Will it be a lot of hard work? Absolutely.
Will it be worth it? Absolutely.

Personally? I'm so disabled I can't leave my house or think very hard for more than a few minutes at a time. So I'll be doing my part by helping establish a road map for you and your referendum. Comment in here if you're interested in taking up the call, and when you need help, I'll help you figure out what you need to do to make it happen.

LFG.

I'll be talking specifically about proportional approval methods here, but the problems exist with ranked methods too. But alternatives are easier to come by with approval methods, so there's less excuse for quota-removal methods with them.

Electing the most approved candidate, removing a quota of votes (e.g. Hare, Droop), and then electing the most approved candidate on the modified ballots (and so on) has intuitive appeal, but I think that's where the advantages end.

First of all the quota size is essentially arbitrary, particularly with cardinal or approval ballots where any number of candidates can be top-rated, and any number of candidates can reach a full quota of votes. This can be considerably more or less than the number of candidates to be elected.

Also adding voters that don't approve any of the candidates that have a chance of being elected can change the result, giving quite a bad failure of Independence of Irrelevant Ballots (IIB), which I'd call an IIB failure with "empty" ballots. Adding ballots that approve all of the candidates in contention and changing the result is a failure of IIB with "full" ballots, but this is harder for a method to pass and not as bad anyway. It is not that hard to pass with empty ballots, but quota-removal methods do fail. I'll give an exaggerated case of where quotas can go badly wrong:

3 voters: A1; A2; A3

1 voter: B1

1 voter: B2

1 voter: B3

6 voters: Assorted other candidates, none of which get enough votes to be elected

4 candidates are to be elected. There are two main parties, A and B, but the B voters have strategically split themselves into three groups. We'll use the Hare quota, but it doesn't really matter. This example could be made to work with any quota.

With 12 voters, a Hare quota is 3 votes. Let's say A1 is elected first. That uses up the entire A vote. All the other seats then go to B candidates, so a 3:1 ratio despite there being a 50:50 split between A and B voters. This example can be made as extreme as you like in terms of the A:B seat ratio. If the 6 "empty" ballots weren't present there would be a 50:50 A:B split.

If you have a fixed quota like this, the voters that get their candidates elected early can get a bad deal because they pay a whole quota, whereas later on, the might not be a candidate with a whole quota of votes and yet you have to elect one anyway, so the voters of this candidate get their candidate more "cheaply".

What you really want to do is look for a quota that distributes the cost more evenly, and that's essentially what Phragmén methods do. They distribute the load or cost across the voters as evenly as it can. So really quota-removal methods are just a crude approximation to Phragmén. Phragmén passes the empty ballot form of IIB and generally would give more reasonable results than quota-removal methods.

Also Thiele's Proportional Approval Voting (PAV) passes all forms of IIB, and has better monotonicity properties than Phragmén, but it is really only semi-proportional, as I discussed here, except where there are unlimited clones, or for party voting.

If you’re not familiar with it, https://resist.bot is amazing. Use it to contact your reps and all levels of local, state, and federal government. I emailed them to ask them to add city council level categories that they don’t currently maintain.

Also, they need help on GitHub to maintain their records in general. The info for my city’s Mayor is out of date and I don’t know how to update it. If you know how to use GitHub, they could use support to update records.

But I’ve used it already to email everyone it would allow me to about a number of issues.

I've read somewhere (I think it might be equal vote coalition) that Condorcet methods might not meet legal requirements on what a vote is.

side question: I've both heard that Condorcet methods are too complex (and won't work on current electoral systems) to be used in an election AND that they can be used through the use of pairwise matrices. Which is correct?

“This subreddit is for promoting activism and discussion related to ending the FPTP voting system internationally.”

That’s the whole purpose of this subreddit.

And yet….every single post on this subreddit is filled with debates over nano-nuances between various alternatives to FPTP instead of actually trying to implement any of them.

There is zero activism here. None.

Well, be the change you want to see in the world. I’ve begun attending virtual meetings for starvoting.org, fairvote, represent.us, equal vote coalition, and a few others. Money where my mouth is. Whoever is most active in my region is getting my effort. They’re all getting my attention. And literally money. I’m donating to them. $10 a month each. But still. It’s what I can afford to do with a new baby in the household.

Everything here is the discussion side of the subreddit and zero activism. I love me some discussion. But even the discussion is off-topic. We’re not even discussing ending FPTP. Instead, we are discussing which non-FPTP is scientifically better. There is no actual discussion about how to end FPTP. We should rename the subreddit because nobody is talking about actually ending FPTP. Nobody is talking about whether a national top-down approach or a bottom-up push to get local chapters of non-profits and their own companies to switch to any one of these acceptable alternatives and then moving to cities and states/provinces (since this isn’t a US-centric sub) and then national.

I have my preferences for which voting method is the right combination of easy to explain vs gets the Condorcet winner most frequently, but why let perfectly be the enemy of good? FPTP isn’t even good. The top 5 alternative proposals to FPTP are better than FPTP.

Instead of dedicating 100% of the subreddit time to discussion, can we shift to 50% maybe even 51% since that’s listed first in the subreddit description? Or maybe let’s start with 14.2% and implement something like “Activism Mondays”? Days where the only posts that are allowed are centered around actual actions related to ending FPTP?

