New electoral systems for Canada

Warren D. Smith, Nov. 2015

PRELIMINARY

Justin Trudeau became Prime Minister in 2015 having pledged to replace Canada's first past the post voting system with something better before the next election, i.e. before 2019. Voting theorists do not employ the silly name "first past the post" (there is no post...) but rather call it "plurality" voting. It is widely agreed among those knowledgeable in voting systems that plurality is a very poor system. It solicits the least possible amount of information from voters ("name one candidate, then shut up"); it often encourages even that small amount of information to be a lie; it suffers from "wasted vote," "vote splitting," and "cloning" pathologies; and it has often yielded quite "disproportional" parliaments that failed to duplicate the true sentiments of Canadians. At the Du Baffy voting procedures workshop held in Normandy France in 2010, zero participants rated plurality acceptable out of 18 systems they considered.

But what should Canada's new system be?

Our purpose here is to outline possible options for Canada. We do not intend to restrict ourselves solely to ideas invented 100 or more years ago. We will try to state the best modern ideas. There is a science of voting systems, with many books and papers on the topic. Unfortunately the field contains many counterintuitive truths, aka "banana peels."

Canada now has the opportunity to lead, and serve as a role model for, the entire democratic world. But it also could severely damage itself, at worst becoming a world laughingstock. Many previous Canada-reform proposals have failed.

Table of contents

Taxonomy of choices

Of the below, I recommend choices I(A), II(C)1, or II(C)2. Those will be explored in more detail later. [Also II(C)3 and II(B) could be good, but I suspect not as good as those three.]

Single-winner in riding system (like Canada already has) except replace plurality with a better single-winner voting system, such as Score voting: each voter rates each candidate on a numerical scale (e.g. from 0=bad to 9=good); highest average wins. (A voter also may leave a candidate intentionally unscored; a voter who does that will not affect that candidate's average.)

This option is simple, easy, and a clear large improvement. (And that is good, since, as misquotations of the Hippocratic Oath inevitably begin, "First, do no harm.") A simpler, but lower quality, option would be approval voting. Neither I(A) nor I(B), nor indeed any deterministic single-winner method, can force proportionality. Nevertheless, I(A) both should allow "third parties" to win and cause Canada to outperform every single-winner-based democracy during the last 300 years (the most obvious examples being USA, UK, Australian House, and India). Proportional representation (PR) systems. These are more complicated, less understood, and would be a bigger change for Canada than, single-winner-in-riding systems. Countrywide "party lists." I do not recommend this option. Essentially, ridings would be abolished, and instead all MPs would be elected nationwide, and each voter would vote, not for individual candidates, but rather for parties. Each party would then be awarded a number of seats proportional to its vote-share (up to errors forced by the need to round to integers; there are several rounding-off methods, one of the best being Sainte-Laguë's). We then fill those seats by proceeding in order down a list pre-provided by that party. The failed state of Germany, during the period between World Wars I and II, used this approach; and later it was adopted by Israel, which also hasn't exactly been universally lauded. Those are two countries Canada should not try to emulate. On the bright side, this approach seems to have worked OK for the Netherlands. But Canada, as the geographically second largest country with 240 times the land area of Netherlands, probably would be insane to eliminate regionality.

