Edit, 2021-12-26: I wrote the bulk of this post back in August. Then I paused to do research; the topic was multi-winner approval voting. It took me a while to go through the articles, and by then I wasn't all that certain about the idea on this post anymore. Still, I want it to see the light of day, so here it is - unfinished. I think it still has some usefulness nonetheless. I'm sorry if you get really curious about the parts that are only vaguely sketched out.

Institutional decision-making, including decisions made by governments, impacts most areas of people's lives. Improving it can have big positive impacts and one way of effecting that is by implementing better ways of voting. In this post, I will offer a new form of voting, intended mainly for national legislatures, and I would appreciate comments and criticism. I will briefly summarize the idea and its motivation, proceed to analyze some flaws I perceive in methods used in most Western democracies, then describe the proposed method in more detail and finally speculate about what might happen if this was implemented in the United States.

I - Summary

NB: throughout this article, I will refer to a geographical area returning one elected member of a legislature as a "district"; these are equivalent to ridings/ circonscriptions in Canada, constituencies in the United Kingdom, Wahlkreise in Germany and so on.

One big benefit of single-member districts is a direct connection between each voter and their representative, supposedly making legislators more responsive to constituents' voices. One big benefit of proportional representation (PR) is giving everyone's vote an equal weight regardless of where they live, according to the principle of "one person, one vote". This proposal aims to combine the benefits of both systems, by having legislators from single-member districts while still being proportional.

In summary, here is how it would work:

1 - Divide  the country into as many districts as there are seats in the legislature. Countries like the US, Britain and France already have that. It does not matter if, like in many US states, these districts are gerrymandered. They don't strictly have to follow state/province boundaries or even all have the same population.

2 - Parties nominate candidates in districts. They have an incentive to nominate candidates in as many districts as possible, even if they don't expect to win there.

3 - Voters vote normally. This could take the form of first-past-the-post (as in the US), or some other system that allows votes from all districts to be meaningfully aggregated.

4 - Each party's votes across all districts are added up; based on each party's vote total across the nation (or at the state/province level, if desired), calculate the number of seats each party wins, according to some proportional representation method - for example, Jefferson's method, aka D'Hondt. So each party's share of the seats in the legislature corresponds to its share of the vote, with minor adjustments due to rounding.

5 - Each district elects the candidate that did better in that district, relative to their party's national /state vote. The candidate who outperformed their party the most is elected, with the other candidates in the district eliminated; once the correct number of representatives for a given party is reached, all other non-elected candidates from the same party are also eliminated. Repeat the process with the remaining non-eliminated candidates until all seats are filled.

By construction, this gives a proportional result, while also electing a single member from each district. The caveat is that the winner might not be the candidate most voters preferred in that particular district, as each winner is also determined by the nation/statewide vote.

II - Problems I intend to solve with this

The idea for this method arose out of noticing a trilemma:

• single-member districts produce results that are not very proportional;
• proportional representation lacks a direct connection between each voter and a specific, individual representative;
• mixed systems like in Germany, with single-member districts supplemented by PR seats, result in unpredictable growth of the size of the legislature.

The three parts are analyzed in more detail below.

II.1 - Problems with FPTP single-winner elections

In the United States, Canada and the UK, whichever candidate  wins the most votes in a given district is elected, even if they are short of a majority, a system called "first past the post" (FPTP).

This has the effect of favoring parties whose support is concentrated in certain regions at the expense of those which enjoy support across many districts. Regional parties like Canada's Bloc Québécois and Scotland's SNP have an outsized influence in their respective countries' Parliaments due to only contesting elections in their respective geographies; a party that got the same number of votes as the Bloc, but spread out over Canada's 338 ridings, would likely win at most a handful of seats, if any. The Bloc, however, got a healthy 32 seats in the 2019 election.

In the United States, where drawing the district maps is a state responsibility rather than the federal government's, gerrymandering is often practiced: state legislatures under control of a single party explicitly draw districts that favor their candidates, allowing them to get strong majorities of seats with much lower vote shares, sometimes less than the other big party. One of the reasons for that is: in a district where, say, Democratic voters are "packed" by a Republican gerrymander, the GOP does not even bother running a candidate. This happens since, by design, the district is carved out in such a way to give the Dems a large majority. This allows the GOP to concentrate its resources into more competitive races.

