# The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions

arXiv: https://arxiv.org/abs/2605.12717 · HTML: https://mlsanity.com/paper/2605.12717/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 50 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 28 of 28)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Interpretable Preferences via Multi-Objective Reward Modeling and Mixture-of-Experts](https://mlsanity.com/paper/2406.12845/) (2024)
- [Arithmetic Control of LLMs for Diverse User Preferences: Directional Preference Alignment with Multi-Objective Rewards](https://mlsanity.com/paper/2402.18571/) (2024)
- [Distributional Preference Learning: Understanding and Accounting for Hidden Context in RLHF](https://mlsanity.com/paper/2312.08358/) (2023)
- [Social Choice Should Guide AI Alignment in Dealing with Diverse Human Feedback](https://mlsanity.com/paper/2404.10271/) (2024)
- [A Voting-Based System for Ethical Decision Making](https://mlsanity.com/paper/1709.06692/) (2017)
- [Axioms for AI Alignment from Human Feedback](https://mlsanity.com/paper/2405.14758/) (2024)
- [Adapting a Kidney Exchange Algorithm to Align with Human Values](https://mlsanity.com/paper/2005.09755/) (2020)
- [Moral Machine or Tyranny of the Majority?](https://mlsanity.com/paper/2305.17319/) (2023)
- [Direct Alignment with Heterogeneous Preferences](https://mlsanity.com/paper/2502.16320/) (2025)
- [Proportional Aggregation of Preferences for Sequential Decision Making](https://mlsanity.com/paper/2306.14858/) (2023)
- [Jackpot! Alignment as a Maximal Lottery](https://mlsanity.com/paper/2501.19266/) (2025)
- [A Computational Model of Commonsense Moral Decision Making](https://mlsanity.com/paper/1801.04346/) (2018)
- [Truthful Aggregation of Budget Proposals](https://mlsanity.com/paper/1905.00457/) (2019)
- [Clone-Robust AI Alignment](https://mlsanity.com/paper/2501.09254/) (2025)
- [Strengthening Proportionality in Temporal Voting](https://mlsanity.com/paper/2505.22513/) (2025)
- [On the Pros and Cons of Active Learning for Moral Preference Elicitation](https://mlsanity.com/paper/2407.18889/) (2024)
- [On The Stability of Moral Preferences: A Problem with Computational Elicitation Methods](https://mlsanity.com/paper/2408.02862/) (2024)
- [Settling the Score: Portioning with Cardinal Preferences](https://mlsanity.com/paper/2307.15586/) (2023)
- [Moral Change or Noise? On Problems of Aligning AI With Temporally Unstable Human Feedback](https://mlsanity.com/paper/2511.10032/) (2025)
- [The Squared Kemeny Rule for Averaging Rankings](https://mlsanity.com/paper/2404.08474/) (2024)
- [Truthful Aggregation of Budget Proposals with Proportionality Guarantees](https://mlsanity.com/paper/2203.09971/) (2022)
- [Project-Fair and Truthful Mechanisms for Budget Aggregation](https://mlsanity.com/paper/2309.02613/) (2023)
- [Optimal bounds for dissatisfaction in perpetual voting](https://mlsanity.com/paper/2501.01969/) (2024)
- [Truthful Budget Aggregation: Beyond Moving-Phantom Mechanisms](https://mlsanity.com/paper/2405.20303/) (2024)
- [Proportional Representation in Rank Aggregation](https://mlsanity.com/paper/2508.16177/) (2025)

## Built on by (most cited first, 1 of 1)

- [Internal Pluralism and the Limits of Pairwise Comparisons](https://mlsanity.com/paper/2607.02672/) (2026)

## Abstract (arXiv)

AI alignment and participatory design motivate a new democratic design problem: how to collectively choose a decision rule to use repeatedly. We study this problem for linear ranking rules, which repeatedly rank items xj within batches X=(x1,dots,xm)∈(ℝd)m, where each item's ranking is dictated by its score langle θ^*,xjrangle according to a fixed scoring vector θ^*. Given voters' preferred scoring vectors θ(1),dots,θ(n) and their population fractions α(1),dots,α(n), we ask how to choose a collective vector θ^* satisfying individual proportionality (IP): every voter type i should agree with the resulting rankings to an α(i)-proportional degree, either on average over time (long-run IP) or even within each batch (per-batch IP). The default rule, the arithmetic mean of the θ(i), has been shown to be severely majoritarian; more generally, it is not clear that any fixed linear rule can balance many voters' disparate opinions. Our main result is that, surprisingly, there is a simple rule that does satisfy long-run IP: the angular mean, the spherical analog of the arithmetic mean. We then show that exact per-batch IP is impossible for fixed linear rules, but that the gap between per-batch and long-run IP shrinks quickly with batch size. Experiments on three real-world preference datasets show that all rules perform similarly when voters' preferences are homogeneous, while the angular mean substantially improves proportionality in high-disagreement regimes.