It also would be possible (although I also would not recommend this) to abolish ridings and replace party-lists with the candidate-based asset voting PR system. "Improved Ireland." Ireland elects its Dail (parliament) by multiwinner elections within each riding. Specifically, each Irish riding elects 3 to 5 MPs, usually 3. (Originally the range was 3-to-9, but the unwieldy 9-member ridings were later split up, so that today it is 3-to-5, and probably Ireland is moving toward all-3s.) Ireland for some time appeared to be one of the best, perhaps even the best, performing country in Europe... until a giant bank crash plunged their country into a hole from which they won't be able to emerge for a generation. The PR system Ireland uses to elect the 3 (or 5) winners from each riding is the "Hare/Droop STV system." Variants of this system have also been employed in Malta and for the Australian Senate, in both cases seeming to work worse than for Ireland. 98% of Australian Senate voters do not provide their own rank ordering, but rather simply regurgitate a pre-provided party-recommended massively-strategic ordering, thus mooting the entire goal of the system. This is probably since their 12-winner ridings, featuring 50-120 candidates, make it nearly impossible for voters to provide a full ordering. In Malta, despite "PR," no independent candidate has been elected since 1950 and it's massively 2-party dominated. Ireland has settled into fairly stable 3-party domination, albeit 10 parties hold nonzero numbers of seats as of late 2015, including 18 independent MPs among the 166 total seats, which if "independents" were a party would make them the 4th most powerful. This suggests the Irish system disproportionately hurts small parties, but nevertheless allows independents to survive. A better and simpler modern replacement for Hare/Droop STV would be optimum harmonic-voting and should assure outperforming Ireland. "Improved Germany and New Zealand." Today's Germany and New Zealand both employ mixed member proportional (MMP) systems, also called "top up" systems. The idea is two-stage. In the first stage, MPs are elected from ridings using a single-winner voting system. The resulting parliament probably would be disproportional, i.e. parties' seat-shares would differ from their vote-shares. In the second stage, a pre-specified number of additional seats would be filled by a new kind of MP – "top-up," or "nonlocal," MPs – elected in a manner designed to give under-represented parties additional seats to bring the parliament back to proportionality (or at least, to approach it more nearly). By replacing Germany and New Zealand's outdated first- and second-stage methods by more modern ones, we can obtain several new MMP approaches simpler and/or better than theirs: Electing local MPs via score voting and top-up MPs using harmonic voting, with asset voting as the top-up method, with open party lists as the top-up method (agglomerating independent candidates into a pseudo-party to try to remove unfair bias against them). With correct "tuning," those should assure outperforming Germany and New Zealand. "Double proportional" approaches involving "linear programming." In this system – which is entirely new and has never been tried in any country – ridings still elect single winners, but candidates receive differing numbers of "bonus votes" that depend on their party and riding. The riding-bonus and party-bonus amounts are chosen by an algorithm which (a) assures that the winners (based on both genuine and bonus votes) in each riding result in a proportional parliament, and (b) does so in an "optimum" manner, i.e. among all possible parliaments meeting both the party-seat-share and 1-MP-per-riding constraints, the one rated the highest by voters (based on genuine votes only), is used. Unfortunately, I have reason to believe that any country using such a system would be devastated by a new kind of strategic voting I call "targeted killing." I therefore do not recommend methods of this class. Approaches based on integer programming, or on mixed integer linear programs (MILP). Ideas of this nature have been suggested by Monroe and by Ebert. So far, all proposals of this nature seem utterly computationally infeasible to solve exactly, and/or succumb to embarrassing simple election bad-behavior examples. I therefore cannot recommend any of them as they presently (Nov. 2015) stand. But it remains possible that some future development in this line of thinking, will force me to alter that judgment. And it does seem feasible to solve some of these formulations approximately – if it were deemed politically acceptable to produce "a non-optimum parliament which is at least 64% of optimum and likely 96-99%" via an extremely complicated and messy and highly arbitrarily alterable algorithm. Ideas based on "weighted MPs." That is, make the MPs have unequal voting weights in parliament. Their weights are somehow based on how much support they each got during their election. This enables a finer degree of proportionality than possible with all-equal MPs, but at the cost of sacrificing some simplicity and perhaps making the everyday lives of parliamentarians more painful. E.g: what should we do about the salaries and staff-sizes of MPs with different voting-weights? How does this interact with parliamentary rules (and how should they all be redone?). What new avenues does this unfortunately open to "game the system"? The weighted-MPs idea potentially is employable in many combinations with other ideas.

How can we judge which methods are "better"?

Simplicity. Simpler systems (both to describe, understand, count, and vote with) are preferable, all else being equal. I think a complete definition of the method (and from which its most important properties should be easy to see), must fit on one side of one page. Simplicity and comprehensibility can be assessed via experimental testing with human subjects (error rates and speed can be measured) – and sometimes the results disagree with people's naive guesses!

Expressiveness. The more information a voter can provide, the better. Plurality voting is extremely bad in the sense that the voter provides nearly the least amount of information she possibly could ("name one candidate, then shut up and say nothing else"). The best ballot-types are score and approval (with score being more expressive, but approval being simpler). Interestingly, this 26 Nov 2015 screen shot reveals that Canada's liberal party already had discovered (and used!) score voting for itself.

Historical test. One can try to examine the history of different countries to see which systems have worked better or worse in the past. That's an excellent mission, but it can be problematic because of subjectivity, paucity of data, and lying. But one important question which can be answered pretty objectively for any particular country, is whether a system tends to collapse entirely, devolve into 1-party rule, or sink into permanent 2-party "duopoly" (with "third parties" and/or independents effectively killed or maimed). Obviously these possibilities have been listed worst-first, but all three are very harmful to ideals of democratic choice. For present purposes I like the approach of designing a voting system highly analogous to some actual country's, except better in certain respects (e.g. property-theorems, simplicity). Then we can argue with some confidence that the new system should outperform that country.

Biological test. Certain social insects, e.g. honeybees, are the only known animals that employ a (single-winner) voting system for a very crucial purpose – choosing the location for the next year's hive. These creatures employ a highly refined algorithm essentially implementing score voting. That strikes me as a very strong endorsement of it, since it is based on hundreds of trillions of elections (far more than humans, or even their computer helpers, have ever done) judged in a very cruelly objective way: the hive survives, or not.

Properties. Theorems can often be proven saying a voting system obeys (or fails to obey) some simple and desirable-sounding property. Quite often, naive advocates of some voting system are startled when confronted with some failed-property example, aka "paradox." They then often claim (with no evidence besides their – obviously poor – intuition) that paradox is uncommon, or will not matter much in "real life." But they often are wrong about that too. Here are some important properties:

Monotonicity. If a voter raises the score of a candidate on her ballot while leaving all else unaltered, that should not hurt that candidate, i.e. should not convert his election into a loss. Similarly a voter by lowering a candidate on her ballot should not be able to cause his election. (Seems obvious... but several proposed methods, including the STV and IRV systems used in Ireland, Malta, and Australia, disobey this.) Participation. By voting honestly, you should not be able to make the election result worse (from your perspective) than if you had not voted at all. Without this or without monotonicity, the meaning and usefulness of voting become questionable, plus each time such a paradox-election occurs, it is quite embarrassing. (Again, disobeyed by STV and IRV in Ireland, Malta, and Australia.) Favorite-safe. It ought to be "safe" to vote your honestly-favorite candidate top, in the sense that there is always a way to do so which will not worsen the election result (from your perspective) versus if you had lied by pretending somebody else was your favorite. (Disobeyed by plurality voting. Disobeyed by IRV and hence by the STV systems used in Ireland, Malta, and Australia. Obeyed by score and approval voting.) Cloneproof. Suppose a "clone" of some candidate enters or leaves the race. (Voters regard clones as virtually identical.) That should not alter the winner (except by replacement by clones). Without this, elections become readily manipulable by, e.g, simply sponsoring clones to run, thus ensuring defeat for any targeted candidate. Plurality and Borda fail this test. Precinct countability. It should be possible for precincts to publish compact "subtotals tables" such that the entire countrywide election can be counted by using those precinct tables, alone, as its input. (Again, disobeyed by IRV & STV in Ireland, Malta, and Australia; those systems inherently must be counted at a single central location, diminishing transparency and increasing the risk of fraud conspiracies.) NESD property. Suppose voters employ the following "naive exaggeration strategy" (NES – which is highly appealing and appears to be used by about 85% of Australian voters): they (1) identify the top two parties, (2) score them on their ballots MAX and MIN, which is an exaggeration of their true feelings, (3) score all remaining parties in some manner. If, when nearly all voters use NES, that forces 2-party domination, then the voting method fails NESD property and conjecturally causes high risk your country will fall into a permanent 2-party-domination (2PD) trap. Many proposed (and used) voting methods fail this test, including plurality (which indeed led to massive 2-party domination in the USA) and instant runoff voting (IRV, which indeed yielded massive 2-party domination in the Australian House), and, to perhaps a somewhat lesser extent, STV (which indeed led to massive 2-party domination in Malta). NESD-failure tends to be much harder to avoid if rank-ordering ballots are used, than if score-type (rate the candidates) ballots are used. Thus score and approval voting both pass the NESD test. That is because, even if 100% of voters apply NES, then with score and approval it is still entirely possible for a third party to win seats (indeed, in principle, even all of the seats). No country that has used score or approval voting is known to have suffered 2-party domination, albeit 2PD seems less likely with score than with approval. Biases in favor of extremists or centrists. This isn't really a precisely defined "property," but computer experiments with voting methods often pretty clearly show they tend to favor extremists (IRV & plurality), or tend to favor centrists (approval voting with some voter-behavior models). It seems best if the method has little or no such built-in bias (score voting).

Computer simulation test. Several studies of single-winner systems have been done via computer simulations of elections. A quality measure called "Bayesian Regret" (BR) can be measured in this way in widely varying situations controllable by the simulator (e.g. one can adjust the numbers of candidates, voters, and issues; the utility models; voter "honesty- and strategy-levels"; and other "knobs"). The result of my original such study (in 2000) was that score voting outperformed all other commonly-proposed rival single-winner systems, and did so highly "robustly" i.e. regardless of the knob-settings. This reinforced the findings of the honeybees.

For comparing multiwinner voting systems – necessary to allow proportional representation – however, no Bayesian-regret computer study has yet been done. (In principle I know how to do such a computer-aided study, and expect it would take me ≤1 year of work; but as of year 2015, nobody, including myself, has actually done this.) And there are no known biological examples. Therefore, we are for the present forced back to the older methodology of property-theorems and historical examination. And even those have been done quite poorly.

Some "tunable trade offs"

PR versus single-winner: two competing philosophies. The ideal of proportional representation is that if the parliament perfectly mirrors the population of a country, then any binary decision that country as a whole would make in a (usually impractical) referendum, automatically would be the same decision the parliament would make via a binary vote.

That's an excellent ideal. But in practice, no PR system so far in the history of the world has ever managed to create a parliament that perfectly mirrored its people. Usually looking at some simple characteristics such as gender, economic situation, and experience and expertise in various areas, quickly reveals large differences between the parliament and whole population.

The best that PR systems so far have managed to do is to duplicate the party composition of the population. That's a crude and inadequate measure of similarity. (And for that reason it is pointless to "overdo" PR by trying to assure proportionality to within ±1% accuracy, say. Other error-effects will dwarf that 1% rendering this a fool's mission.) PR systems based on explicit party labels can guarantee this. But it also is possible to design PR systems which are entirely candidate-based, i.e. ignore party labels (if any) and still work just as well even if no political parties exist. Those systems' claims to be PR depend on "proportionality theorems" which say that if the voters behave in specified (usually rather extreme and "racist") ways, then the system guarantees electing a parliament whose "color" composition is the same as the electorate's (up to small errors forced by integer roundoff, and assuming enough candidates of whatever color are available so that we do not run out). In these theorems, it does not matter what "color" actually means. It could be anything. If it happens to be "party label" (i.e. if the voters are "partisan" not "racist") these systems would assure PR in the old sense of party composition; but they also would assure proportionality in any other sense – gender, militarism, whatever corresponded most closely to whatever voters cared about most intensely in that election.

If, on the other hand, the voters do not all vote in the "maximally racist" manners assumed by the proportionality theorems, then it becomes unclear what, exactly, proportionality even should mean, and these theorems do not really apply. PR advocates then hope that the system because it works well for all extreme voter behaviors, also presumably produces good results for more-realistic less-extreme intermediate voter behaviors.