In short, the one big problem with FPTP is that it throws away millions of votes, since parties with support of maybe 1 in 5 voters might end up out of Parliament altogether, or having much fewer than 1/5 of the seats. In the US, the system is also rigged by gerrymandering.

II.2 - Problems with proportional representation

Many European countries adopt some form of proportional representation (PR). Parties present lists of candidates, voters vote for a list or individual candidates within the list, and seats are allocated to the various parties in proportion to their share of the vote. Three ways in which these systems differ from each other are in how they deal with the fractionary numbers that inevitably arise when calculating proportionality, whether votes for individual candidates within a list or for independents are allowed and how they are counted, and what the vote share is below which a party gets no seats at all - the threshold.

To be quite honest, the countries that already use PR are not my main focus of change here. I like PR, I think countries using it generally have it together to a greater extent than FPTP countries, and I would be far from suggesting that, say, Finland change from their open-list system to the single-winner PR system I am proposing.

My focus is on how to bring the advantages of PR to current FPTP countries, and possibly single-winner countries that use other systems (like Australia with instant-runoff voting); this is done by addressing the potentially biggest qualm voters in, say, America might have with PR. Namely, this is the lack of a direct connection between a voter and their representative. For four years, Americans living in a certain area of Manchester, New Hampshire had EA movement member Elizabeth Edwards-Appell as their representative; these were the people she needed to listen to the most when considering what to say and how to vote in the New Hampshire State House, as their vote decided whether she got re-elected or not (which she did, serving two terms of two years each then retiring from the House. She's just so cool.)

I believe this system could be a selling point for PR in America because it preserves this connection. If anyone decides to advocate for it, a good framing is needed to draw away focus from the fact that the top vote-getter in the district is not necessarily elected - but I get ahead of myself.

II.3 - Problems with the German system

The postwar German constitution sought a compromise between these two systems. The Bundestag, the German parliament, is elected via a mixed system. The country is divided into 299 single-winner seats; in addition to voting for a candidate in their local seats, voters have a second vote, which can be cast for any party (not necessarily the same as the first vote). Second votes are used to allocate a further 299 seats, so as to obtain a nationwide proportional representation.

However, it might be that 299 proportional seats is not enough, as one party might often win a large share of the first-vote seats, requiring a large number of seats for other parties to achieve proportionality. In the 2017 election, the latest as of writing, Angela Merkel's CDU/CSU won 231 mandates, sweeping five of the nation's sixteen states. In a June opinion poll for the election coming up on September 26, the CDU/CSU was forecast to increase its strength to 243 seats.

The solution the Germans chose is to create extra seats, as many as necessary to reach proportionality. So, instead of the planned 598 members, the current Bundestag rose to 709 members. It seems that this is a disadvantage; it increases costs and reduces the power of each individual member, counteracting to some extent the effect of having single-winner districts in the first place. After all, a voter has less impact by calling their representative when there are 708 others, than when there are 597 others.

III - Single-winner meets proportional representation

In this section, I will present a more fully fleshed out description of my idea. I will start by getting at the core principle, then suggest some considerations that affect the election as a whole while being independent from that core idea.

III.1 - Divide and conquer?

The method focuses on a single polity to be divided into districts and returning a certain delegation to a legislature; the method is agnostic as to whether the polity is an entire nation, with the delegation corresponding to the whole of the legislature, or if the polity is a state/region/province, with the caveat that any proportional representation at the level of smaller polities tends to be less precisely proportional than larger ones due to compounding errors in rounding.

To be precise, if "election results" are defined as each party's vote share in each district, each of their vote shares in the whole polity and the number of seats each is entitled to, then the method is designed to produce a list of elected candidates out of election results.

The method proposed here assumes the polity adopting it is already divided into districts, that parties have already presented their candidates for each district they can and want, and that voters have already voted.

It also assumes that each party has had a total vote calculated by adding up the votes for its candidates from each of the districts in the polity. Furthermore, it is assumed that the proportional part of the results has already been calculated. I recap one method this can be done in section III.3.

The first step of the algorithm is to produce a list of tuples of the form:

$(districtName;partyName;relativeVoteShare)$

The relative vote share for party A in district X is defined as A's vote share in X, divided by A's vote share in the entire polity.

So, if X gave A 35% of the vote, and A got 20% of the polity-wide vote, then the relative vote share for A in X results in 1.75. The resulting tuple looks like $(DistrictX,PartyA,1.75)$.