And sorry, I don’t want to see the word Condorcet in a discussion anymore. Can we also implement Condorcet Saturdays? Where we leave the minutiae to a single day of the week? Let’s actually shift this subreddit to be about how to actually mobilize a Girl Scout troupe, a PTA board, your house party’s vote about pizza toppings, the company you work for, your local planning commission, city council, citywide elections, political party elections, county elections, state elections, and national elections away from FPTP toward ANY of the more effective alternatives.

Thanks for reading my rant.

There are various proportional methods that use approval voting and they can be turned into more general cardinal methods by allowing scores or stars instead of a simple yes/no. But as well as all the different approval methods, there are different ways to convert these methods into score voting methods, so you can end up with a proliferation of possible methods with these two essentially independent choices you have to make (which approval method, how to deal with scores).

First of all, I should say that I'm talking about methods that use the actual values of the scores, not where scores are used as a proxy for ranks.

For example, you have methods like Allocated Score, Sequential Monroe and Sequentially Spent Score. As far as I understand, if everyone voted approval-style (so only max or min scores), these methods would all be essentially the same. The highest scoring candidate is elected, and a quota of votes is removed, as so on.

All of these methods are actually quite messy, not to mention arbitrary, and you can end up with a lot of discontinuities and edge cases when you make small changes in the vote. Scores are an inconvenience in this sense (which is why all these similar but different methods were invented) and it would be much better if you could just make them behave more predictably and continuously from the start, so you can then just apply your favourite approval method knowing things will run smoothly.

And the way to do this? Well, as far as I'm concerned, it's the KP transformation. It turns the score ballots into approval ballots in a consistent manner, so you then only have to worry about what approval method you want to use. For e.g. scores out of 5, this essentially splits each ballot into 5 parts with their own approval threshold for each candidate. The "top" part will only approve those given 5, the next part will approve those given 4 and 5, and so on. The highest scoring candidate overall automatically becomes the most approved candidate, and so on. The total scores are proportional to the total approvals they've been converted to.

This makes methods far more continuous than the above ad hoc score conversions, so the weird discontinuities they cause will go away.

The KP transformation has nice properties. For example, for an approval method that passes Independence of Irrelevant Ballots, the KP transformed method will pass multiplicative and additive scale invariance. That means that if you multiply the scores on all ballots by a constant, or add a constant, or both, the result will still be the same. So you could multiply the scores by 7 and add 3. It would not affect the result.

Taking Thiele's Proportional Approval Voting as an example, Reweighted Range Voting and Single Distributed Vote are both conversions that cause a failure in one or both forms of scale invariance. However, Harmonic Voting, or it's sequential variant, which both use the KP transformation, pass.

Also, this means that electing two candidates that a voter has given a 2 and a 3 respectively is not the same as a single 5 (and 0 for any others). But I see this as a feature, not a bug. It means that someone's ballot will never be "used up" by candidates they don't give their full support to. With scores out of 5, electing candidates a voter gives 3 or less to means that 2/5 of their vote will be completely protected until a 4 or 5 is elected.

Pretty self-explanatory. Given a sufficient number of list seats, can fixed-seat MMP work well?

I see most answers on the question of open v. closed lists prefer the open list option because it reduces the power of party elites chosing the order of list. However, what if the closed list is combined with a primary-like system where party members/base vote to decide the order of members on the list before the election. Would this system be more preferable to open list system?

While Approval is not my first choice and I still generally prefer ordinal systems to cardinal, I have found myself advocating for approval ballots or straight up single winner approval voting in certain contexts.

I'd like to raise two points:

• Vote totals
• Electoral fraud

1. Vote totals

We are used to being given the results of an election, whether FPTP, list PR or even IRV/IRV by first preference vote totals per party. Polls measure partisan support nationally or regionally. People are used to seeing this in charts adding up to 100%.

Approval voting would change this. You cannot add up votes per party and then show from 100%, it's meaningless. If that was common practice, parties would run more candidates just so they can claim a larger share of total votes for added legitimacy in various scenarios (campaigns, or justifying disproportional representation).

You could add up the best performing candidates of each party per district and then show it as a % of all voters, but then it won't add up to 100%, so people might be confused. I guess you can still show bar sharts and that would kind of show what is needed. But you can no longer calculate in your head like, if X+Y parties worked together or voters were tactical they could go up to some % and beat some other party. It could also overestimate support for all parties. Many people could be dissuaded from approving more if it means actually endorsing candidates and not just extra lesser evil voting.

What do you think? Would such a change be a welcome one, since it abandons the over-emphasis on first preferences, or do you see more downsides than upsides?

2. Electoral fraud

Now I think in many cases this is the sort of thing people overestimate, that people are just not as rational about, such as with fear of planes and such. But, with advocacy, you simply cannot ignore peoples concerns. In fact, even the the electoral reform community, the precinct summability conversation is in some part about this, right?

People have reacted sceptically when I raised approval ballots as an option, saying that at least with FPTP you know a ballot is invalid if there are 2 marks, so if you see a suspicious amount, you would know more that there is fraud going on, compared to a ballot that stays valid, since any of that could be sincere preferences. I have to assume, it would indeed be harder to prove fraud statistically with approval.