For example, an extreme voter might claim her party was wonderful while all others were horrible. A more realistic assessment would be that the other parties were worse, but had varying quality levels. If all voters are extreme, then it is easy to assess the party composition of the electorate and verify it matches that of parliament. But there is no simple and good way to pigeonhole more realistic voters. (Is she "70% Tory and 30% Liberal"?).

Meanwhile, the opposed philosophy of single-winner has a different ideal: try to elect the best candidates, based on quality-assessments by the voters. That also is an excellent ideal!

The flaws of the PR ideal are most dramatically exhibited by the "closed party list" PR systems, in which all votes are for parties, not candidates, and the list-orderings used by each party are chosen by party bosses in "smoke filled rooms." In those systems, individual accountability almost vanishes. It is virtually impossible to kick a corrupt party boss out of power.

But the single-winner ideal also is flawed. If 51% of voters are Republican, this could result in a 100%-Republican parliament, which would be quite disproportional. Small parties with important views (such as the Green Party as defenders of the environment) could be shut out of power entirely.

So which is better: PR countries, or single-winner-based countries? That debate has raged for 100 years. Attempts have been made to settle it by collecting economic and health statistics from different kinds of countries, but successful counterattacks have been made against all those attempts. Furthermore, there is plenty of room to improve the designs of all the PR systems, and all the single-winner systems, versus the designs that unfortunately have been used by all countries during the last 300 years. So even if someday, say, PR were judged superior, then somebody could justifiably argue that, if only the single-winner countries had used a better single-winner voting system, then they would have been superior! So in my opinion this debate remains unresolved.

I believe the best practical answer is to try to design a "tunable hybrid" system which incorporates both principles and in an adjustable way (so we can "turn a knob" to make the system "more PR").

Regions versus parties. Another (related) conflict is the question of what MPs should be representing. Should an MP represent a geographic region? That has been the view in Canada, at least up to year 2015. Or should he represent a party or ideology (more the view in Israel)? Norway's solution is to give each inhabitant one "shekel" and each square kilometer 1.8 shekels, then MPs each represent an equal number of shekels! While I happen to think both views have merit, I think that the region-based countries have tended to exhibit better overall historical performance. Parties often are regarded as a bad thing. (In the early USA, George Washington famously railed against their appearance as a harbinger of doom. Strong measures were undertaken in renaissance Venice to try to prevent their appearance.)

A good reason for the anti-party view is that if every MP were to reach their decision independently that ought to cause high-quality decisions. Parties tend to abolish independent thinking, yielding lower-quality decisions.

Simple model in which this is provable: Suppose 100 people independently assess the bias of a coin via tossing it once, and then vote on whether the coin will come up heads, or has a pro-head bias. Either way, they will get a better chance of correctness than if 4 people assess it and each get 24 others to copy them.

Unfortunately parties offer candidates substantial advantages in campaigning and in battles involving parliamentary rules, hence it is probably impossible to abolish them. But that makes it desirable for voting system designs to try to counteract party-forming and independent-destroying tendencies.

Financing. Money corrupts. It's historically exerted a huge corrupting influence on politics. So it is desirable for voting system designs to try to counteract and diminish its importance.

Plans [such as "II(B) improved Ireland"] involving enlarged ridings would increase the cost of campaigns. Also, designs like the USA's plurality voting system that make it useful for candidates to create the illusion that they are highly likely to win, increase the importance of money. Creating this impression is expensive, but with plurality voting it then becomes a "wasted vote" to vote for anybody who is not among the top-2 most-likely-to-win. So voters avoid it. (Similarly for money donations being "wasted.") That gives anybody creating such an illusion a large artificial advantage, causing votes to be a huge distortion of popular sentiment. With score voting, on the other hand [I(A)], voters can honestly give a candidate a high or low score, not caring whether said candidate seems likely to win. So with score voting, there is reason to hope that high-quality candidates can win even with little money; that is almost impossible with plurality voting. Money is thus inherently less important under score voting.

Now let's examine some system proposals using those techniques

Enough abstract principles. Let's start designing and analysing voting systems.

I(A): Score voting to elect MPs within ridings.

Each vote consists of a numerical score within some range (most simply 0 to 9) for each candidate. Voters may also indicate "X" or "NO OPINION" if they wish not to express any opinion about that candidate. Such votes don't affect that candidate's average. The candidate with the highest average score wins. Some people worry a little-known candidate could organize a band of fanatics to all give him 9s, while the other voters would all rate him "no opinion" causing Hitler to win. This actually is not a concern because it is an empirical fact that about twice as many voters "play it safe" by giving unknowns 0s than give them "no opinion." But if this actually had been a problem, then we could solve it by pre-agreeing that each candidate is to be awarded T fake votes using score S (for some pre-agreed values of S and T such as S=2 and T=1500) before the election begins. The highest average based on both the real and fake votes, wins. This is fair since S and T are the same for every candidate.

This is a very simple and easy change for Canada. Surveys suggest it would (if put to Canada's voters via a referendum) be enacted right now by about 60-40. Meanwhile every other voting-system reform Canada tried to enact by referendum was more complicated and failed massively by votes of 64-36 (Prince Edward Island 2005), 63-37 (Ontario 2007), and 61-39 (British Columbia 2009).

