📄 social science

Multi-Member-District System with Single Non-Transferable Voting

By applying the $\alpha$ -divergence decomposition method to Japanese local elections, this study demonstrates that the Multi-Member District system with Single Non-Transferable Voting (MMD-SNTV) achieves proportionality comparable to or exceeding list proportional representation by offsetting wasted votes through intra-party unequal representation, thereby functioning as a quasi-proportional system driven by candidate rationality rather than mechanical vote transfers.

Original authors: Junichiro Wada, Yuta Kamahara, Yukihiro Nakajima

Published 2026-10-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Junichiro Wada, Yuta Kamahara, Yukihiro Nakajima

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the machinery of democracy, the way votes are counted and turned into seats is the engine that determines who speaks for the people. For decades, political scientists have debated which engine runs smoothest. On one side sits the system used in many places, where voters pick a single candidate in a small district, and the person with the most votes wins everything. On the other side is a more complex system used in places like Ireland and Malta, where voters rank candidates in order of preference, and votes that would otherwise be wasted are passed along to other candidates until every seat is filled. This ranking system is designed to ensure that the final group of representatives mirrors the diversity of the voters' choices as closely as possible. But there is a third, often overlooked engine: a system where voters pick just one candidate in a larger district that elects multiple representatives, but they cannot transfer their vote if their first choice fails. This method, known as single non-transferable voting, has been used in Japan for local elections and was once the standard for the national parliament. It sits in a strange middle ground, seemingly simple yet capable of producing surprisingly fair results, leaving researchers to wonder if it is a flawed relic or a hidden gem of proportionality.

A team of researchers from three Japanese universities set out to settle this question by looking at the actual numbers behind the system. They focused on twenty major cities in Japan that still use this single-vote method for their local assemblies, covering a population of roughly 27.7 million people. To see how well this system works, they compared it against two other standards: the ranked-choice system used in Ireland and Malta, and the party-list systems used in Japan's own national elections. The goal was not just to count how many votes were "wasted"—meaning votes that did not help elect anyone—but to understand the hidden mechanics that make the final result look the way it does. They wanted to know if the system's simplicity came at the cost of fairness, or if something else was happening behind the scenes to keep the results balanced.

The researchers discovered a counterintuitive truth about how this system functions. When they looked at the raw numbers, the single-vote system in Japanese cities actually produced more wasted votes within individual districts than the national party-list systems, yet it still achieved a more proportional outcome overall. In the ranked-choice system used in Ireland, voters can list their preferences, and votes are transferred from losing candidates to winners. However, this very feature leads to a chaotic spread of initial votes that often ends up far from the final seat distribution. In fact, the study found that wasted votes in Ireland under this ranked system are substantially higher than those in Japanese cities using the single-vote method. In contrast, the Japanese system, where voters cast a single, non-transferable ballot, forces political parties to be extremely careful and strategic. Because they cannot rely on votes being passed along, parties must calculate exactly how many candidates to run in each district to avoid splitting their own support. This rational behavior acts as a natural brake, preventing the system from becoming too messy.

The study revealed that the apparent fairness of the single-vote system comes from a specific balancing act. While individual districts might see some inefficiency, where a party wins a seat with slightly more votes than necessary or loses one with slightly too few, these local errors cancel each other out across the entire country. When a party performs slightly better than expected in one city, it often performs slightly worse in another. This cross-district balancing creates a national result that is remarkably close to the ideal of proportional representation, where the share of seats matches the share of votes. The researchers found that when they stripped away the noise caused by independent candidates running alone, the system performed even better, often outperforming the party-list systems used in Japan's national parliament.

This finding challenges the long-held belief that complex systems with vote transfers are the only way to achieve true proportionality. The paper suggests that the single non-transferable voting system achieves what the authors call "superproportionality," a state where the results are even more aligned with the popular vote than the standard benchmarks used by political scientists, though they note this is a point one might argue rather than a settled fact. The key is not in the mechanical transfer of votes from one candidate to another, but in the strategic behavior of the politicians themselves. By limiting the number of candidates to match the number of available seats, parties inadvertently create a system where the collective outcome is highly efficient. The study concludes that while the system does not capture the nuanced preferences of voters through transfers, it achieves a high degree of effectiveness through the calculated decisions of the candidates, making it a viable, if unusual, form of proportional representation.

Technical Summary: Multi-Member-District System with Single Non-Transferable Voting

Problem Statement
The Multi-Member-District system with Single Non-Transferable Voting (MMD-SNTV) occupies an ambiguous position in electoral system classifications. While often categorized alongside plurality systems (such as SMD-FPTP or Block Voting) due to its single-vote constraint, it possesses unique characteristics that allow smaller parties and independents to win seats, leading some scholars to classify it as "semi-proportional." Despite its historical use in Japan, Korea, and Taiwan, and its continued application in Japan's Upper House and local assemblies, MMD-SNTV has received less scholarly attention than Single Transferable Voting (STV) or List Proportional Representation (PR). A central paradox exists: while SNTV is prone to generating "wasted votes" due to the lack of vote transfers and potential miscalculations in candidate nomination, it is historically praised for achieving high proportionality. This paper seeks to resolve this ambiguity by rigorously evaluating the proportionality of MMD-SNTV, specifically comparing it against List PR and STV, and identifying the mechanisms that drive its electoral outcomes.

