Smooth Partial Lotteries for Stable Randomized Selection

Alexander Goldberg; Giulia Fanti; Nihar B. Shah

arxiv: 2605.20069 · v2 · pith:62PZUHJJnew · submitted 2026-05-19 · 💻 cs.LG · cs.GT

Smooth Partial Lotteries for Stable Randomized Selection

Alexander Goldberg , Giulia Fanti , Nihar B. Shah This is my paper

Pith reviewed 2026-05-22 09:08 UTC · model grok-4.3

classification 💻 cs.LG cs.GT

keywords partial lotteriessmooth selectionregret boundsrandomized selectionstabilityLipschitz conditionpeer reviewClipped Linear Lottery

0 comments

The pith

The Clipped Linear Lottery achieves near-optimal regret for any smooth partial lottery by scaling selection probabilities linearly with scores between fixed thresholds.

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

Many selection processes randomize final choices among top-scoring candidates to reduce the impact of arbitrary fine distinctions. Current lottery methods can still be unstable, with one candidate's probability jumping sharply from a tiny score adjustment. The paper establishes smoothness, defined as a Lipschitz bound on how probabilities respond to score changes, as a stability criterion. It then shows that the Clipped Linear Lottery meets this criterion while its regret is within a factor of (1 - k/n) of the best any smooth rule can achieve. Experiments on actual conference and grant review data verify that the new design is more stable and has a superior tradeoff than previous approaches.

Core claim

We introduce the Clipped Linear Lottery, a partial lottery in which a candidate's selection probability is set to 1 above an upper threshold on their score, to 0 below a lower threshold, and interpolated linearly in between. We prove that this mechanism's worst-case regret matches the lower bound for any smooth selection rule up to a multiplicative factor of (1 - k/n), where k/n is the acceptance rate. We further show that it attains a better smoothness-regret tradeoff than alternatives based on individual fairness or differential privacy.

What carries the argument

The Clipped Linear Lottery mechanism, which enforces linear scaling of selection probabilities with estimated quality between reject and accept thresholds.

If this is right

Any smooth selection rule cannot beat the Clipped Linear Lottery's regret by more than the factor 1/(1-k/n).
The linear form provides a concrete, simple way to implement stable lotteries in practice.
Comparisons show superiority over fairness and privacy based alternatives in the smoothness-regret tradeoff.
Real-world tests on peer review datasets confirm reduced instability under score perturbations.

Where Pith is reading between the lines

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

If smoothness is adopted, borderline candidates would see more predictable outcomes even if scores fluctuate slightly due to reviewer variability.
This approach could extend to non-peer-review settings like job hiring where evaluators give noisy scores.
Future work might explore whether nonlinear smooth functions could improve the regret factor further while keeping the Lipschitz property.

Load-bearing premise

That requiring the selection probability map to be Lipschitz continuous in the review scores adequately captures the desired stability against small perturbations.

What would settle it

Finding a smooth selection rule whose worst-case regret is more than (1 - k/n) times smaller than that of the Clipped Linear Lottery, or observing in experiments that probability changes exceed the Lipschitz bound under small score perturbations.

Figures

Figures reproduced from arXiv: 2605.20069 by Alexander Goldberg, Giulia Fanti, Nihar B. Shah.

**Figure 1.** Figure 1: Clipped Linear Lottery responds smoothly to score perturbations. Marginal selection probabilities before and after a one-point increase in proposal 5’s review score, with 2 awards for 8 candidates. Under the thresholded partial lottery, proposal 8 is selected deterministically, while the remaining award is allocated with equal probability among proposals 6 and 7; the score increase lifts proposal 5, split… view at source ↗