Have you encountered such concerns and what is the general take on this?

Additionally, what would be the probability of this method electing a Condorcet winner? What about the VSE? (If the top two candidates are selected using a proportional approval voting method.)

For example, recording the ranking of all ballots (such as 100 ballots are A > B > C, 50 are C > B > A), and then comparing all candidates one by one—is this really more difficult to count than ranked-choice voting?

Hello. There are a few things I want to discuss about proportional approval/cardinal methods. First of all I want to discuss proportionality criteria for approval methods.

There are quite a few criteria that have been discussed in the literature, and this paper by Martin Lackner and Piotr Skowron gives a good summary. On page 56 it has a chart showing which criteria imply which others. However, most of them imply lower quota, which says that under party voting no party should get fewer than their exactly proportional number of seats rounded down. While this might sound reasonable it would actually throw away all methods that reduce to Sainte-Laguë party list under party voting as can be seen on this page. And Sainte-Laguë is considered by many to be the most proportional method. The authors of the paper acknowledge this shortcoming on page 121.

Most axiomatic notions for proportionality are only applicable to ABC rules that

extend apportionment methods satisfying lower quota (see Figure 4.1). This excludes, e.g., ABC rules that extend the Sainte-Lagu¨e method. As the Sainte-Lagu¨e

method is in certain aspects superior to the D’Hondt method (Balinski and Young

[2] discuss this in detail), it would be desirable to have notions of proportionality

that are agnostic to the underlying apportionment method.

The question is whether we need all these criteria and how many of them are really useful. If I want to know if a particular approval method is "proportional", I don't want to have to check it against 10 different criteria and then weigh them all up. And since they mostly throw out Sainte-Laguë-reducing methods - e.g. var-Phragmén - they are not ultimately fit for purpose.

There are two criteria in that table that don't imply lower quota. They are Justified Representation, which is not considered a good criterion in general and Perfect Representation, which is too restrictive since it's incompatible with what I would call strong monotonicity. Consider these approval ballots:

x voters: A, B, C

x voters: A, B, D

1 voter: C

1 voter: D

With two to elect, a method passing Perfect Representation will always elect CD regardless of the value of x despite both A and B having near unanimous support for high values of x. But Perfect Representation can still make the basis of a good criterion. Perfect Representation In the Limit (PRIL) says:

As the number of elected candidates increases, then for v voters, in the limit each voter should be able to be uniquely assigned to 1/v of the representation, approved by them, as long as it is possible from the ballot profile.

This makes sense because the common thread among proportionality criteria is the notion that a faction that comprises a particular proportion of the electorate should be able to dictate the make-up of that same proportion of the elected body. But this can be subject to rounding and there can be disagreement as to what is reasonable when some sort of rounding is necessary. However, taken to its logical conclusions, each voter individually can be seen as a faction of 1/v of the electorate for v voters, and as the number of elected candidates increases the need for any sort of rounding is eliminated in the limit.

In fact any deterministic method should obey Perfect Representation when Candidates Equals Voters (PR-CEV): when the number of elected candidates equals the number of voters there should be Perfect Representation as long as it is possible from the ballot profile.

I think most approval methods purporting to be proportional would pass these criteria. However, Thiele's Proportional Approval Voting (PAV) fails them so can really only be described as a semi-proportional method. Having said that, with unlimited clones, PAV is proportional again, so it would be completely acceptable for e.g. party-list approval voting.

Finally, one could argue that PRIL is not specific enough because it doesn't define the route to Perfect Representation, only that it must be achieved in the limit, which could potentially allow for some very disproportional results with a low number of candidates. The criticism is valid and further restrictions could be added. However, PRIL is similar to Independence of Clones in this sense, which is a well-established criterion. Candidate sets are only clone sets if they have the same rating or adjacent rankings on all ballots (which is essentially never). However, we would also want a method to behave in a sensible manner with near clones, and it is generally trusted that unless a method passing the criterion has been heavily contrived then it would do this. Similarly, one would expect the route to Perfect Representation in a method passing PRIL to be a smooth and sensible one unless a method is heavily contrived and we'd be able to spot that easily.

In any case, I think PRIL gets closer to the essence of proportionality than any of the criteria mentioned in Lackner and Skowron's paper.

Proportionally Swayed Favourite Voting System

Abstract

This paper introduces the "Proportionally Swayed Favorite" (PSF) voting system, a hybrid electoral method designed to balance proportional representation with local accountability. In PSF, each vote serves as both a preference for an individual candidate and their affiliated party. Seats are awarded iteratively based on a combination of individual vote percentages and adjusted party-wide support, ensuring a dynamic and equitable allocation process. Importantly, PSF requires no changes to how voters cast their ballots; each voter still casts a single vote, making the system intuitive and easy to implement. A simulation of the 2021 Canadian federal election under PSF demonstrates its potential to produce results closer to proportional representation while retaining strong local representation. This system addresses key shortcomings of both first-past-the-post (FPTP) and pure proportional representation systems, offering a compelling alternative for modern democracies.

