Context

I have written a series of longer posts about voting theory: a general primer, single-winner pathologies, multi-winner method building blocks, and a longer, harder-to-read piece where I struggled through to a metric for multi-winner methods.

I believe that I should write a simpler, (relatively) shorter piece on proportional multi-winner methods. This is partly because my own thinking has evolved and sharpened, and partly because I think that current events in the US, Canada, and UK make these ideas more timely:

In the US, the Democratic party is poised to have the trifecta they'd need to pass Federal laws, and they seem to be motivated to look at fundamental political reforms due to such events as the recent (ineptly) attempted coup.
In Canada, various people, including Jagmeet Singh (the head of the third-largest party by votes) have called for a Citizens' Assembly on election method reform.
The UK is also in political crisis due to Brexit and coronavirus, and election reform seems to me to be on the agenda.

In all three of these cases, the situation favors grassroots calls for reform, but it seems to me that a substantial number of existing political incumbents will have to be on board for this reform to succeed. This means that reform that is especially hostile to such incumbents would probably be vetoed.

It is utterly clear to me that election method reform, based on some form of proportional representation, could be a key step towards healthier politics in all three places. It would not be a panacea; politics is, and always will be, a battleground of competing interests, and whatever your sympathies are, I suspect you see some of those interests as less valid than others. But it would help.

This article only mentions the best methods. As a co-organizer of the 2018 British Columbia Symposium on Proportional Representation; a committee-member of the Equal.Vote committee on proportional representation methods; and the primary designer of the proportional voting method used for Hugo Award nominations; I could explain dozens of different (and interesting) #ProRep methods if I wanted. But this is focusing on those I think are worth knowing from a practical standpoint.

Note: there is a short summary of this article at the end.

Pros and Cons of #ProRep in General

Any form of proportional representation would have several huge advantages over the current "first past the post" (FPTP, aka single-winner plurality) method used to elect the lower house of the legislature in all three countries (US House of Representatives, Canadian and UK National Parliaments). These advantages include:

Decreased ideological polarization in the legislature, and thus lower chance of spiteful (lose/lose) outcomes.
- Though there might be a few representatives further out on a fringe than currently, there would be far more closer to the center (median voter). IMO, this would increase the chances of healthy negotiation and win/win compromise, even in cases where I disagree ideologically with the centrist position.
Drastically fewer wasted votes. Nearly every vote would help elect a representative.
- This increases voters' incentive to vote, and politicians' incentive to care about all voters.
Increased diversity (demographic, ideological, and cognitive) in the legislature.
- I believe that, on balance, this would lead to better outcomes; though it's true that this is mediated by the procedures for deliberation within the legislatures themselves, which also could and should be reformed.
A total fix to any ideological bias that's due to the particularities of voter distribution across districts.
- In the US in particular, this problem is at crisis levels, because it's been deliberately weaponized through partisan gerrymandering. But even in the other two countries, it's a problem, largely because of the natural asymmetry of the urban/rural divide in politics.

Note that while some of these are particularly desirable from certain partisan standpoints, at least the first three are desirable on meta-ideological grounds. In other words, I believe there's reason to want this kind of reform almost no matter where you are on the ideological spectrum.

This is not to say that there are no potential downsides to proportional representation. For instance, some proportional methods, such as Israel's closed-list one, seem to lead to excessive party fragmentation, which in my opinion can result in myopic single-issue parties getting disproportionate power over their issue of interest. But even at its worst, I think this downside is small relative to the upsides above; and well-designed voting methods can minimize this downside even further.

Four #ProRep methods

Here are the four reasonably-good proportional methods that I think it's worth knowing about. For each, I'll briefly explain how it works, then mention a few pros and cons relative to the other three. Note again: the "downsides" are only relative to other methods here; all these methods are better than the current FPTP in almost every way. All of them share all the advantages listed above, and I've left out #ProRep methods like Israel's which are strictly worse than the other options. The cons I will list are only relative to the other options listed here, not relative to the status quo.

STV/RCV5

This base method is called "Single Transferable Vote", and has also been rebranded in some cases as "multi-seat Ranked Choice Voting". (It's not the same as the single-winner version of RCV, though; single-winner reform is out of the scope of this article.) At the bottom of this section, I'll quickly sketch a proposed improvement on this method, known as Sequential Monroe, which is slightly better but shares most of the primary pros and cons. But for now, here's how the basic STV works:

The state/province/country is divided up into subregions, each with enough population to elect 3-5 legislators. (Annoyingly, there are different names for these regions: "districts" in the US, "ridings" in Canada, and "constituencies" in the UK. I'll use "districts" from here on. The number of seats in a district is known as its "magnitude" M.)
- Note that states/provinces with fewer than 3 seats total cannot really use this or other proportional methods.
Within each district, voters rank the local candidates in order of preference.
Votes are tallied for their top choice.
Any candidate with over a "quota" — 1/(M+1) of all the district votes — is seated. If so, a quota of their votes are "used up", and the rest of them are transferred to the next candidate in that ballot's preference order.
The lowest-tallying candidate is eliminated, and those ballots are transferred.
The above two steps (4 & 5) are repeated until all seats have been filled.