**Figure 2.** Figure 2: A concrete example of the Clipped Linear Lottery with [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Regret–smoothness tradeoff for existing partial lottery mechanisms. Down and to the left [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** Figure 4: Regret vs. smoothness L at acceptance rate 10%. Down and to the left indicates a better tradeoff. The Clipped Linear Lottery achieves lower regret than softmax at every smoothness level across all datasets. 7.2 Regret–Smoothness Tradeoff We next measure regret as a function of the smoothness parameter L. For each dataset and acceptance rate, we compute the regret of the Clipped Linear Lottery and top-k sof… view at source ↗

**Figure 5.** Figure 5: Worst-case empirical smoothness vs. target global smoothness [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

**Figure 6.** Figure 6: CCDF of mean normalized utilities across datasets. Swiss NSF and Beta have heavier [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗

**Figure 7.** Figure 7: Regret vs. Beta shape parameter α = β at acceptance rate 10%. As the distribution concentrates around 0.5 (larger α), regret decreases for both selection rules because the worst-case utility gap narrows. 0.2 0.4 0.6 0.8 1.0 L (lower is smoother) 0.00 0.05 0.10 0.15 0.20 Regret / k Beta 0.2 0.4 0.6 0.8 1.0 L (lower is smoother) 0.00 0.02 0.04 0.06 0.08 ICLR 0.2 0.4 0.6 0.8 1.0 L (lower is smoother) 0.00 0.0… view at source ↗

**Figure 8.** Figure 8: Regret vs. smoothness L at acceptance rate 33%. advantage over the softmax rule, consistent with the theoretical gap of O(log n). Figures 8 and 9 show the tradeoff between regret and smoothness at higher acceptance rates. In both cases, the Clipped Linear Lottery still outperforms softmax, although the gap is smaller. Intuitively, as the acceptance rate increases, regret normalized by the number of selecte… view at source ↗

**Figure 9.** Figure 9: Regret vs. smoothness L at acceptance rate 50%. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗

**Figure 10.** Figure 10: Regret–smoothness tradeoff at acceptance rate 50%. Points show MERIT and Swiss NSF [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗

read the original abstract

Competitive selection processes, from scientific funding to admissions and hiring, use evaluations to score candidates, and eventually choose a subset of them based on those scores. Recently, many organizations have adopted partial lotteries, which randomize selection based on evaluation scores. However, existing lottery designs are inherently unstable, as a small change to a single candidate's score can cause large shifts in their selection probabilities. This instability undermines a key goal of lotteries: reducing the influence of fine-grained score distinctions near the decision boundary. We propose smoothness as a design principle for partial lotteries, formalizing it as a Lipschitz condition on the mapping from review scores over candidates to selection probabilities. We introduce the Clipped Linear Lottery, a simple mechanism in which selection probabilities scale linearly with estimated quality between an upper threshold, above which we always accept, and a lower threshold, below which we always reject. We prove that the Clipped Linear Lottery's worst-case regret matches a lower bound for any smooth selection rule up to a factor of $(1 - k/n)$, where $k/n$ is the acceptance rate. We compare smooth selection to other stability notions like Individual Fairness and Differential Privacy, showing that the Clipped Linear Lottery achieves a better smoothness-regret tradeoff than alternatives. Experiments on real peer review data from ICLR 2025, NeurIPS 2024, and the Swiss National Science Foundation demonstrate that existing lottery designs are highly unstable in practice even under perturbations to a single score. Our experiments also confirm the tightness of our theoretical analysis and show that our proposed Clipped Linear Lottery achieves a better smoothness-utility tradeoff than alternatives in practice.

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 defines smoothness for partial lotteries as Lipschitz continuity and gives a clipped linear mechanism whose regret nearly matches the lower bound for smooth rules, with real-data experiments showing practical instability in existing designs.

read the letter

The main thing here is a new smoothness criterion for partial lotteries, formalized as a Lipschitz condition on the map from scores to selection probabilities, plus the Clipped Linear Lottery that scales linearly between fixed thresholds. They prove its worst-case regret comes within a (1 - k/n) factor of the lower bound that holds for any smooth rule, and they show it beats alternatives on the smoothness-regret tradeoff.