This system would not be proportional. It would simply be a better single-winner system. It would satisfy cloneproofness, monotonicity, participation, favorite-safety, NESD property (hopefully avoiding 2-party-domination trap), and would reduce the importance of money. It's precinct-countable. It has little or no pro-extremist or pro-centrist bias. It's highly expressive, but at the same time allows voters who wish intentionally not to express an opinion about one or more candidates, to do so in a non-distortionary manner. It has been heavily tested by honeybees, over hundreds of trillions of elections and tens of millions of years. It also has performed excellently in computer simulation quantitative Bayesian Regret (BR) testing, which indicates the improvement got by adopting it would be comparable to, or exceed, the improvement got by inventing democracy in the first place. That is, BR with score voting should be about 5 times better than with plurality voting, which in turn is about 2 times better than "random winner" (a crude model of non-democracy). Score formed the basis of government in ancient Sparta and renaissance Venice, both of which lasted longer than any modern democracy, despite apparently-tougher challenges and worse circumstances.

I repeat. The improvement got by adopting score would be comparable to, or exceed, the improvement got by inventing democracy in the first place.

I(B): Approval voting to elect MPs within ridings. This is a simpler but cruder version of score voting in which a voter can only give one of two allowed scores to each candidate: "thumbs up" (1) and "thumbs down" (0); no intermediate scores are permitted. (Also called "approval" and "disapproval.") The candidate with the greatest approve/disapprove ratio wins their riding.

This enjoys all the same properties as score voting with finer scales, but: it's less accurate, less expressive, about twice as bad Bayesian regret, and empirically less favorable for small parties, hence more likely to fall into a 2-party domination trap. Approval was the voting system most-approved (!) by the participants at the Du Baffy voting procedures workshop held in Normandy France in 2010. (Incidentally, I was not one of those participants and I do not exactly think they should be regarded as the last word. But still, their view that approval is top, and plurality worst, among the 18 systems they considered, is a good starting point.)

Both approval and score are simple changes that would be clear large improvements over Canada's present system.

Some alternative single-winner-in-riding suggestions that have been made in Canadian media, but which I disparage, are to use IRV (instant runoff) voting (which employs a rank-ordering ballot), or to use two-round plurality (the second round, held at a later date, is a "runoff" between the two top finishers from the first round). Two-round systems force Canadians to vote twice, which is more expensive and annoying. IRV fails precinct-countability, monotonicity, participation, NESD, favorite-safety, and is pro-extremist biased; and it has a less-expressive-than-score ballot which takes voters longer to fill out than either score or approval ballots, does not permit voters to express "no opinion" about a candidate (at least not without doing so in an inherently massively distortionary way) and causes greater voter error rates.

PR systems all necessarily are more complicated than, and more of a change for Canada than, and also currently less well-understood than, single-winner systems can be. But if you are willing to accept that greater complexity and risk, then you can enjoy the (probably real) benefits of proportionality. Further, the proposals we shall outline below have better simplicity and/or quality than all previous PR-system designs. Indeed, the systems below will be designed to be hopefully-clear improvements over Germany & New Zealand (IIC) and over Ireland (IIB).

II(C)1: Score voting to elect MPs within ridings, followed by harmonic-voting-based "top-up" stage to restore proportionality within regions.

Divide Canada into "ridings" and "regions," where each region consists of 13 ridings. Each riding elects a single local MP using score voting just like I(A). Each region elects 5 top-up MPs chosen (by exhaustive consideration of all possible choices) so that its total set of 18 MPs – that is, the 13 local plus 5 top-up MPs – maximizes the harmonic quality measure:

Q = ∑ 1≤v≤V ∑ 1≤j≤W (jth greatest S v,c among winning c) / (j-½)

Here V is the number of voters, C is the number of candidates, W the number of winners (0<W<C; for us W=18), and S v,c is the score given to candidate c by voter v on her ballot.

The magical point of this quality formula is that maximizing Q guarantees proportional representation, and in about the simplest conceivable manner. (Worried that formula seems a bit complicated? Well, when you think about it, any PR method needs some formula somewhere, and this is pretty nearly as simple a formula as anybody could hope for. So you really could not ask for anything much simpler.)

As far as each voter is concerned, she simply provides a score (on an 0-9 numerical scale, if M=9) for each candidate in her riding, i.e. each ballot is no more and no less complex than scheme I(A).

The number of seats in parliament would need to be a multiple of 18; as of year 2015 it currently is 308 seats while the nearest number divisible by 18 is 306=18×17. The total parliament then would consist of 13/18=72.2% local MPs and 5/18=27.8% top-up MPs, which (based on historical data) should be enough top-ups to achieve proportionality in at least 50% of elections. The other 50% (or less) of the time, the parliament still would exhibit some disproportionality, albeit much reduced versus the present system.