Introduction

The design of electoral systems profoundly shapes democratic governance. First-past-the-post (FPTP) systems, widely used in countries like Canada, the United States, and the United Kingdom, have long been criticized for their tendency to produce disproportional outcomes. Parties with concentrated regional support may be overrepresented, while smaller parties with broad national appeal often struggle to gain seats. Conversely, systems based on proportional representation (PR) can dilute the link between elected officials and their local constituents, weakening the accountability that comes from direct voter-candidate relationships.

The "Proportionally Swayed Favorite" (PSF) system seeks to address these challenges by combining elements of proportional and constituency-based representation. This hybrid approach ensures that electoral outcomes reflect both the preferences of local voters and the overall distribution of party support nationwide. Additionally, PSF maintains the simplicity of the voting process: voters cast a single ballot as they would in FPTP elections, ensuring ease of understanding and efficient vote counting. In this paper, we describe the PSF system, compare it to existing electoral methods, and present a case study of its application to the 2021 Canadian federal election.

The simulation results reveal how PSF delivers a more proportional seat allocation while preserving the local dynamics critical to effective representation. By leveraging both individual and party-level support, PSF offers a nuanced and equitable solution for electoral reform.

Background and Motivation

Electoral systems are the backbone of representative democracies, translating voter preferences into seats in legislative bodies. However, no single system perfectly balances the competing goals of proportionality, local accountability, and simplicity. Each widely used system has its strengths and weaknesses, which have sparked debates about electoral reform worldwide.

First-past-the-post (FPTP) systems, for instance, prioritize local representation by electing the candidate with the most votes in each riding. While this fosters a direct connection between voters and their representatives, it often leads to disproportionate outcomes where a party’s share of seats in the legislature significantly deviates from its share of the popular vote. This distortion can result in "wasted votes" and discourage voter participation.

Proportional representation (PR) systems, on the other hand, address this by allocating seats based on the share of the vote each party receives. Although this approach ensures fairer representation for smaller parties, it often severs the link between voters and specific local representatives, potentially reducing accountability and regional engagement.

Mixed-member proportional (MMP) systems attempt to bridge these gaps by combining elements of FPTP and PR. However, MMP can introduce complexity for voters, as they must cast multiple votes, and for administrators, who must manage distinct processes for constituency and list seats.

The Proportionally Swayed Favorite (PSF) system emerges as a novel solution to these challenges. By preserving the simplicity of FPTP—where voters cast a single ballot—while incorporating proportional adjustments at the party level, PSF seeks to achieve a fairer balance between proportionality and local representation. This system is particularly suited to contexts where voter familiarity with FPTP is high, but there is significant demand for more proportional outcomes. PSF’s iterative allocation process ensures dynamic seat distribution without compromising the voter’s experience or the administrative ease of the electoral process.

The following sections will detail the mechanics of PSF, demonstrate its application through a case study of the 2021 Canadian federal election, and explore its potential advantages and challenges in comparison to existing systems.

Description of the System

The Proportionally Swayed Favorite (PSF) system is designed to balance proportional representation and local accountability while maintaining simplicity for voters. Below, we describe the mechanics of the system using a detailed example.

1. Ballot Casting Voters cast a single vote for their preferred candidate, as in the first-past-the-post (FPTP) system. This vote counts for both the individual candidate and the party they represent.
2. Initial Vote Calculation Two values are calculated for each candidate:
   1. The individual vote percentage, representing the share of votes the candidate received in their constituency.
   2. The party vote percentage, representing the party's share of the national vote.
3. Combined Score and Lead The combined score is the sum of the individual vote percentage and the party vote percentage. The lead is the margin by which a candidate's combined score exceeds their opponents in the same riding.Iterative Seat AllocationSeats are awarded iteratively based on the highest lead they have over other candidates in their riding:
   1. The candidate with the highest lead first seat. Once a candidate wins a seat, their party's vote percentage is reduced by an amount proportional to one seat. For example, if 100 seats are available, the reduction is 1% per seat. This adjustment ensures proportionality while accounting for local popularity.
   2. The combined scores are recalculated for all remaining candidates after each seat allocation.
   3. The process continues until all seats are filled, ensuring that the final allocation reflects both local candidate popularity and national party support.
4. Example Scenario

|| || |Riding|Party A|Party B|Party C| |North|50%|30%|20%| |South|40%|35%|25%| |East|30%|30%|40%| |West|35%|45%|20%| |Center|42%|40%|18%| |Overall|40%|35%|25%|

Initial Combined Scores and Leads

|| || |Riding|Party A|Party B|Party C|Lead| |North|90|65|45|35| |South|80|70|50|10| |East|70|65|65|5| |West|75|80|45|5| |Center|82|75|43|7|

Round 1: Party A wins the North riding. Party A's national vote share decreases by 20% (from 40% to 20%).

|| || |Riding|Party A|Party B|Party C|Lead| |South|60|70|50|10| |East|50|65|65|15| |West|55|80|45|25| |Center|62|75|43|13|

Round 2: Party B wins the West riding. Party B's national vote share decreases by 20% (from 35% to 15%).

|| || |Riding|Party A|Party B|Party C|Lead| |South|60|50|50|10| |East|50|45|65|15| |Center|62|55|43|7|

Round 3: Party C wins the East riding. Party C's national vote share decreases by 20% (from 25% to 5%).

|| || |Riding|Party A|Party B|Party C|Lead| |South|60|50|30|10| |Center|62|55|28|7|

Round 4: Party A wins the South riding.

