Warren D. Smith, Nov. 2015
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.
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.]
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.
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.
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:
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.
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.
Enough abstract principles. Let's start designing and analysing voting systems.
I(A): Score voting to elect MPs within ridings.
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.
- 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.
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:
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 Sv,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:
But those two gains come at a cost:
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:
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
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.
The three election system options I'm recommending for Canada are:
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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