Here are the upsides of this method:

All the power remains with the voters. Parties play no explicit role in deciding who wins.
Because districts have a relatively small magnitude of 3-5 seats, the effective quota to get representation is relatively high — 1/6 to 1/4 of the voters in a given district. This means that the method naturally tends to give few or no seats to fringe groups, unless those groups are geographically concentrated. While this could be seen as a downside from the point of view of maximizing diversity and minimizing wasted votes, it is an important upside in terms of making this proposal politically viable with existing partisan incumbents, as it would tend to reduce the disruption to the existing party landscape, especially in the short term.
In the US in particular, there is a long history of activism in support of this method, so it's relatively familiar to reformers.

Here are the downsides:

The bottom-up elimination process can prematurely eliminate good compromise candidates in some cases.
Because parties play no explicit role, the winners may not have enough incentive to build strong relationships (such as parties and coalitions), especially before they are seated. This may lead to an even less-coherent overall vision than you'd get from the negotiated platform of a big-tent party or coalition.
Voters can only choose candidates within their district, but must rank them all. This puts relatively high demands on voter engagement/ballot complexity, with relatively less effective voter choice than the two at-large-proportional methods I'll discuss (MMP and PLACE).
Voters who do not rank enough candidates may have their votes wasted. This may have disparate impact on certain groups (especially if, as Yeats says, "the best lack all conviction, while the worst are full of passionate intensity".)
Wasted votes, while substantially lower than the almost-50% they are under the status quo, would still be around 15% (a bit under one quota).

Interlude: Allocated Score

Allocated Score is an STV-like method which uses 0-5 star ballots (that is, voters can rate candidates independently on a 0-5 star scale, with tied and skipped ratings allowed), and without the need for bottom-up eliminations. This would remove the first downside above and slightly mitigate the complexity of voting; but it would also remove the last upside (familiarity).

OLPR (in 3-5 seat districts)

This is known as Open List Proportional Representation. Here's a blog post proposing it, by Jack Santucci, an expert in the US history of voting reform. Here's how it works:

As above, the state/province/country is divided up into districts (see above for discussion of terminology), each with enough population to elect 3-5 legislators. The number of seats in a district is known as its "magnitude" M.
Voters each choose one candidate from their district. Each candidate is either associated with a party, or independent. (The risk of wasted or inefficient votes is slightly higher when voting for an independent candidate.)
Parties are assigned a number of seats based on the number of "quotas" of votes for all their candidates combined. As above, one quota is 1/(M+1) of the total votes in the district. Each independent candidates are treated as being in a party by themself.
1. In case this leaves seats un-assigned, a least-remainder method is used to assign those.
A party's seats go to the candidates in that party in descending order of votes.

Here are the upsides:

Simple ballots and easy implementation.
Like STV/RCV5, because districts have a relatively small magnitude of 3-5 seats, the effective quota to get representation is relatively high — 1/6 to 1/4 of the voters in a given district. This means that the method naturally tends to give few or no seats to fringe groups, unless those groups are geographically concentrated. While this could be seen as a downside from the point of view of maximizing diversity and minimizing wasted votes, it is an important upside in terms of making this proposal politically viable with existing partisan incumbents, as it would tend to reduce the disruption to the existing party landscape, especially in the short term.

Here are the downsides:

Votes for smaller parties and/or independent candidates — those that get less than a quota — are wasted. Total wasted votes might be over 20%, and certain kinds of voters would be more likely to have their votes wasted. This can lead to some amount of bias/mis-representation, though probably far less than under FPTP.
Voters can only choose candidates within their district. This means less choice than the two methods below.

MMP (Modified-Bavarian style)

This is known as Mixed Member Proportional. In particular, the method I'll explain is based on the open-list MMP method used in the German region of Bavaria, with some STV-inspired modifications to further equalize voting power. Here's how it works:

Each state/province/country is divided into single-seat districts corresponding to 50%-70% of the total number of seats. The remainder of the seats will be assigned proportionally. (In the US case, existing single-seat districts could be used, and the House simultaneously expanded to 650 or more members to make room for the proportional seats.)
Ballots list all candidates in the state/province/country, with one column of candidates per party for the largest parties, and independents and smaller parties on the back. The local district candidates would be listed at the top of their respective columns, probably in a larger font.
- Yes, this is a lot to pack onto a ballot, but they manage 17 columns for 17 parties in Bavaria, so it's definitely possible.
- Note that the largest states/provinces/countries, such as California in the US or England in the UK, would probably divide into a few sub-elections of around 20 seats each, to keep the ballot from being much much too big.
Voters choose one local candidate, and one non-local candidate. These can be from the same or different parties.
In each district, the local candidate with the larga plurality is seated.
The correct number of seats for each party is calculated. Ballots which did not vote for the local winner are counted as one vote each for the party of their chosen local and non-local candidates. Ballots which did vote for the local winner are counted as two votes for that candidate's party (this first modification to the Bavarian method reduces the chances of "overhang" and thus the incentives for strategic clone parties). Ballots which are half-blank (only local or only non-local) are counted as two votes for the candidate and party on the other half.
Parties under a certain threshold (such as 5%) are eliminated and, for any of their votes that came from ballots voting for two different parties, those votes are transferred to the other party (this second modification to the Bavarian method reduces the amount of wasted votes.)
Each party's ideal number of seats is calculated using a least remainders method. The seats they've won locally are subtracted from those numbers, and the leftover seats are filled by any of that party's candidates who haven't won yet, in descending order of their total number of personal votes received.

