arxiv: 2605.10452 · v1 · submitted 2026-05-11 · 🧮 math.PR

Recognition: no theorem link

A secretary for Messrs. Luce and Mallows

Ross G. Pinsky

Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:10 UTC · model grok-4.3

classification 🧮 math.PR

keywords secretary problemLuce distributionMallows distributionpermutationsoptimal stoppingasymptotic analysisnon-uniform arrivals

0 comments

The pith

The secretary problem success probabilities are identical under Luce distributions with weights {q_j} and corresponding Mallows distributions for every n and every strategy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when candidates arrive according to a Luce distribution on permutations defined by a sequence of weights, the chance of selecting the best one with any stopping rule matches exactly the chance under a related Mallows distribution. This identity holds when the smallest label counts as the highest rank. The author uses the match to obtain the optimal strategy and its limiting success rate as the number of candidates grows large, and carries out the same limiting analysis for Mallows models with parameter greater than one and for other Luce weight families such as Sukhatme weights.

Core claim

For the Luce distribution on S_n induced by the weight class {q_j}, the probability of selecting the overall best item under any strategy in the secretary problem is exactly the same as the probability under the Mallows distribution with parameter q, when the smallest number is interpreted as the top rank. This identity holds for every finite n and transfers known results between the two families. Explicit asymptotic analysis of the optimal threshold strategy and the limiting success probability is then performed for the Mallows model with q greater than one and for Luce models using other weight sequences.

What carries the argument

The exact coincidence, for every n and every stopping strategy, of the secretary-problem success probability under the Luce distribution with weights {q_j} and under the Mallows distribution with parameter q, when the smallest label ranks highest.

If this is right

Any explicit formula or bound derived for one model applies unchanged to the other.
The optimal stopping threshold and the limiting success probability exist and are the same for the matched Luce and Mallows families.
The limiting analysis extends immediately to the previously untreated regime q>1 for Mallows models and to Luce models with Sukhatme weights.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The identity suggests that the Luce construction may be viewed as a weighted generalization that collapses to Mallows probabilities under the secretary-problem evaluation.
The same probability-matching technique could be tested on other optimal-stopping problems or under different rank interpretations.
Explicit enumeration for small n would produce tables of optimal strategies that are identical across the paired models.

Load-bearing premise

The arrival order is distributed exactly according to the stated Luce or Mallows probability measure on the set of all permutations of n items.

What would settle it

Compute the success probability for n=3 and the strategy of accepting the first left-to-right minimum after position 1; the two distributions must give identical numbers if the claim holds.

read the original abstract

We analyze the secretary problem in the case that the $n$ ranked items arrive not in uniformly random order but rather according to a certain type of Luce distribution or according to a Mallows distribution on the set $S_n$ of permutations of $[n]$. The secretary problem for the Mallows distributions with parameter $q\in(0,1)$ was analyzed in a previous paper; in this paper the case $q>1$ is also analyzed. The Luce distribution with the class $\{q_j\}_{j=1}^\infty$ of weights is related in a certain sense to the Mallows distribution with parameter $q$, but is more difficult to analyze. It turns out that for every $n$ and every strategy, the probabilities for the secretary problem when the smallest number is considered of highest rank for the Luce distribution with this class of weights coincides with those for the corresponding Mallows distribution. We analyze the asymptotic optimal strategy and corresponding limiting probability for the above cases, as well as for the Luce distributions with other classes of weights, such as the Sukhatme weights.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's main new result is an exact match in secretary problem success probabilities between a class of Luce distributions and the corresponding Mallows models for any n and any strategy, once the smallest label is fixed as highest rank.

read the letter

The central result here is that the success probabilities in the secretary problem coincide exactly for the Luce distribution with a certain class of weights and the matching Mallows distribution, for every n and every online strategy, under the convention that the smallest label ranks highest. This equivalence lets the author handle the Mallows case for q greater than 1, which was missing from the earlier work, and then extract asymptotics for the optimal strategy and limiting probability. The new piece is this probability coincidence, plus the extension to q>1 and the treatment of other Luce weights such as the Sukhatme class. The paper does a clean job of using the match to simplify the harder Luce analysis by borrowing from the Mallows side. The abstract presents the claims directly without overstatement. The soft spot is that the whole thing rests on the arrival process following exactly those distributions and on fixing the ranking interpretation in that specific way. If either changes, the equivalence disappears. The impact stays narrow because it is a targeted extension inside optimal stopping with biased permutations rather than a broad new method. This paper is for specialists who already know the uniform secretary problem and the Mallows model from the prior paper. A reader working on non-uniform variants will find the equivalence useful for computation and limits. It deserves a serious referee because the result is precise and the asymptotics add concrete value. I would send it to peer review. The derivations need checking, but the stated claims look coherent and worth verifying.