Referee Report

1 major / 2 minor

Summary. The paper proposes smoothness, formalized as a Lipschitz condition on the mapping from review scores to selection probabilities, as a stability principle for partial lotteries in selection processes. It introduces the Clipped Linear Lottery, in which probabilities scale linearly between fixed upper and lower thresholds, and proves that this mechanism's worst-case regret matches a lower bound applying to any smooth selection rule up to a multiplicative factor of (1 - k/n). The work compares smoothness to Individual Fairness and Differential Privacy, and validates the approach via experiments on real peer-review data from ICLR 2025, NeurIPS 2024, and the Swiss National Science Foundation.

Significance. If the central regret bound holds for feasible smooth rules, the paper supplies a concrete, simple mechanism with a near-optimal smoothness-regret tradeoff and demonstrates its practical stability advantage over existing lottery designs on real data. The real-data experiments and explicit comparison to alternative stability notions are clear strengths that ground the theoretical contribution.

major comments (1)

[§3] §3 (Definition of smoothness): the Lipschitz condition is stated with respect to a norm on R^n without explicit projection onto the hyperplane where probabilities sum to k. Because any valid partial lottery must satisfy this linear constraint, the lower bound may be derived over a strictly larger function class than the feasible smooth rules; this directly affects whether the (1 - k/n) factor establishes near-optimality for the intended domain. The proof of the lower bound and the regret analysis for the Clipped Linear Lottery should clarify how (or whether) the constraint is incorporated into the Lipschitz constant.

minor comments (2)

[Experiments] The experimental section would benefit from reporting the precise perturbation magnitudes used in the single-score sensitivity tests and the number of random seeds for the regret estimates.
[Introduction] Notation for the acceptance rate k/n is introduced late; an early global notation table or consistent use from the abstract onward would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the positive assessment of our work and for the detailed comments. We are grateful for the opportunity to clarify the technical point raised regarding the smoothness definition and its relation to the sum-to-k constraint. We address this below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

Referee: [§3] §3 (Definition of smoothness): the Lipschitz condition is stated with respect to a norm on R^n without explicit projection onto the hyperplane where probabilities sum to k. Because any valid partial lottery must satisfy this linear constraint, the lower bound may be derived over a strictly larger function class than the feasible smooth rules; this directly affects whether the (1 - k/n) factor establishes near-optimality for the intended domain. The proof of the lower bound and the regret analysis for the Clipped Linear Lottery should clarify how (or whether) the constraint is incorporated into the Lipschitz constant.

Authors: We thank the referee for this precise observation. In the manuscript, the selection rule is defined as a mapping f from score vectors in R^n to selection probability vectors p in R^n satisfying sum p_i = k and 0 ≤ p_i ≤ 1 for all i. The Lipschitz condition is ||f(s) - f(s')|| ≤ L ||s - s'||, where the norm on the left is taken in R^n (typically Euclidean). Since f(s) always lies in the affine hyperplane H = {p | sum p_i = k}, the differences f(s) - f(s') lie in the parallel subspace orthogonal to the all-ones vector. The lower bound is established by constructing specific score vectors s and s' differing in a way that forces any smooth f to incur regret, and these constructions are chosen so that the worst-case p remains feasible (i.e., in H). The factor (1 - k/n) arises naturally from the geometry of this (n-1)-dimensional subspace. For the Clipped Linear Lottery, the mechanism explicitly normalizes to ensure sum p_i = k after clipping and linear scaling. We will revise §3 to explicitly state the codomain as the constrained set and add a paragraph in the proof of the lower bound (and in the regret analysis) clarifying that the Lipschitz condition is with respect to the ambient norm but restricted to functions with range in H. This restriction does not invalidate the lower bound, as the adversarial instances are feasible, and the (1 - k/n) factor precisely quantifies the near-optimality within the feasible class. revision: yes

