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Proportional Representation in Practice: Quantifying Proportionality in Ordinal Elections
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Quantifies proportionality of ordinal multiwinner voting rules by relaxing size thresholds of PSC, LS, EJR+ and priceability into alpha-parameterized measures, and evaluates them on 1070 real Scottish STV elections.
Problem
Proportionality axioms such as PSC rarely constrain outcomes in real-world ordinal elections because large enough solid coalitions almost never occur, so it is unclear how to assess proportionality of voting rules in practice.
Approach
Parameterize proportionality axioms with a multiplicative factor alpha on the group-size threshold and define each committee's proportionality as the smallest alpha at which it satisfies the axiom, yielding instance-specific quantitative measures (alpha-PSC, alpha-LS, alpha-EJR+, alpha-priceability) that can be computed or optimized.
Key results
- Under PSC, 27.5% of the 1070 Scottish elections have every size-k committee satisfying PSC, so PSC places no restriction there.
- SNTV achieves the optimal PSC value in 901 of 1070 instances, more than S-STV (856) and EAR (677).
- seq-RCV violates 1-PSC in only 9 instances, yet performs poorly on alpha-values, with local stability alpha-values up to 1.6 on completed ballots.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SNTV rather than the other evaluated rules | Scottish local elections · Scottish local elections | PSC optimal instances: 901 vs. 856 (Scottish STV), 819 (Meek STV), 677 (EAR), and 552 (Sequential RCV) | Helped |
| Use SNTV rather than the other evaluated rules | Scottish local elections · Scottish local elections | LS distance from optimal: 0.13 vs. 0.23 (Scottish STV), 0.30 (Meek STV), 0.40 (EAR), and 0.60 (Sequential RCV) | Helped |
| Use Scottish STV or Meek STV rather than EAR | Scottish local elections · Scottish local elections | PSC optimal instances: 856 (Scottish STV), 819 (Meek STV), 677 (EAR) | Helped |
| Use Sequential RCV rather than Scottish STV | Scottish local elections · Scottish local elections | PSC value: seq-RCV fails 1-PSC in 9 instances | Hurt |
| Complete truncated ballots rather than use the original truncated ballots | Scottish local election data · Scottish local elections | elections where every committee satisfies PSC: 27.5% in the truncated case; 21.8% in the completed case | No clear effect |
| Complete truncated ballots rather than use the original truncated ballots | Sequential RCV · Scottish local elections | PSC violations: 9 instances; 55 elections | Hurt |
Surprising findings
- SNTV, a rule without proportionality guarantees, performs as well as or better than STV and EAR on the alpha-PSC and alpha-LS measures.
- EAR, which satisfies the strongest axioms, does slightly worse than other proportional rules on these measures.
Limitations
- Axiom-based measures do not identify a unique outcome in any election studied
- Synthetic completion of truncated ballots is an approximation of true complete preferences
- Computing the minimal alpha for an instance is NP-hard
Experimental setup
Datasets: McCune and Graham-Squire (2024) Scottish local council election ballots (1070 multiwinner elections)
Benchmarks: Scottish local council elections 2007-2022 (1070 instances)
Metrics: alpha-PSC value alpha-LS value alpha-EJR+ value alpha-priceability value number of axiom-satisfying committees distance from optimal alpha
Baselines: Meek-STV Expanding Approvals Rule SNTV Sequential Ranked-Choice Voting
Builds on: PSC (Dummett 1984) local stability (Aziz et al. 2017) EJR+ (Brill and Peters 2023) priceability (Peters and Skowron 2020)
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