Round 5: Party B wins the Center riding.

This example illustrates how PSF balances local preference and national proportionality through iterative adjustments.

Analysis and Discussion

The Proportionally Swayed Favorite (PSF) voting system offers a unique approach to addressing the longstanding challenges of electoral systems. Its design ensures a fair balance between local representation and proportional outcomes, addressing the key weaknesses of both first-past-the-post (FPTP) and proportional representation (PR) systems. This section analyzes the strengths, limitations, and potential implications of PSF, drawing comparisons with existing methods and examining its application in real-world scenarios.

1. Strengths of the PSF System

• Enhanced Proportionality: By incorporating national party support into the allocation of seats, PSF reduces the significant disparities between vote share and seat share that often occur under FPTP. This approach ensures that smaller parties with broad national appeal receive more equitable representation.
• Retention of Local Accountability: PSF preserves the direct connection between voters and their local representatives. Candidates must still garner significant support within their constituencies to win, ensuring accountability and responsiveness to local concerns.
• Simplicity for Voters: Unlike mixed-member proportional (MMP) or other complex systems, PSF does not require voters to cast multiple ballots or make unfamiliar decisions. Each voter casts a single vote, minimizing confusion and maintaining the familiarity of FPTP elections.
• Flexibility in Implementation: PSF can be integrated into existing electoral frameworks with minimal changes to ballot design and voting procedures. This reduces the administrative burden and facilitates its adoption in jurisdictions seeking electoral reform.

2. Limitations and Challenges

• Potential Complexity in Seat Allocation: While the process of awarding seats is straightforward for voters, the iterative calculation and adjustment of combined scores may be seen as complex by election administrators or the general public. Transparent and well-documented procedures would be essential to ensure trust in the system.
• Reduced Weight of Local Votes Over Time: As party vote percentages are adjusted iteratively, the influence of individual candidate support may diminish in later rounds. This could lead to perceptions that early victories unduly impact subsequent allocations.
• Limited Accommodation for Independents: PSF relies on party-level adjustments to achieve proportionality. Independent candidates, who do not belong to a party, might face disadvantages under this system unless additional provisions are made.

3. Comparative Insights

• FPTP vs. PSF: Compared to FPTP, PSF significantly improves proportionality while retaining the simplicity of a single vote and strong local representation. In the 2021 Canadian federal election simulation, PSF’s results demonstrated a closer alignment with national vote shares than FPTP, reducing overrepresentation of major parties and increasing seats for smaller parties.
• PR vs. PSF: While PR systems achieve near-perfect proportionality, they often weaken the link between voters and local representatives. PSF’s hybrid nature addresses this gap, ensuring that local preferences remain integral to the electoral process.
• MMP vs. PSF: MMP combines FPTP and PR elements but requires voters to navigate two separate votes and introduces dual categories of representatives. PSF simplifies this by consolidating proportional adjustments within a single-vote framework, streamlining both voter experience and administrative processes.

4. Implications for Electoral Reform

PSF’s innovative approach holds significant promise for jurisdictions seeking to modernize their electoral systems. By balancing proportionality and local representation, it addresses key public demands for fairness and accountability. However, successful implementation would require public education campaigns to build understanding and confidence in the system. Furthermore, its performance in diverse electoral contexts should be explored through additional simulations and pilot programs.

5. Simulation Insights

The application of PSF to the 2021 Canadian federal election highlights its potential to deliver more equitable outcomes. The following observations emerged from the simulation:

• Redistribution of Seats: PSF significantly altered the seat distribution compared to FPTP. Notably:
  • Liberal Party: Reduced from 160 to 132 seats.
  • Conservative Party: Increased from 119 to 125 seats.
  • New Democratic Party: Increased from 25 to 45 seats.
  • Bloc Québécois: Increased from 23 to 34 seats.
  • Green Party: Retained its 2 seats.
• Riding-Level Changes: A total of 30 ridings (approximately 10% of the total) changed representatives under PSF. These changes highlight the system’s sensitivity to both local preferences and proportionality.
• Vote Margin Analysis: On average, the FPTP winner led the PSF winner by 3.95% in the ridings where outcomes differed. This indicates that PSF predominantly overturned results in close contests, where local and national dynamics diverged.
• Popular Vote Alignment: The popular vote percentages for major parties were as follows: Liberal (32.62%), Conservative (33.74%), NDP (17.82%), Bloc Québécois (7.64%), and Green (2.33%). The PSF seat distribution closely aligns with these percentages, reflecting its proportional nature.

Below is a chart that compares the popular vote, FPTP seat share, and PSF seat share:

Conclusion

The Proportionally Swayed Favorite voting system represents a promising advancement in electoral design. By seamlessly integrating proportionality with local accountability, it addresses the core deficiencies of traditional systems. While challenges remain, its innovative mechanics and adaptability position PSF as a compelling alternative for democratic reform. Future research and real-world applications will be crucial to refining and validating its potential to enhance representative democracy.