The list to be produced should be based on the Cartesian product of districts by parties; that is, it will have one element for each combination of district and party that ran in that district, accompanied by the corresponding relative vote share. Once that is done, the list is sorted in descending order of relative vote shares. It does not matter if entries for various districts or parties get mixed together, and in fact that is to be expected.

The next step produces the winning party in each district. This happens in steps that look at the first element of the list that is not yet elected or eliminated (we'll get to how entries can get eliminated).

If we are focused on a tuple like $(DistrictX,PartyA,1.75)$, that means the candidate from party A wins district X. Mark them as elected; the district already has a representative, so eliminate all other entries on the list that refer to the same district. Since the list is sorted, these other parties must have received a smaller vote share in that district, relative to their performance in the whole polity.

We must also keep track of how many seats each party is still entitled to. So, in our list of seats per party, generated by the PR calculation, we reduce the number for the party winning the current seat by one. Eventually, at this point of the process, a party's remaining allocation will reach 0; when that happens, eliminate all remaining candidates from that party.

Repeat the process until every element of the list has been either elected or eliminated.

III.2 - Properties

It can be seen that the method does produce the correct number of winners from each party, as determined by the PR calculation; the iteration through the list of tuples starts from those correct numbers, and decreases them by one for every candidate that gets elected, stopping when it reaches zero, so the end result must obey the PR calculation.

The method does elect exactly one member by district, since every district is present several times in the list that is iterated through; no district being absent means they all get at least one representative, and all remaining tuples for a district being removed when it gets its first winner ensures that the winner is unique. This also entails that the size of the legislature corresponds to the number of districts.

Therefore, this addresses the lack of proportionality of FPTP, the lack of a direct connection between voter and elected official of other forms of PR, and the house size inflation of the German mixed system. It also solves a problem present in an existing proposed system, one which, given $N$ seats for a party, simply elects its $N$ candidates that got the most votes; in that system, it is possible for some districts to have more than one representative, leaving other districts unrepresented, but the method described here avoids that.

III.3 - Additional considerations

In this section, I will deal with aspects of the voting process that are not directly related to the main topic, but still have an influence. If this whole article is an opinionated brain dump, this section is the opinionated-brain-dumpiest of them all.

First, regarding polities. I personally think that elections for the national legislature should be of national scope; in other words, the polity used should be the entire country. This is not a new idea; the Netherlands and Israel already do that. Candidates still come from specific provinces, in the Dutch system at least, but the proportionality is calculated at the nationwide level. This seems like a good idea because the laws and decisions the legislature makes affect the entire nation; even when, for example, some special measure is taken to support a specific state, that is still coming from the collective purse of all other states.

So, ideally, all of America would be a single polity, for the purposes of this system, as opposed to 50 polities corresponding to the 50 states. (Maybe that might remove the taboo that currently exists about increasing the House of Representatives from its century-long cap at 435 members?) In the UK, Northern Ireland has a completely separate party system from Great Britain; also, Scotland and Wales have their respective nationalist parties, the SNP and Plaid Cymru. In that case, it might seem wise to use the constituent countries as the polities. I am not sure how to deal with the Québec case in Canada, though.

Second, regarding districts. The process of redistricting would become less critical, as the stakes in districting are lower when all votes from all districts are added together anyway. This shows up in a couple of ways: district sizes can vary more, and gerrymandering for political gain ceases to be important.

To the extent that I have considered the issue, it would not matter to this method if the districts varied significantly in size. All votes count equally for the PR calculation, and vote totals being larger in one district than another does not matter because they are all divided by the polity-wide vote share; the result that counts is the share, not the total number.

So, on the one hand, this reduces the need for redistricting as population changes cause districts to gain and lose population relative to each other, but clearly there are other considerations such as fairness and democratic legitimacy that favor equally-sized districts. Still, one might point out that countries using districts already allow for quite significant discrepancies in district size, with the largest UK parliamentary constituency having 5 times as many voters as the smallest one; even excluding the numerical and geographical outliers that are islands off the British mainland, one can still find ratios like 7:3 between constituency sizes.

It would seem that under this system parties would have little incentive to gerrymander. This outcome is not unique to this particular method, but common to most if not all PR methods. The main point of gerrymandering is for a party to win more seats than it would under PR, so adopting proportionality addresses the root of that problem.

