Research map / Learning theory and algorithms / Submodular and game algorithms
Proportional Aggregation of Preferences for Sequential Decision Making
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Proposes proportionality axioms (PJR, EJR and weak versions) for sequential multi-round decisions and shows Sequential Phragmén, Method of Equal Shares and PAV satisfy different subsets, with synthetic, U.S. election and Moral Machine experiments.
Problem
Fair sequential decision making from repeated approval votes, where a fixed group of voters should influence a proportional share of decisions across rounds rather than a simple majority always winning.
Approach
Adapts multi-winner proportionality axioms (JR, PJR, EJR and weak variants) to sequences of decisions where groups that agree in k rounds deserve utility proportional to group size and agreement fraction; shows which online, semi-online and offline rules satisfy them, including a polynomial-time local-search PAV.
Key results
- Sequential Phragmén satisfies PJR online; MES satisfies Weak EJR semi-online; PAV and local-search PAV satisfy EJR offline.
- On Moral Machine, Approval Voting and the Combined Model agree on nearly identical metrics, while the five proportional rules have much smaller Gini coefficients and higher 25th-percentile satisfaction.
- On U.S. political data, PAV gives a more equal utility distribution than Approval Voting with almost negligible loss in average utility.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Approval Voting instead of proportional voting rules | Synthetic clustered-voter data · Synthetic data, all distributions and parameter settings | average utility | Helped |
| Use Round Robin instead of the other rules | Synthetic clustered-voter data · Synthetic data, all distributions and parameter settings | average utility | Hurt |
| Use PAV instead of other proportional rules | Synthetic clustered-voter data · Synthetic data, most distributions and parameter settings | average utility | Helped |
| Use Approval Voting instead of proportional rules | Synthetic clustered-voter data · Restricted and Many Groups distributions | 25th-percentile utility | Hurt |
| Use PAV instead of other proportional rules | Synthetic clustered-voter data · Synthetic data, generally | 25th-percentile utility | Helped |
| Use Perpetual Quota or Perpetual Consensus instead of PAV | Synthetic clustered-voter data · Balanced and Nearby distribution | 25th-percentile utility | Helped |
| Use Approval Voting instead of other rules | Synthetic clustered-voter data · Across distributions | Gini coefficient | Hurt |
| Use Perpetual Quota or Perpetual Consensus instead of other proportional rules | Synthetic clustered-voter data · Balanced and Nearby distribution | Gini coefficient | Helped |
| Use proportional rules instead of Approval Voting | Political election data · Shasta County and Colorado elections | Gini coefficient | Helped |
| Use proportional rules instead of Approval Voting | Political election data · Shasta County and Colorado elections | 25th-percentile utility | Helped |
| Use PAV instead of Approval Voting | Political election data · Shasta County and Colorado elections | average utility | No clear effect |
| Aggregate country preference models with proportional voting rules instead of using the Combined Model | Country-level Moral Machine preference models · Moral Machine, high-disagreement alternatives | Gini coefficient | Helped |
Surprising findings
- Method of Equal Shares fails PJR and EJR despite satisfying Weak EJR, unlike its typical multi-winner performance.
- No semi-online rule can satisfy EJR, and no semi-online rule can satisfy epsilon-Stronger PJR whenever such a decision sequence exists.
Limitations
- No known online rule satisfies Weak EJR, which would resolve an open problem in multi-winner voting.
- Moral Machine experiment is a small proof-of-concept and not suitable for automating moral reasoning.
- Political dataset reinterprets multi-candidate votes as approval sets, so the fit to the model is imperfect.
Experimental setup
Datasets: Moral Machine dataset U.S. county cast vote records (2020, 2022) synthetic 2D voter-alternative data
Benchmarks: synthetic 2D distributions U.S. political elections (Shasta County CA, 14 Colorado counties) Moral Machine
Metrics: average utility 25th percentile utility Gini coefficient
Baselines: Approval Voting Round Robin Perpetual Quota Perpetual Consensus Combined Model
Builds on: Perpetual voting Sequential Phragmén Method of Equal Shares Proportional Approval Voting
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