Methodology
The authors apply the $\alpha$ -divergence decomposition method developed by Wada and Kamahara (2024) to evaluate electoral proportionality. Unlike traditional indices (e.g., Gallagher or Loosemore-Hanby) based on $L_p$ -norms, $\alpha$ -divergence allows for the explicit selection of value premises (political philosophies) via the parameter $\alpha$ .

Parameters: The study utilizes $\alpha=2$ to evaluate proportionality under the Sainte-Laguë (Webster) criterion and $\alpha=10$ (approximating $\alpha \to \infty$ ) for the D'Hondt (Jefferson) criterion.
Decomposition: The core analytical innovation is the decomposition of total disproportionality ( $D_\alpha$ $D_{α}$ ) into three distinct components:
1. Malapportionment: Disproportionality arising from district magnitude variations.
2. Wasted Votes: Disproportionality within individual districts (surplus votes for winners and shortfall votes for losers).
3. Intra-Party Unequal Representation: Disproportionality arising from a party's varying seat-to-vote ratios across different districts.
Data Scope: The analysis focuses on local-assembly elections in 20 ordinance-designated cities in Japan (employing MMD-SNTV). These are compared against:
- List PR tiers of Japan's Lower House (closed list) and Upper House (open list).
- National elections in Ireland and Malta (employing MMD-STV).
Treatment of Independents: The methodology explicitly decomposes disproportionality between "single-candidate parties" (independents) and "multi-candidate parties" to isolate the specific contribution of independents to the system's overall proportionality.

Key Results

Superproportionality of Multi-Candidate Parties: When analyzing only multi-candidate parties, MMD-SNTV demonstrates proportionality comparable to, or exceeding, List PR. Under the Sainte-Laguë criterion ( $\alpha=2$ ), SNTV is more proportional than List PR. Under the D'Hondt criterion ( $\alpha=10$ ), it is comparable to List PR. This suggests SNTV achieves "superproportionality" (Cox 1994) relative to the D'Hondt standard when independent candidates are excluded from the comparison.
The Role of Intra-Party Unequal Representation: The paper identifies a counter-intuitive mechanism where "wasted votes" (inequality within districts) are largely offset by "intra-party unequal representation" (inequality across districts). As the number of electoral districts increases, the variance in a party's win-loss patterns across districts increases. This cross-district smoothing effect reduces the overall degree of disproportionality, effectively canceling out the inefficiencies of wasted votes within specific districts.
Comparison with STV: Wasted votes in STV systems (Ireland, Malta) are substantially higher than in Japanese SNTV systems. This is attributed to the STV mechanism, which encourages a larger number of candidates to contest elections. The dispersed distribution of first-preference votes before transfers creates a significant divergence between initial preferences and final seat allocation. In contrast, SNTV's lack of transfers forces parties to limit the number of candidates rationally, reducing the initial dispersion of votes.
Impact of Independents: The primary source of disproportionality in SNTV systems is the disparity within single-candidate parties (independents). When this component is isolated, the remaining system (multi-candidate parties) performs exceptionally well in terms of proportionality.

Key Contributions

Reclassification of MMD-SNTV: The paper provides empirical evidence to provisionally position MMD-SNTV alongside PR, or at least as semi-PR. It demonstrates that the system's proportionality is not merely a result of mechanical seat allocation but is achieved through the rational behavior of candidates and parties (limiting candidate numbers) rather than the mechanical transfer of votes, suggesting the system could arguably be considered a "quasi-PR" system.
Decomposition of Disproportionality: By applying the $\alpha$ -divergence decomposition, the authors clarify that the "paradoxical advantage" of SNTV over List PR stems from the offsetting relationship between wasted votes and intra-party unequal representation. This offers a new theoretical lens for understanding how electoral systems can achieve high proportionality without vote transfers.
Evaluation of "Vote Effectiveness": The study challenges the notion that vote transfers (STV) are the sole mechanism for vote effectiveness. It argues that SNTV achieves vote effectiveness through strategic coordination (the $(M+1)$ rule), where parties limit candidates to avoid vote splitting, thereby transforming votes into seats efficiently without mechanical transfers.

Significance and Claims
The authors claim that MMD-SNTV could arguably be considered a viable "quasi-PR" electoral system. The significance of the findings lies in the identification of a trade-off: the high proportionality of the multi-candidate party sector in SNTV comes at the cost of allowing independent candidates (single-candidate parties), which introduces disproportionality. The paper suggests that this disproportionality within the independent sector may be a necessary "cost" to avoid the "dark side" of List PR—specifically, the strengthening of party control and the emergence of "cartel parties" that detach from constituency preferences.

The paper concludes that while SNTV does not capture nuanced voter preferences through transfers like STV, it achieves a high degree of proportionality and vote effectiveness through the rational, strategic behavior of political actors. Consequently, the system warrants reconsideration as a reform option, particularly in contexts where maintaining independent candidacies and preventing cartelization are prioritized over the mechanical precision of vote transfers.

Technical Summary: Multi-Member-District System with Single Non-Transferable Voting

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