Another reason to gerrymander in America, creating minority-majority districts, would be left unaffected. There is a chance those districts would be less necessary since the point of creating them, which is not to spread out minority voters so much that they cannot elect anyone from their group, is negated to some extent by all of their votes counting. One would still need minority candidates, though; but they would not necessarily have to come from minority-majority districts.

Third, regarding nominations. Under the system proposed here, there would be no incentive for parties not to nominate candidates in districts they thought they had no chance to win, as their votes in those districts are still added to the party's polity-wide total. It could be that they would campaign little in these districts, but voters belonging to its hard core of constituents could still vote for them and have their vote improve their preferred party's polity-wide seat share.

One thing that would need to be decided is who can present candidacies, and what are the requirements to do so. Some countries (I am thinking of my home country, Brazil) have officially-registered political parties that are the only ones that can nominate candidates for political office. Individual party members cannot self-nominate, as the party conventions decide on candidacies and there are no independents. This seems somewhat similar to the British system, where disobedient MPs can "lose the whip" and not be nominated by their parties in the subsequent election.

Conversely, I think presenting party lists of candidates should be very easy, so as to make representation widely accessible; if registering as a party for the purpose of having votes added across districts is easy, it seems that purely party-independent candidacies would not have much point to them. Maybe parties would be even allowed to be on the ballot in a given district, without nominating a specific candidate, just so that their votes could count for other districts where they do field actual candidates. This allows "independent" candidates under this system. On the other hand, in order to prevent "fake" campaigns that divert votes from real ones, there should perhaps be a deposit that gets forfeited if a party/candidate fails to reach, say, even 0.1% of the vote.

Fourth, regarding actual voting. In many PR systems, each voter is allowed to select only one party, sometimes also picking a candidate from that party's list. This goes counter to other voting systems advocated by voting reforms in general, like instant-runoff voting (IRV) and by the people in the EA community that have looked the deepest into the voting issue - approval voting. For clarity, IRV is a more precise term to describe what was labeled as "Alternative Vote" in a 2011 referendum in the UK, and what is often referred to as "Ranked Choice Voting" in the US, having been adopted recently by jurisdictions such as New York City and Maine. While IRV certainly fits the description of ranked choices, there are many different ways of counting ranked ballots, so it is imprecise to use RCV as a synonym for IRV.

In general terms, this method assumes voters would vote for a single party in their district. Apparently, a recent theoretical development is party-list approval voting, which might allow voters to cast approval-style votes in their districts, which are then combined to produce a PR result (using a modified criterion of proportionality called justified representation) that could then, hopefully, be fed into this method. I still haven't read the article and I predict its math will go over my head, so I cannot guarantee that it works with my proposal.

As for ranked votes like in IRV, I do not know if it is possible to count ballots that are ranks of parties, as opposed to ranks of candidates, in such a way as to produce a proportional result. Furthermore, it is unclear how the relative vote share stage would work. To be clear, IRV is just the single-winner special case of the single transferable vote (STV). STV is a multi-winner method, but it apportions seats between candidates that voters ranked in small districts - 3 to 5 seats each in the case of Ireland, the only country to use it.

The difference that makes STV not very suitable for our purposes of achieving a PR result similar to that a party list would produce is that candidates in STV can only be in one of three states - elected, eliminated, or still in the race. Once a candidate is either elected or eliminated, their votes are transferred, following the order on the ballots, to candidates still in the race; but it is unclear how to apply that to parties, as it is unclear when to transfer votes, to whom and how many. Maybe that is possible and someone will tell me how in the comments?

Fifth, regarding the calculation of proportionality. Since numbers of seats virtually always have to be rounded ((D'Hondt better than other methods; remind reader that thresholds exist))

--

As an aside, it could well be that the unrepresented districts of that system are the most competitive ones, the ones with the smallest difference between candidates, which might mean they are the most moderate ones; if that is the case, then the system privileges extremist candidates, which sounds even worse than just disenfranchising some voters! So taking into account relative vote shares really seems like an improvement in this case.

• embedding party primaries in the general election by allowing more than one candidate by party

IV - What if this actually happened

• criticism about top vote-getter not winning
• probably higher churn in the legislature
• speculate about representatives' incentives in different scenarios
• multi-party system by breaking loyalty between different factions of each major party
• possible tactical voting?