Circularity Check

0 steps flagged

No significant circularity; regret bounds derived independently of fitted inputs or self-citations

full rationale

The paper's central result establishes a lower bound on worst-case regret that holds for any selection rule satisfying the stated Lipschitz smoothness condition on the score-to-probability map, then shows that the Clipped Linear Lottery achieves an upper bound within a (1-k/n) factor of that lower bound. This is a standard minimax-style approximation argument with no reduction of the claimed prediction to a fitted parameter or to a self-citation chain. The abstract and reader's summary give no indication that the lower bound or the smoothness definition is constructed from the mechanism itself; the comparison to Individual Fairness and Differential Privacy is presented as an external benchmark. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central claim rests on the Lipschitz smoothness definition and the existence of a regret lower bound for smooth rules.

axioms (1)

domain assumption Smoothness is formalized as a Lipschitz condition on the mapping from review scores over candidates to selection probabilities.
Introduced as the design principle for stable partial lotteries.

pith-pipeline@v0.9.0 · 5823 in / 1076 out tokens · 31510 ms · 2026-05-22T09:08:53.748215+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel / Jcost uniqueness unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

p(X) = arg min_{p in C_{n,k}} 1/2 ||p - w u(X)||_2^2 ; clip[0,1](w u_i + b) with sum p_i = k
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

L-smoothness: ||p(X) - p(X')||_1 <= L ||X - X'||_{1,1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 2 internal anchors

[1]

Calibrating noise to sensitivity in private data analysis

Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. InTheory of Cryptography, 3rd Theory of Cryptography Conference, TCC 2006, volume 3876 ofLecture Notes in Computer Science, pages 265–284. Springer,

work page 2006
[2]

doi: 10.1007/11681878

work page doi:10.1007/11681878
[3]

Ferric C Fang and Arturo Casadevall

Accessed: 2025-05-07. Ferric C Fang and Arturo Casadevall. Research funding: The case for a modified lottery,

work page 2025
[4]

Ref. no. 2025–01621. Alexander Goldberg, Giulia Fanti, and Nihar B Shah. A principled approach to randomized se- lection under uncertainty: Applications to peer review and grant funding. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems,

work page 2025
[5]

Rankmax: An adaptive projection alternative to the softmax function

Weiwei Kong, Walid Krichene, Nicolas Mayoraz, Steffen Rendle, and Li Zhang. Rankmax: An adaptive projection alternative to the softmax function. InAdvances in Neural Information Processing Systems 33 (NeurIPS 2020),

work page 2020
[6]

NeurIPS 2024 Conference Submissions.https://openreview.net/submissions? venue=NeurIPS.cc/2024/Conference,

19 OpenReview. NeurIPS 2024 Conference Submissions.https://openreview.net/submissions? venue=NeurIPS.cc/2024/Conference,

work page 2024
[7]

ICLR 2025 Conference Submissions.https://openreview.net/submissions? venue=ICLR.cc/2025/Conference,

OpenReview. ICLR 2025 Conference Submissions.https://openreview.net/submissions? venue=ICLR.cc/2025/Conference,

work page 2025
[8]

Nihar B Shah, Melisa Bok, Xukun Liu, and Andrew McCallum

Accessed: 2025-05-07. Nihar B Shah, Melisa Bok, Xukun Liu, and Andrew McCallum. Identity theft in ai conference peer review.Communications of the ACM, 68(12):32–34,

work page 2025
[9]

SIGMOD 2025 call for research papers.https://2025

SIGMOD 2025 Program Committee. SIGMOD 2025 call for research papers.https://2025. sigmod.org/calls_papers_sigmod_research.shtml,

work page 2025
[10]

Tight lower bounds for differentially private selection

Thomas Steinke and Jonathan Ullman. Tight lower bounds for differentially private selection. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, pages 552–

work page 2017
[11]