Why the "magic numbers" 5+13=18? I actually suspect (5,13) is optimum, i.e. any other integer pair (x,y) is worse. First of all, we could not make x bigger than 5 (maybe 6 or 7 if you went to a superduper computer, but I prefer to avoid that; even my circa 2005 desktop with single core definitely could handle 5,13 by brute force exhaustive search), because that makes the computation too heavy. (For example, using harmonic voting at nationwide scale would become computationally infeasible.) Now, staying with x=5, if we went to y>13 then the percentage of top-ups would be too small, so the parliament would fail to reach proportionality too often. I also think y<12 is a bad idea since then there would be too large a percentage of top-up MPs; it is best to keep them riding-based as much as possible to maximize quality while keeping proportionality ok. The choice between y=12 and y=13 is somewhat debatable. If we used 12 we'd get a bit higher top-up fraction allowing more disproportionality to be topped-away; but would sacrifice small parties somewhat – a party has to get 1/(x+y+1) of the votes, roughly speaking, to be assured seats. I considered the latter the greater price for Canada to pay, so I went with y=13, assuring any party with >1/19≈5.26% support will get proportional seats (such as the Greens in 2008; and often less support would suffice due to "Sainte-Lague small-number effects," e.g. often 3% support would suffice).

Later note: with score voting used to select local MPs instead of (as in Canada's prior history) plurality voting, it is plausible that fewer top-up MPs will be needed to obtain proportionality, than what appears needed based on prior-historical data. In that case (4,12) or (5,20) might be superior choices.

Also, I should remark that many believe proportionality within regions is actually better than nationwide (even though, naively, this sacrifices some PR accuracy). Why? Well, there are psychological speculations, which probably contain truth, about the benefits of regional identity/responsibility... but also, fully objectively, there is the "rain problem." That is, suppose the Liberals live mainly in the East. And it rains in the East on election day, depressing turnout there. Result: Canada suffers an unhealthy liberal-deficit purely because of random weather! But with regional proportionality, not nationwide, regional rain becomes irrelevant.

Optional modification: We could allow voters to score the candidates from the 12 other ridings in her region too (listed in same-party bunches, with "independent" regarded as a pseudo-party; you'd be allowed to coequally score all members of an entire-bunch at a time, if you did not want to score all the individuals); those scores would not affect the local-MP elections, but would affect the harmonic-scores for region-wide top-up MPs. Adding this feature unfortunately would make ballots a factor 13 larger, which might not be a price worth paying.

EXPLAIN PEREIRA'S BALLOT SCHEME FOR THAT?? PROS & CONS??

II(B): Optimum-harmonic-voting within multiwinner ridings.

Ridings would have K times their present populations (and there would be a factor K fewer of them) each electing K, not one, MPs. Here 1≤K≤5, usually 3≤K≤5. These K winners would be chosen within each riding by optimum-harmonic-voting.

Discussion: This II(B) scheme seems to me clearly worse than II(C)1 because it offers worse proportionality accuracy, enlarges ridings 3-5 times hence increasing campaign costs thus increasing the importance of money and decreasing regionality, and it makes ballots 3-5 times larger. Nevertheless, since optimum harmonic voting is better and simpler than Hare/Droop STV, II(B) still ought to be good enough to assure that Canada would outperform Ireland!

II(C)1 and II(B) both have the disadvantage that they would require centralized counting done by computer – albeit in II(C)1 the local MPs could still be counted in precincts and even (if desired) manually; only the top-up MPs would need to be found by computer. Both schemes are extremely simple to define, and extremely simple, especially II(C)1, for the voter. They both enjoy monotonicity, cloneproofness, and NESD properties.

Proofs: With II(C)1, any voter by raising her ballot's score for candidate X, can elect him as a local MP, and, whether or not that happens, all 18-member winner-sets containing X will increase their Q, while all those not containing X will have unaltered Q. This proves that the voter, by raising X's score on her ballot, can help elect X but cannot hurt him – i.e. proves monotonicity. To prove cloneproofness, suppose a clone Y of X enters the race. Then all 18-member winner sets containing X and not Y, or Y and not X, or neither, will have unaltered Q scores, and all candidates will have unaltered average scores. Hence the same winners will arise as before Y entered the race (up to possible replacement of X by Y) unless X was a winner and now both X and Y are winners. That exceptional case could never arise with a single-winner voting system and is only possible in multiwinner voting systems, which makes it unclear what we should call this, since the inventor (N.Tideman) of cloneproofness had only defined the concept in the single-winner case. Anyhow, it's cloneproof as far as that definition had gone. To prove NESD, note that even if 100% of voters exaggerate the two major parties' candidates to MAX and MIN in a naive attempt to maximize the impact of their vote, then still a third-party candidate, even one who never receives the top score 9, only 8s, can win either his local MP seat, or a top-up seat (both are possible). Indeed, it even would be mathematically possible for 100% of both kinds of seats to be won by third parties, if enough voters gave the third party candidates 8s while giving the major party candidates 9s and 0s in approximately equal proportions. [With II(C)2 the proofs are analogous but easier.]

There is a simple and quite understandable measurable sense – maximizing the quality function Q – in which both II(C)1 and II(B) yield "optimum" results. Further, in II(C)1 the local MPs enjoy participation and favorite-safety properties too, as well as lack of any noticeable built in pro-extremist or pro-centrist bias. And both would make it beneficial for MPs to increase voter turnout within their own ridings, unlike (e.g.) the 2015 USA where recently Republicans have intentionally tried to discourage voter turnout.

In my view II(C)1 has only two disadvantages. And the following two schemes II(C)2 and II(C)3 remove them! Namely:

Heavy computation is no longer needed

Precinct-countability is entirely restored.