Here are the upsides:

Voter choice is maximized; voters have a say in the winner among, and a choice between, not just the candidates in a 5-seat district, but those in their whole state/province/country.
The part of the method that elects local winners is essentially identical to the status quo, so incumbents have little to fear.
Almost all votes count fully and equally to help empower a party or parties of the voter's free choice; almost no partisan-wasted votes. Most also can help elect a member of the party sub-faction of the voter's choice; relatively few faction-wasted votes, at least compared to the two methods above.
Allows an adjustable party threshold, but sub-threshold votes are not necessarily wasted. A high threshold (though not what I'd choose in its own right) might help allay party insiders' concerns about excessive party fragmentation, allowing this to pass. (To be clear: Under this method, I'd far prefer a 5% threshold to a 20% one; but I'd also far, far prefer this method with a 20% threshold over the FPTP status quo. And even with a transferrable 20% threshold, I think this method would lead to fewer overall wasted votes than high non-transferrable thresholds, such as Poland's 5-8%, do.)

Here are the downsides:

This reform, as stated above, would require expanding the size of the legislature. While, at least for the US case, this is probably desirable in its own right from most points of view, it also increases the "surface area" for opponents to attack the idea on, reducing its viability.
Ballots are large. Though they're simpler for voters than STV/RCV5, they're still more complex than OLPR or PLACE.
Creating two different ways to win seats — locally and at-large — could lead to some legislators being seen as "second-class", wasting their talents.

PLACE voting

PLACE voting stands for "Proportional, Local, Accountable Candidate Endorsement". This is a voting method I designed in 2016 to show off what's possible in terms of mechanism design. I am serious about this in that I have done enough simulations and analysis to be confident that this method would work well, but I do realize that it's highly unlikely that a country of tens or hundreds of millions would adopt an untried voting method solely on my advice.

Here's how it would work:

Candidates run in specific single-seat districts, and are affiliated with parties. Before the election, each candidate may designate some of the other candidates from their party as "faction allies", and some of the candidates from other parties as "coalition allies". These pre-designations are public information and are available for voters to look up in voting locations.
To vote, you either choose one of the local-district candidates, explicitly listed on the ballot; just choose a party, without choosing a candidate from that party; or choose a party and write in the name and/or district of another candidate from that party.
Each vote is converted to a preference order, starting with the chosen candidate, then their faction allies, then other candidates from their party, then their coalition allies. Ties are broken by direct vote tallies. (Thus, the preference order follows the chosen candidate's predeclarations and party affiliation for the general distinctions, but other voters for the fine distinctions; this impedes detailed vote-trading agreements between candidates.)
Any candidates who do not get 25% of the votes from their local district are eliminated (unless that would leave fewer than 2 remaining candidates from that district).
Winners are chosen using an STV-like process, with the guarantee that there will be exactly one winner per district.
Each winner is assigned a "territory" which includes their district and possibly others, so that each district is in the territory of exactly one winner per winning party. Thus, even if your favorite party did not win in your district, you will have a representative from your party who's responsible to you.

Here are the upsides:

Ballots are at least as simple as anything above.
Almost no votes are wasted, and average voting power and choice is higher than anything above.
Using a threshold at the level of local votes for an individual candidate, rather than total votes for a party, allows for a high threshold while still keeping wasted votes very low.
- This high threshold discourages highly-polarizing or contentious candidates. Fringe voters still get a representative, but it's one who's shown at least some willingness to work with others in a viable party.
- This also discourages excessive party fragmentation, making the proposal potentially more acceptable to existing incumbents. Diverse opinions still get represented, but groups smaller than around 15-20% tend to be there as factions within larger parties, not as their own separate parties.
Predeclaration/delegation means that even losing candidates have an impact and voice, encouraging and aiding effective political organization of groups even smaller than one district-worth of voters.
It gives each voter "their own representative". (In fact, it does an even better job at this than FPTP.)

Here are the downsides:

It's completely untried.
It's pretty complex, with a lot of "moving parts" that can have non-obvious effects in combination.

TLD̦R ("Too Long, Didn't Read"; Summary)

I believe that politics in the US, Canada, and UK are broken — even more contentious and stuck than politics inevitably is — and that moving to a proportional representation method would help find more positive-sum outcomes. I've discussed four options, which have different relative advantages, both in terms of potential viability and of probable consequences. I've tried to focus this article on the concrete; if you want more abstract or theoretical discussions of voting theory, there's plenty in my other articles.

I have more hope, right now, of seeing at least one of these good options implemented in at least one of the countries I've mentioned in the next decade, than I have had at any time in the last 25 years I've been a voting reform activist.