Tight Lower Bounds for Differentially Private Selection

doi: 10.1109/FOCS.2017.57.https://arxiv.org/abs/ 1704.03024. The British Academy. Partial randomisation trial extended after diversity of ap- plicants and award holders increases.https://www.thebritishacademy.ac.uk/news/ partial-randomisation-trial-extended-small-research-grants/, April

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1109/focs.2017.57.https://arxiv.org/abs/ 2017
[12]

Giulia Trentacosti

Accessed: 2025-05-07. Giulia Trentacosti. Using lotteries to allocate research funding: Perspec- tives from switzerland. Open Science Blog, University of Groningen Li- brary, September

work page 2025
[13]

Interview with Marco Bieri, Swiss National Science Foundation

URLhttps://www.rug.nl/library/open-access/blog/ using-lotteries-to-allocate-research-funding-perspectives-from-switzerland? lang=en. Interview with Marco Bieri, Swiss National Science Foundation. UK Metascience Unit. A year in metascience (2025). Technical report, Department for Sci- ence, Innovation and Technology, June 30

work page 2025
[14]

Published 30 June

URLhttps://www.gov.uk/government/ publications/a-year-in-metascience-2025. Published 30 June

work page 2025
[15]

USENIX security 2025 conference format.https: //github.com/USENIX-Security-2025/conference-format, April

USENIX Security 2025 Program Committee. USENIX security 2025 conference format.https: //github.com/USENIX-Security-2025/conference-format, April

work page 2025
[16]

Projection onto the capped simplex

Accessed: 2025-05-07. Weiran Wang and Canyi Lu. Projection onto the capped simplex.arXiv preprint arXiv:1503.01002,

work page internal anchor Pith review Pith/arXiv arXiv 2025
[17]

Conditioning onG −j, there is a deterministic thresholdθ j such thatjis selected iffv j +G j ≥θ j, namely thek-th largest among{v ℓ+Gℓ :ℓ̸=j}

Combining, the column sum equals P i |∂jpi|= 2∂ jpj. Conditioning onG −j, there is a deterministic thresholdθ j such thatjis selected iffv j +G j ≥θ j, namely thek-th largest among{v ℓ+Gℓ :ℓ̸=j}. Hencep j(v) =E G−j[1−FG(θj−vj)]. Differentiating and usingf G(x)≤1/(eτ),∂ jpj =E G−j[fG(θj −v j)]≤1/(eτ).Thus every column ofJhasℓ 1 norm at most 2/(eτ), so∥p(u)...

work page 2025
[18]

Since log P i∈Rt eui/τ ≥M t/τand H(p t)≤log|R t|,M t −E[u it |R t]≤τlog|R t|

The conditional one-step regretM t −E[u it |R t], whereM t = max i∈Rt ui, is bounded via the variational form of log-sum-exp:E[u it |R t] = τlog P i∈Rt eui/τ −τ H(p t) wherep t is the softmax onR t. Since log P i∈Rt eui/τ ≥M t/τand H(p t)≤log|R t|,M t −E[u it |R t]≤τlog|R t|. Summing overtand bounding log(n−t+ 1)≤logn yields R(D;u)≤kτlogn. A.7 Individual ...

work page 2023
[19]

For completeness, Algorithm 2 outlines an exact, non-iterative implementation with runtimeO(n 2), adapted from Wang and Lu (2015)

and water-filling methods (Bairaktari et al., 2023). For completeness, Algorithm 2 outlines an exact, non-iterative implementation with runtimeO(n 2), adapted from Wang and Lu (2015). The algorithm first finds the 2nbreakpoints at whichS(b) changes (where a candidate’s probability hits the floor of 0 or the ceiling of 1), then locates the interval of brea...

work page 2023
[20]