Proportionally Swayed Favourite Voting System

Abstract

This paper introduces the "Proportionally Swayed Favorite" (PSF) voting system, a hybrid electoral method designed to balance proportional representation with local accountability. In PSF, each vote serves as both a preference for an individual candidate and their affiliated party. Seats are awarded iteratively based on a combination of individual vote percentages and adjusted party-wide support, ensuring a dynamic and equitable allocation process. Importantly, PSF requires no changes to how voters cast their ballots; each voter still casts a single vote, making the system intuitive and easy to implement. A simulation of the 2021 Canadian federal election under PSF demonstrates its potential to produce results closer to proportional representation while retaining strong local representation. This system addresses key shortcomings of both first-past-the-post (FPTP) and pure proportional representation systems, offering a compelling alternative for modern democracies.

Introduction

The design of electoral systems profoundly shapes democratic governance. First-past-the-post (FPTP) systems, widely used in countries like Canada, the United States, and the United Kingdom, have long been criticized for their tendency to produce disproportional outcomes. Parties with concentrated regional support may be overrepresented, while smaller parties with broad national appeal often struggle to gain seats. Conversely, systems based on proportional representation (PR) can dilute the link between elected officials and their local constituents, weakening the accountability that comes from direct voter-candidate relationships.

The "Proportionally Swayed Favorite" (PSF) system seeks to address these challenges by combining elements of proportional and constituency-based representation. This hybrid approach ensures that electoral outcomes reflect both the preferences of local voters and the overall distribution of party support nationwide. Additionally, PSF maintains the simplicity of the voting process: voters cast a single ballot as they would in FPTP elections, ensuring ease of understanding and efficient vote counting. In this paper, we describe the PSF system, compare it to existing electoral methods, and present a case study of its application to the 2021 Canadian federal election.

The simulation results reveal how PSF delivers a more proportional seat allocation while preserving the local dynamics critical to effective representation. By leveraging both individual and party-level support, PSF offers a nuanced and equitable solution for electoral reform.

Background and Motivation

Electoral systems are the backbone of representative democracies, translating voter preferences into seats in legislative bodies. However, no single system perfectly balances the competing goals of proportionality, local accountability, and simplicity. Each widely used system has its strengths and weaknesses, which have sparked debates about electoral reform worldwide.

First-past-the-post (FPTP) systems, for instance, prioritize local representation by electing the candidate with the most votes in each riding. While this fosters a direct connection between voters and their representatives, it often leads to disproportionate outcomes where a party’s share of seats in the legislature significantly deviates from its share of the popular vote. This distortion can result in "wasted votes" and discourage voter participation.

Proportional representation (PR) systems, on the other hand, address this by allocating seats based on the share of the vote each party receives. Although this approach ensures fairer representation for smaller parties, it often severs the link between voters and specific local representatives, potentially reducing accountability and regional engagement.

Mixed-member proportional (MMP) systems attempt to bridge these gaps by combining elements of FPTP and PR. However, MMP can introduce complexity for voters, as they must cast multiple votes, and for administrators, who must manage distinct processes for constituency and list seats.

The Proportionally Swayed Favorite (PSF) system emerges as a novel solution to these challenges. By preserving the simplicity of FPTP—where voters cast a single ballot—while incorporating proportional adjustments at the party level, PSF seeks to achieve a fairer balance between proportionality and local representation. This system is particularly suited to contexts where voter familiarity with FPTP is high, but there is significant demand for more proportional outcomes. PSF’s iterative allocation process ensures dynamic seat distribution without compromising the voter’s experience or the administrative ease of the electoral process.

The following sections will detail the mechanics of PSF, demonstrate its application through a case study of the 2021 Canadian federal election, and explore its potential advantages and challenges in comparison to existing systems.

Description of the System

The Proportionally Swayed Favorite (PSF) system is designed to balance proportional representation and local accountability while maintaining simplicity for voters. Below, we describe the mechanics of the system using a detailed example.

1. Ballot Casting Voters cast a single vote for their preferred candidate, as in the first-past-the-post (FPTP) system. This vote counts for both the individual candidate and the party they represent.
2. Initial Vote Calculation Two values are calculated for each candidate:
   1. The individual vote percentage, representing the share of votes the candidate received in their constituency.
   2. The party vote percentage, representing the party's share of the national vote.
3. Combined Score and Lead The combined score is the sum of the individual vote percentage and the party vote percentage. The lead is the margin by which a candidate's combined score exceeds their opponents in the same riding.Iterative Seat AllocationSeats are awarded iteratively based on the highest lead they have over other candidates in their riding:
   1. The candidate with the highest lead first seat. Once a candidate wins a seat, their party's vote percentage is reduced by an amount proportional to one seat. For example, if 100 seats are available, the reduction is 1% per seat. This adjustment ensures proportionality while accounting for local popularity.
   2. The combined scores are recalculated for all remaining candidates after each seat allocation.
   3. The process continues until all seats are filled, ensuring that the final allocation reflects both local candidate popularity and national party support.
4. Example Scenario

|| || |Riding|Party A|Party B|Party C| |North|50%|30%|20%| |South|40%|35%|25%| |East|30%|30%|40%| |West|35%|45%|20%| |Center|42%|40%|18%| |Overall|40%|35%|25%|

