Smooth Partial Lotteries for Stable Randomized Selection

Competitive selection processes, from scientific funding to admissions and hiring, use evaluations to score candidates, and then choose a subset 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 propose using a Linear Lottery, a simple mechanism in which selection probabilities scale linearly with aggregated review scores between an upper threshold, above which we always accept and a lower threshold, below which we always reject. We prove that the 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 Linear Lottery achieves a better smoothness--regret tradeoff than alternatives. Experiments on real peer review data from ICLR 2025, NeurIPS 2024, and the Swiss NSF confirm our theoretical analysis is tight and demonstrate that existing lottery designs are highly unstable in practice even under changes to a single review score.