánchez Galán
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text