But those two gains come at a cost:

For II(C)2 the costs include both increased complexity, and a (dangerous?) weakening of the NESD property. II(C)2 as a consequence of its use of "asset voting" for its top up instead of optimum harmonic voting offers the interesting possible new benefit of superior proportionality due to negotiated seats-for-concessions deals, which can cause the effect of "fractional seats" worth of power. That can be better than any "conventional" PR scheme based on whole-number seat counts only. But some people worry that those deals might create more problems than they solve. [Another worry is the perception of possible problems, even if not real. As Justice Gordon Hewart famously wrote, "justice should not only be done, but should manifestly and undoubtedly be seen to be done."]

proportionality due to negotiated seats-for-concessions deals, which can cause the effect of "fractional seats" worth of power. That can be better than any "conventional" PR scheme based on whole-number seat counts only. some people worry that those deals might create more problems than they solve. [Another worry is the of possible problems, even if not real. As Justice Gordon Hewart famously wrote, "justice should not only be done, but should manifestly and undoubtedly be to be done."] For II(C)3 the costs include explicit use of (and inherent need for) party-labels – which is a kind of PR that I consider inferior to candidate-based PR systems like asset and optimum-harmonic – and there is risk of interaction between the party- and candidate-choices causing strategic-voting problems (i.e. system could encourage voters to lie).

In view of that, my guess is II(C)1 is, over all, the best PR choice for Canada.

II(C)2: Score voting to elect MPs within ridings, followed by an asset-voting-based "top-up" stage to restore proportionality.

Here is a separate web page describing the "score+asset hybrid" system, its properties, and listing criticisms & responses.

II(C)3: Score voting to elect MPs within ridings, followed by open-party-list-based "top-up" stage to restore proportionality.

Here's the plan to elect a W-member parliament of which W-T seats are local MPs and T seats are top-up MPs:

1. Each voter rates each candidate in her riding, on an 0-to-9 numerical scale. (Voters also allowed to leave some candidates unrated, if that voter wishes not to express any opinion about that candidate.)

2. In each riding, the candidate with the highest average score wins that riding's MP seat.

[At this point we have elected W-T of the W seats in Parliament, each via the excellent "score voting" single-winner system. The "top up" now commences, whose mission now is to fill the remaining T seats.]

3. We now compute how many seats each party "deserves" in the full, "topped-up" parliament. This is done by the Sainte-Laguë method: each party's total vote V is the number of voters who rated their candidate top (if a voter had rated K candidates coequal top, then each party gets 1/K share of her vote). We successively assign seats as follows: whichever party has the greatest V/(2S+1) wins the next seat (provided it hasn't run out of candidates). Here V is the votes owned by that party, and S is the number of seats that party's members currently occupy (initial S-values arise from the seats elected in step 2, and increment each time we award a new seat to that party). We keep going in this way until each of the T top-up seats has been assigned a party label. [Each candidate had pre-specified his party label. Parties can expel people by 2/3 supermajority vote.]

[At this point we have elected W-T of the W seats in Parliament, and assigned party labels to the remaining T seats. We now must assign actual people to those T seats.]

4. Each party's candidates are sorted into decreasing order by their average scores they received from the voters in their ridings. We simply elect, from each party deserving S top-up seats, the top S as-yet-unelected members of their list. [Note: independent candidates are regarded for the purposes of steps 3 & 4 as members of the "independents' (pseudo)party."]

Discussion: II(C)3 is very similar to the MMP systems currently used by Germany and New Zealand, but with the following differences, all of which seem to be improvements:

Score voting, not (which Germany and New Zealand use) plurality voting, within ridings. The use of an "independents' (pseudo)party" for Canada to try to redress unfair pro-party anti-independent biases. These have afflicted Germany & New Zealand so severely that New Zealand has never elected any independent MP ever since they adopted MMP in 1996, while Germany elected three in their first MMP election in 1949 and zero ever after. Germany's Bundestag's 1:1 ratio of local:top-up MPs seems top-heavy. I suggest for Canada more like an 8:3 ratio. (Or 13:5 or 18:7.) Optionally: we could add "regions" to the Canada II(C)3 plan. Simply divide Canada into regions, each electing only a portion (based on its population) of parliament. The system we just described is used separately within each region, and then all the resulting sub-parliaments are combined to obtain Canada's total parliament. In this way the proposed scheme for Canada can be made "tunable" by adjusting the number of regions, as well as by adjusting the relative numbers of ordinary and top-up seats. These two "knobs" tune how much regionality to have and how much PR to have. Many people believe MPs that exist without regions, are inherently worse. Whether or not you believe that, a second argument for why regionality could be good, is the rain problem.

So if Canada adopted II(C)3 and tuned it decently, it should become similar to, but better than, Germany and New Zealand.