Initial Combined Scores and Leads

|| || |Riding|Party A|Party B|Party C|Lead| |North|90|65|45|35| |South|80|70|50|10| |East|70|65|65|5| |West|75|80|45|5| |Center|82|75|43|7|

Round 1: Party A wins the North riding. Party A's national vote share decreases by 20% (from 40% to 20%).

|| || |Riding|Party A|Party B|Party C|Lead| |South|60|70|50|10| |East|50|65|65|15| |West|55|80|45|25| |Center|62|75|43|13|

Round 2: Party B wins the West riding. Party B's national vote share decreases by 20% (from 35% to 15%).

|| || |Riding|Party A|Party B|Party C|Lead| |South|60|50|50|10| |East|50|45|65|15| |Center|62|55|43|7|

Round 3: Party C wins the East riding. Party C's national vote share decreases by 20% (from 25% to 5%).

|| || |Riding|Party A|Party B|Party C|Lead| |South|60|50|30|10| |Center|62|55|28|7|

Round 4: Party A wins the South riding.

Round 5: Party B wins the Center riding.

This example illustrates how PSF balances local preference and national proportionality through iterative adjustments.

Analysis and Discussion

The Proportionally Swayed Favorite (PSF) voting system offers a unique approach to addressing the longstanding challenges of electoral systems. Its design ensures a fair balance between local representation and proportional outcomes, addressing the key weaknesses of both first-past-the-post (FPTP) and proportional representation (PR) systems. This section analyzes the strengths, limitations, and potential implications of PSF, drawing comparisons with existing methods and examining its application in real-world scenarios.

1. Strengths of the PSF System

• Enhanced Proportionality: By incorporating national party support into the allocation of seats, PSF reduces the significant disparities between vote share and seat share that often occur under FPTP. This approach ensures that smaller parties with broad national appeal receive more equitable representation.
• Retention of Local Accountability: PSF preserves the direct connection between voters and their local representatives. Candidates must still garner significant support within their constituencies to win, ensuring accountability and responsiveness to local concerns.
• Simplicity for Voters: Unlike mixed-member proportional (MMP) or other complex systems, PSF does not require voters to cast multiple ballots or make unfamiliar decisions. Each voter casts a single vote, minimizing confusion and maintaining the familiarity of FPTP elections.
• Flexibility in Implementation: PSF can be integrated into existing electoral frameworks with minimal changes to ballot design and voting procedures. This reduces the administrative burden and facilitates its adoption in jurisdictions seeking electoral reform.

2. Limitations and Challenges

• Potential Complexity in Seat Allocation: While the process of awarding seats is straightforward for voters, the iterative calculation and adjustment of combined scores may be seen as complex by election administrators or the general public. Transparent and well-documented procedures would be essential to ensure trust in the system.
• Reduced Weight of Local Votes Over Time: As party vote percentages are adjusted iteratively, the influence of individual candidate support may diminish in later rounds. This could lead to perceptions that early victories unduly impact subsequent allocations.
• Limited Accommodation for Independents: PSF relies on party-level adjustments to achieve proportionality. Independent candidates, who do not belong to a party, might face disadvantages under this system unless additional provisions are made.

3. Comparative Insights

• FPTP vs. PSF: Compared to FPTP, PSF significantly improves proportionality while retaining the simplicity of a single vote and strong local representation. In the 2021 Canadian federal election simulation, PSF’s results demonstrated a closer alignment with national vote shares than FPTP, reducing overrepresentation of major parties and increasing seats for smaller parties.
• PR vs. PSF: While PR systems achieve near-perfect proportionality, they often weaken the link between voters and local representatives. PSF’s hybrid nature addresses this gap, ensuring that local preferences remain integral to the electoral process.
• MMP vs. PSF: MMP combines FPTP and PR elements but requires voters to navigate two separate votes and introduces dual categories of representatives. PSF simplifies this by consolidating proportional adjustments within a single-vote framework, streamlining both voter experience and administrative processes.

4. Implications for Electoral Reform

PSF’s innovative approach holds significant promise for jurisdictions seeking to modernize their electoral systems. By balancing proportionality and local representation, it addresses key public demands for fairness and accountability. However, successful implementation would require public education campaigns to build understanding and confidence in the system. Furthermore, its performance in diverse electoral contexts should be explored through additional simulations and pilot programs.

5. Simulation Insights

The application of PSF to the 2021 Canadian federal election highlights its potential to deliver more equitable outcomes. The following observations emerged from the simulation:

• Redistribution of Seats: PSF significantly altered the seat distribution compared to FPTP. Notably:
  • Liberal Party: Reduced from 160 to 132 seats.
  • Conservative Party: Increased from 119 to 125 seats.
  • New Democratic Party: Increased from 25 to 45 seats.
  • Bloc Québécois: Increased from 23 to 34 seats.
  • Green Party: Retained its 2 seats.
• Riding-Level Changes: A total of 30 ridings (approximately 10% of the total) changed representatives under PSF. These changes highlight the system’s sensitivity to both local preferences and proportionality.
• Vote Margin Analysis: On average, the FPTP winner led the PSF winner by 3.95% in the ridings where outcomes differed. This indicates that PSF predominantly overturned results in close contests, where local and national dynamics diverged.
• Popular Vote Alignment: The popular vote percentages for major parties were as follows: Liberal (32.62%), Conservative (33.74%), NDP (17.82%), Bloc Québécois (7.64%), and Green (2.33%). The PSF seat distribution closely aligns with these percentages, reflecting its proportional nature.