,0| {z } h+1 , δ,

1 1 (0, . . . ,0| {z } h+1 , δ, . . . , δ| {z } h ) flip one 0 toδ Remark D.2.For each of the above utilities,d u =D u. In general,D u is a worst-case global upper bound on the Lipschitz quotient over all row pairs, whereasd u is a guarantee on the maximum achievable quotient for any distance budget up to 1/du. It always holds thatd u ≤D u, with equality ...

work page 2019
[21]

It effectively requires the funder to relax their ex post validity constraint

This resolution suggests that a funder could preserve exact smoothness by padding intervals to be wide enough relative to 1/w(equivalently, setting a stricter standard for asserting dominance). It effectively requires the funder to relax their ex post validity constraint. Resolution 2: Projection onto valid marginals.When the core-width condition fails, a...

work page 2025

[1] [1]

Calibrating noise to sensitivity in private data analysis

Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. InTheory of Cryptography, 3rd Theory of Cryptography Conference, TCC 2006, volume 3876 ofLecture Notes in Computer Science, pages 265–284. Springer,

work page 2006

[2] [2]

doi: 10.1007/11681878

work page doi:10.1007/11681878

[3] [3]

Ferric C Fang and Arturo Casadevall

Accessed: 2025-05-07. Ferric C Fang and Arturo Casadevall. Research funding: The case for a modified lottery,

work page 2025

[4] [4]

Ref. no. 2025–01621. Alexander Goldberg, Giulia Fanti, and Nihar B Shah. A principled approach to randomized se- lection under uncertainty: Applications to peer review and grant funding. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems,

work page 2025

[5] [5]

Rankmax: An adaptive projection alternative to the softmax function

Weiwei Kong, Walid Krichene, Nicolas Mayoraz, Steffen Rendle, and Li Zhang. Rankmax: An adaptive projection alternative to the softmax function. InAdvances in Neural Information Processing Systems 33 (NeurIPS 2020),

work page 2020

[6] [6]

NeurIPS 2024 Conference Submissions.https://openreview.net/submissions? venue=NeurIPS.cc/2024/Conference,

19 OpenReview. NeurIPS 2024 Conference Submissions.https://openreview.net/submissions? venue=NeurIPS.cc/2024/Conference,

work page 2024

[7] [7]

ICLR 2025 Conference Submissions.https://openreview.net/submissions? venue=ICLR.cc/2025/Conference,

OpenReview. ICLR 2025 Conference Submissions.https://openreview.net/submissions? venue=ICLR.cc/2025/Conference,

work page 2025

[8] [8]

Nihar B Shah, Melisa Bok, Xukun Liu, and Andrew McCallum

Accessed: 2025-05-07. Nihar B Shah, Melisa Bok, Xukun Liu, and Andrew McCallum. Identity theft in ai conference peer review.Communications of the ACM, 68(12):32–34,

work page 2025

[9] [9]

SIGMOD 2025 call for research papers.https://2025

SIGMOD 2025 Program Committee. SIGMOD 2025 call for research papers.https://2025. sigmod.org/calls_papers_sigmod_research.shtml,

work page 2025

[10] [10]

Tight lower bounds for differentially private selection

Thomas Steinke and Jonathan Ullman. Tight lower bounds for differentially private selection. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, pages 552–

work page 2017

[11] [11]

Tight Lower Bounds for Differentially Private Selection

doi: 10.1109/FOCS.2017.57.https://arxiv.org/abs/ 1704.03024. The British Academy. Partial randomisation trial extended after diversity of ap- plicants and award holders increases.https://www.thebritishacademy.ac.uk/news/ partial-randomisation-trial-extended-small-research-grants/, April

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1109/focs.2017.57.https://arxiv.org/abs/ 2017

[12] [12]

Giulia Trentacosti

Accessed: 2025-05-07. Giulia Trentacosti. Using lotteries to allocate research funding: Perspec- tives from switzerland. Open Science Blog, University of Groningen Li- brary, September

work page 2025

[13] [13]