The most obvious defect of this II(C)3 proposal (at least in my view) is its reliance on party-labels to provide proportionality. If, say, parties did not exist, everybody ran independent, but some candidates were similar and others dissimilar, then this system would be unable to know anything about those similarities and hence would become totally helpless about proportionality. Another problem is that parties can contain subfactions. For example, the USA Republican party as of year 2015 is really two parties masquerading as one, according to Warren Buffett in several media interviews. (In 2008, Buffett was the richest person in the world, and his father was a Republican congressman from Nebraska.) If PR is a good idea, shouldn't it also be a good idea for "effective" parties as well as "real" parties? You cannot get that effect with party-list PR, but can get it with party-free PR schemes. Also, by explicitly inserting party names into the voting system definition, we become inherently unable to treat independent candidates fairly, and also this system definitely will be severely vulnerable to candidates "gaming the system" by switching parties, creating ad hoc parties, etc. (As a partial cure for that, we create a pseudo-party consisting of all the independents. This is misleading in the sense that independent candidates can be quite dissimilar. But it will counteract some of the global unfairness against independents.) To prevent "gaming the system," it probably would be felt necessary to enact laws forbidding candidates from switching parties, but any such anti-switching laws themselves seem quite damaging! (Remember, the ultimate cure for "too-whipped" parties is for unhappy MPs to be able to switch parties.)

Another problem with this II(C)3 system: it can be poor voting strategy to honestly top-rate your favorite. If you did, then you might be helping some party to get more top-up seats, except that your vote would be "wasted" for that purpose since they were doomed to get none, or doomed to be unable to rise from X to X+1 seats in your region... hence you would have been better off lying about your favorite to help boost some other party's seat count. The more lying happens in votes, the worse off Canada is going to be. Those problems do not afflict II(C)1 and II(C)2.

Bottom line: I suspect II(C)2 and II(C)1 are better – but even II(C)3 should assure outperforming Germany and New Zealand.

Miscellaneous other Good Ideas or Silly Gimmicks (take your pick)

Forced gender equality? The Socialist Party of the USA, has 2 copies of every (or nearly every) office, and demands each be occupied by 1 man and 1 woman. It accomplishes this by holding separate elections for the woman- and man-occupant of that office. (All voters, regardless of gender, vote in all elections, but all candidates for any particular seat are of the same gender, since each seat has a pre-assigned gender-label.)

There have been countries (e.g. New Zealand, Fiji) where certain numbers of seats are pre-agreed to be filled with people of only one ethnic type. But, oddly enough, as far as I know no country has ever done that for gender despite immense amounts of propaganda worldwide for both "equal rights for women" and "proportional representation."

But note, "ethnic type," "religion," etc are not clearly defined. "Gender" is much better defined, and also happens to come out naturally in nearly 50-50 proportions. So I would not recommend ethnic allotments. If I were going to recommend any such allotment (and I'm not saying I am), then I'd use gender.

Synchronized terms? Norway makes everybody's life easier and cheaper by insisting all government offices (at all levels of government) have 4-year terms and all be elected in synchrony. If we insist on unequal term lengths, then still we could demand they all be multiples of 2 years, then hold elections only in even-numbered years.

"Optimal" legislature size? Canada presently tries to have about one MP per 100000 inhabitants. If it were to grow to the same population as the year-2015 USA (320 million) then its parliament would need 3200 seats, which would be ridiculously large, about 5 times the size of the next-largest parliament on the planet. (The USA's house size has been fixed at 435 seats since 1929.) There are good reasons to suspect that it is optimum for the parliament size not to grow proportionately to a country's population, but rather as some power μ of that population with 1/4≤μ≤1/2. ACE (Administration and Cost of Elections project started by IDEA, IFES and UNDESA) suggested the formula

LegislatureSize = (LiterateAdultPopulation)1/3.

Assuming year-2015 Canada has 24 million literate adults, their optimum legislature size according to the ACE formula would be 288, which is 93.5% of Canada's actual number 308 of MPs.

Make multi-option votes within legislature employ score voting? In votes within the legislature on more than two choices, they ought to employ score voting to perform that vote.

Conclusion

The three election system options I'm recommending for Canada are:

I(A): Score voting to elect MPs within ridings. Each voter rates each candidate on 0-9 scale (allowed to leave some unrated); candidate with highest average score wins. Same as Canada's old system (and also not PR), except score voting replaces plurality voting.

II(C)1: PR hybrid of score voting to elect MPs within ridings, plus optimum harmonic voting to elect top-up MPs within regions. Ballots same as in I(A). Specifically, each "region" consists of 13 ridings. Each riding elects one MP – the one with maximum average score. Each region then elects 5 top-up MPs – the 5 whose election would cause the full 18=13+5 member set, to maximize "harmonic quality value."

II(C)2: PR hybrid of score voting to elect MPs within ridings, plus asset-voting-based second stage to elect top-up MPs. Scheme described on separate web page.

Links to more

Seats, votes, and other data about Canada.

History of voting-system-reform attempts in Canada and what was wrong with them.

The debate: Proportional Representation versus Single-Winner.

"Harmonic voting" and optimal multiwinner PR voting systems (scientific paper) and part II.

Optimal "double proportional" PR voting systems based on linear programming – and the downfall of Balinski's "fair majority voting" (scientific paper).

The USA is unhappy about being stuck in 2-party domination. This could lead to 1-party domination and collapse. As of year 2014, more of the USA's voters are independent than in any party, but you sure could not tell that from its congress, which is over 99% composed of Democrats and Republicans.

Miscellaneous new Canada proposals contributed by denizens of the Center for Election Science & friends (may be half-baked – caveat emptor – but may be a useful source of ideas): 1, 2, 3, 4, 5, 6, 7, 8, 9.

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