Conclusion

The Proportionally Swayed Favorite voting system represents a promising advancement in electoral design. By seamlessly integrating proportionality with local accountability, it addresses the core deficiencies of traditional systems. While challenges remain, its innovative mechanics and adaptability position PSF as a compelling alternative for democratic reform. Future research and real-world applications will be crucial to refining and validating its potential to enhance representative democracy.

Here's how it works:

- Voters get to rank in order of preference local candidates & the candidates running in other districts in their region (on the same ballot) - all candidates have to run in a specific district

1. Elect local reps under IRV (50% of the total reps in a region, while 50% of reps are region-wide reps)
2. Calculate a "regional quota", Determined by dividing the total number of votes in a region by the number of seats (district representatives + regional representatives) in the region + 1
3. Determine the number of surplus votes for the elected local candidates, which are the first preference votes they received locally that are above the regional quota. If an elected local candidate has received fewer first-preference votes locally than the regional quota, they would not have any surplus votes
4. Order the unelected candidates based on the first preferences votes they received in their district only (this incentivizes candidates to try to get votes from their local district)
5. Transfer the surplus votes from the elected local candidates to one of the unelected candidates (based on how the voter has ranked the other candidates on their own ballot)
6. Conduct the election for the remaining seats in the region under the Single Transferable Vote, with the regional quota being the quota to get elected as a regional representative

https://www.nytimes.com/interactive/2025/01/14/opinion/fix-congress-proportional-representation.html?unlocked_article_code=1.pE4.ETfp.faHEI2R4ZAQd&smid=nytcore-ios-share&referringSource=articleShare

The version of STV used for the Australian Senate (where voters can rank political groups), *but* where voters also get to put an X beside a candidate (who is running for the group they ranked first), with the # of Xs each candidate gets would determine the order for each party's list?

View Poll

Obviously one option is a run-off with those tied. But I'm wondering if there's any info in the ballots for automatic tie-breaking. I guess you could use something like satisfaction approval voting where people who approved multiple options votes get diluted to break the tie. But does that make sense? Should being a "picky" voter be rewarded?

Maybe it's not a big deal and unlikely to happen but just curious.

https://en.wikipedia.org/wiki/Satisfaction_approval_voting

I just realized that even though it would be a data gold mine for analysis of partisanship (on a local level) and voter behavior, I don't know whether plurality bloc voting results are published or even counted and recorded properly in my country (per ballot). I guess they are not, but now I will look into whether there was any attempt to change this or something.

In the meantime, if you live in jurisdictions whether bloc voting (so usually n-approval type ballots) is used, do full results get published?

Also, if you live in a jurisdiction with ranked ballots (IRV, STV) do ranked ballots get published? If you live in jurisdiction with two vote MMP, if there are two votes on a ballot (mixed ballot, like in Germany), are the results available according to ballots, not separately? Or if you live in places with amy other interesting system, like panachage, do you have the full results published?

I'd be very interested in any such data.

Register: https://respectvoters.org/volcall

https://preview.redd.it/nlyv4u83rzbe1.png?width=3840&format=png&auto=webp&s=98e65755cd16bac5db54d29822a85b0da6307de2

Hey, sorry for not promoting this sooner, but today I'll be hosting the Respect Voters Coalition first monthly national organizing call for volunteers. Respect Voters (aka Show Me Integrity) is the team behind the Approval Voting win in Saint Louis, MO, and we want to help bring it (and other democracy reforms) to the rest of the country. While I have a list of things that we need help with, I definitely want to empower volunteers to work on the things they feel need to be done in the space. I know many of you ultimately want to run initiative petitions to get Approval Voting adopted, and we do, too! I hope you can make it.

Register here: https://respectvoters.org/volcall

I like it because it utilizes scored ballots, is quite proportional, and seems simple (according to electowiki atleast, I have only a superficial understanding of proportionality and computational complexity, so am asking here regarding those claims). Is there any obvious advantage(s) that make it arguable (or any other method of cardinal PR in general) over STV? I've asked something like this before in general because I don't understand the matter, but moreso towards which voting methods were worth the fight for adoption against STV.

I came up with a probabilistic proportional STV system inspired by Random ballot.
you rank candidates like normal, transfer votes using the hare quota, then when no more transfers can be done, the probability of the remaining candidates being chosen for a seat is their fraction of the hare quota.

the exact equation for the pdf I haven't found yet, but it does exist.

the degrees of freedom could be used to afford better proportionality with the final seats in some way from the final rankings, but i haven't figured this out yet.

this system is proportional to some degree, monotone, probably consistent, satisfies participation, no favorite betrail and perhaps honest.