Referee Report

0 major / 2 minor

Summary. The paper examines the secretary problem under non-uniform permutation distributions, specifically Luce distributions with a class of weights and Mallows distributions with parameter q. The central claim is that the success probabilities coincide exactly between the Luce distribution (with smallest number as highest rank) and the corresponding Mallows distribution for every n and every strategy. The authors analyze the finite-n case, extend Mallows analysis to q > 1, and derive asymptotic optimal strategies and limiting success probabilities for these models as well as for other Luce weights such as Sukhatme weights.

Significance. This equivalence is a notable result that bridges two important models in ranking and choice theory, enabling the transfer of analytical methods and simplifying the study of the secretary problem under these distributions. The asymptotic analysis contributes to understanding how biased arrival orders affect optimal stopping thresholds and success rates. The exact match for all finite n and all strategies strengthens the paper's contribution, as it provides a parameter-free link between the models.

minor comments (2)

[Abstract] The abstract refers to 'the corresponding Mallows distribution' without specifying the parameter mapping from the Luce weights; this should be clarified for readers.
[Introduction] A reference to the previous paper analyzing Mallows for q in (0,1) is mentioned but not cited; please add the bibliographic reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript, including the recognition of the exact equivalence result and its implications for bridging Luce and Mallows models. The recommendation for minor revision is noted; however, the report lists no specific major comments requiring response. We are prepared to incorporate any minor clarifications or corrections once identified.

Circularity Check

0 steps flagged

No significant circularity; equivalence derived independently

full rationale

The paper establishes that secretary-problem success probabilities coincide exactly between the specified Luce weights and corresponding Mallows distributions for every n and every strategy (once the ranking convention is fixed). This equivalence is presented as a result that 'turns out' to hold and is then applied to finite-n and asymptotic analysis. The citation to the author's prior Mallows work (for q<1) supplies background and the q>1 extension but is not invoked to justify the central coincidence claim itself. No self-definitional loops, fitted inputs renamed as predictions, ansatz smuggling, or load-bearing self-citation chains appear in the derivation chain. The result remains externally falsifiable via direct computation on S_n and is therefore scored as self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard definitions of Luce and Mallows distributions on the symmetric group together with the usual axioms of probability.

axioms (1)

standard math Standard axioms of probability on the symmetric group S_n
Invoked throughout the definitions of the arrival distributions and the secretary problem formulation.

pith-pipeline@v0.9.0 · 5476 in / 1111 out tokens · 37901 ms · 2026-05-12T05:10:19.105356+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Borga, J., Chatterjee, S and Diaconis, P.Permuton and local limits for the Luce model, preprint

work page
[2]

Probab.28(2000), 1384-1391

Bruss, F.T.,Sum the odds to one and stop, Ann. Probab.28(2000), 1384-1391

work page 2000
[3]

and Nolen, J.,Block size in Geometric(p)-biased permutations, Electron

Cristali, I., Ranjan, V., Steinberg, J., Beckman, E., Durrett, R., Junge, M. and Nolen, J.,Block size in Geometric(p)-biased permutations, Electron. Commun. Probab. 23 (2018), Paper No. 80, 10 pp

work page 2018
[4]

and Kim, G

Chatterjee, S., Diaconis, P. and Kim, G. B.,Enumerative theory for the Tsetlin library, J. Algebra 655 (2024), 139–162

work page 2024
[5]

II, 3rd ed., Wiley, New York, (1971)

Feller, W.,An Introduction to Probability Theory and its Applications, vol. II, 3rd ed., Wiley, New York, (1971). 28 ROSS G. PINSKY

work page 1971
[6]

and Olshanski, G

Gnedin, A. and Olshanski, G. (2010).q-Exchangeability via quasi-invariance, Ann. Probab. 38 2103–2135

work page 2010
[7]

Pinsky R.,The secretary problem with biased arrival order via a Mallows distribution, Adv. in Appl. Math. 140 (2022), Paper No. 102386, 9 pp

work page 2022
[8]

Pinsky, R.,Comparing the inversion statistic for distribution-biased and distribution- shifted permutations with the geometric and the GEM distributions, ALEA Lat. Am. J. Probab. Math. Stat.,19(2022), 209-229

work page 2022
[9]

and Tang, W.,Regenerative Random Permutations of Integers, Ann

Pitman, J. and Tang, W.,Regenerative Random Permutations of Integers, Ann. Probab.,47(2019), 1378-1416

work page 2019
[10]

Sukhatme, P.,Tests of significance for samples of the chi square population with two degrees of freedom, Ann. Eugen. 8 (1937) 52–56. Department of Mathematics, Technion—Israel Institute of Technology, Haifa, 32000, Israel Email address:pinsky@technion.ac.il URL:https://pinsky.net.technion.ac.il/

work page 1937