Interview with Marco Bieri, Swiss National Science Foundation

URLhttps://www.rug.nl/library/open-access/blog/ using-lotteries-to-allocate-research-funding-perspectives-from-switzerland? lang=en. Interview with Marco Bieri, Swiss National Science Foundation. UK Metascience Unit. A year in metascience (2025). Technical report, Department for Sci- ence, Innovation and Technology, June 30

work page 2025

[14] [14]

Published 30 June

URLhttps://www.gov.uk/government/ publications/a-year-in-metascience-2025. Published 30 June

work page 2025

[15] [15]

USENIX security 2025 conference format.https: //github.com/USENIX-Security-2025/conference-format, April

USENIX Security 2025 Program Committee. USENIX security 2025 conference format.https: //github.com/USENIX-Security-2025/conference-format, April

work page 2025

[16] [16]

Projection onto the capped simplex

Accessed: 2025-05-07. Weiran Wang and Canyi Lu. Projection onto the capped simplex.arXiv preprint arXiv:1503.01002,

work page internal anchor Pith review Pith/arXiv arXiv 2025

[17] [17]

Conditioning onG −j, there is a deterministic thresholdθ j such thatjis selected iffv j +G j ≥θ j, namely thek-th largest among{v ℓ+Gℓ :ℓ̸=j}

Combining, the column sum equals P i |∂jpi|= 2∂ jpj. Conditioning onG −j, there is a deterministic thresholdθ j such thatjis selected iffv j +G j ≥θ j, namely thek-th largest among{v ℓ+Gℓ :ℓ̸=j}. Hencep j(v) =E G−j[1−FG(θj−vj)]. Differentiating and usingf G(x)≤1/(eτ),∂ jpj =E G−j[fG(θj −v j)]≤1/(eτ).Thus every column ofJhasℓ 1 norm at most 2/(eτ), so∥p(u)...

work page 2025

[18] [18]

Since log P i∈Rt eui/τ ≥M t/τand H(p t)≤log|R t|,M t −E[u it |R t]≤τlog|R t|

The conditional one-step regretM t −E[u it |R t], whereM t = max i∈Rt ui, is bounded via the variational form of log-sum-exp:E[u it |R t] = τlog P i∈Rt eui/τ −τ H(p t) wherep t is the softmax onR t. Since log P i∈Rt eui/τ ≥M t/τand H(p t)≤log|R t|,M t −E[u it |R t]≤τlog|R t|. Summing overtand bounding log(n−t+ 1)≤logn yields R(D;u)≤kτlogn. A.7 Individual ...

work page 2023

[19] [19]

For completeness, Algorithm 2 outlines an exact, non-iterative implementation with runtimeO(n 2), adapted from Wang and Lu (2015)

and water-filling methods (Bairaktari et al., 2023). For completeness, Algorithm 2 outlines an exact, non-iterative implementation with runtimeO(n 2), adapted from Wang and Lu (2015). The algorithm first finds the 2nbreakpoints at whichS(b) changes (where a candidate’s probability hits the floor of 0 or the ceiling of 1), then locates the interval of brea...

work page 2023

[20] [20]

,0| {z } h+1 , δ,

1 1 (0, . . . ,0| {z } h+1 , δ, . . . , δ| {z } h ) flip one 0 toδ Remark D.2.For each of the above utilities,d u =D u. In general,D u is a worst-case global upper bound on the Lipschitz quotient over all row pairs, whereasd u is a guarantee on the maximum achievable quotient for any distance budget up to 1/du. It always holds thatd u ≤D u, with equality ...

work page 2019

[21] [21]

It effectively requires the funder to relax their ex post validity constraint

This resolution suggests that a funder could preserve exact smoothness by padding intervals to be wide enough relative to 1/w(equivalently, setting a stricter standard for asserting dominance). It effectively requires the funder to relax their ex post validity constraint. Resolution 2: Projection onto valid marginals.When the core-width condition fails, a...

work page 2025