Proportional Aggregation of Preferences for Sequential Decision Making

TL;DR

This work formulates fair, proportional sequential decision making under approval votes by introducing axioms inspired by multi-winner voting (PJR and EJR variants) and showing how three natural rules—Sequential Phragmén (online), Method of Equal Shares (MES, semi-online), and Proportional Approval Voting (PAV, offline)—satisfy different combinations of these guarantees. It proves that online Sequential Phragmén satisfies PJR but not Weak EJR, semi-online MES achieves Weak EJR but may fail PJR, and offline PAV attains all axioms (including EJR) with a polynomial-time local-search variant. The paper also establishes impossibility results, indicating that stronger proportionality guarantees cannot be achieved in online/semi-online settings, and supports these findings with extensive experiments on synthetic data, U.S. political datasets, and Moral Machine-derived preferences. The results demonstrate that proportional aggregation improves fairness (e.g., lower inequality in utilities and better bottom-quartile outcomes) relative to using a single global model or plurality-based methods, highlighting practical implications for long-horizon, multi-issue decision processes in governance, virtual democracy, and democratic AI systems.

Abstract

We study the problem of fair sequential decision making given voter preferences. In each round, a decision rule must choose a decision from a set of alternatives where each voter reports which of these alternatives they approve. Instead of going with the most popular choice in each round, we aim for proportional representation across rounds, using axioms inspired by the multi-winner voting literature. The axioms require that every group of $α\%$ of the voters that agrees in every round (i.e., approves a common alternative), must approve at least $α\%$ of the decisions. A stronger version of the axioms requires that every group of $α\%$ of the voters that agrees in a $β$ fraction of rounds must approve $β\cdotα\%$ of the decisions. We show that three attractive voting rules satisfy axioms of this style. One of them (Sequential Phragmén) makes its decisions online, and the other two satisfy strengthened versions of the axioms but make decisions semi-online (Method of Equal Shares) or fully offline (Proportional Approval Voting). We present empirical results for these rules based on synthetic data and U.S. political elections. We also run experiments using the moral machine dataset about ethical dilemmas: We train preference models on user responses from different countries and let the models cast votes. We find that aggregating these votes using our rules leads to a more equal utility distribution across demographics than making decisions using a single global preference model.

Figures (12)

• Figure 1: Implications between axioms. Perpetual priceability and (perpetual) lower quota for closed groups are discussed in \ref{['app:moreaxioms']}.
• Figure 2: Example illustrating our axioms.
• Figure 3: Example illustrating how MES fails PJR (and JR) without empty approval sets. The cell "all disjoint" indicates that each of the last 30 voters approves a unique alternative not approved by anyone else in the last 3 rounds. On this example, MES selects in the first 7 rounds and then terminates prematurely. Most completion methods select in the remaining rounds (including Phragmén completion, utilitarian completion, and Add1).
• Figure 4: Performance of the different rules for $n = 20$ voters, $T = 50$ rounds, $20$ alternatives in each round, with $f = 1.5$. The length of the bar represents the median across all the trials while the error bars represent the $25$th and $75$th percentile with the numeric text after the bar showing the mean value. The blue pictures in the right-most column illustrates the underlying distribution of voter locations.
• Figure 5: Performance of the different rules for $n = 50$ voters, $T = 50$ rounds, $40$ alternatives in each round, with $f = 1.5$. The length of the bar represents the median across all the trials while the error bars represent the $25$th and $75$th percentile with the numeric text after the bar showing the mean value. The blue pictures in the right-most column illustrates the underlying distribution of voter locations.

Theorems & Definitions (57)

• Definition 3.1: Weak PJR
• Definition 3.2: PJR
• Definition 3.3: Weak EJR
• Definition 3.4: EJR
• Definition 3.5: Weak JR
• Definition 3.6: JR
• Definition 3.7: Local-Search PAV
• Proposition 3.8: aziz2018complexity
• proof
• Theorem 4.1