# What electoral system should Canada have?

Once again, Canadians have voted as if they had a proportional representation (PR) electoral system, but obtained almost exactly the party system they should be expected to get, given the first-past-the-post (FPTP) system that they actually use.

If voters are voting as if they had PR already, why not just give them PR? Of course, it does not work that way, as the decision to adopt a new electoral system is rarely separable from party politics. Nonetheless, it is worth asking what electoral system the country should have, based on how voters are actually voting. They certainly are not playing the game as if it were FPTP. Even though it is.

To get at an answer to this question, we can start with the average value of the effective number of vote-earning parties over recent elections. (For those just tuning in or needing a refresher, the effective number of parties is a size-weighted count, where each party’s “weight” in the calculation is its own size–we square the vote (or seat) share of each party, sum up the squares, and take the reciprocal. If there were four equal size parties, the effective number would be 4.00. If there are four parties of varying sizes, the effective number will be smaller than four. For instance, if the four have percentages of 40%, 35%, 20%, and 5%, the effective number would be 3.08.) From the effective number, we can work backwards through the Seat Product Model (SPM) to determine what electoral system best fits the distribution of parties’ votes that Canadians have actually been providing. The SPM lets us estimate party system outputs based on a country’s mean district magnitude (number of seats elected per district (riding)) and assembly size. As noted above, Canada currently tends to have a distribution of seats among parties in the House of Commons consistent with what the SPM expects from a district magnitude of 1 and a House size of 338. The puzzle is that it does not have a distribution of votes consistent with the SPM. Instead, its distribution of votes across parties looks more like we would expect from a PR system. But what sort of PR system? That is the question the following calculations aim to answer.

Over the past eight elections, going back to 2000, the mean effective number of vote-earning parties (dubbed NV in systematic notation) has been 3.70. During this time, it has ranged from a low of 3.33 (2015 when Justin Trudeau won his first, and so far only, majority government) to a high of 3.87 (the second Conservative minority government of the period under leadership of Stephen Harper). In 2019 it was 3.79 and in 2021 it was very slightly higher (3.84, based on nearly complete results). Even the lowest value of this period is not very “two party” despite the use of FPTP, an electoral system allegedly favorable to two-party systems. (I say allegedly, because given FPTP with a House of 338 seats, we actually should expect NV=3.04, according to the SPM. In other words, a “two-party system” is not really what the current electoral system should deliver. Nonetheless, it would not be expected to be associated with as fragmented a voting outcome as Canadians typically deliver.)

How to get from actual voting output to the PR system Canadians act as if they already had

The SPM derives its expectation for NV via a phantom quantity called the number of “pertinent” vote-earning parties. This is posited in Shugart and Taagepera (2017), Votes from Seats, to be the number of parties winning at least one seat, plus one. It is theoretically expected, and empirically verifiable, that the effective number of seat-winning parties (NS) tends to equal the actual number of seat winning parties (NS0, with the 0 in the subscript indicating it is the unweighted, raw, count), raised to the exponent, 2/3. That is, NS=NS02/3. The same relationship logically would hold for votes, meaning NV=NV02/3, where NV0 is the aforementioned number of pertinent vote-earning parties. We can’t measure this directly, but we take it to be NV0=NS0+1, “strivers equal winners, plus one.” In Votes from Seats we show that this assumption works for estimating the impact of electoral systems on NV.

Thus we start with the recently observed mean NV=3.7. From that we can estimate what the number of pertinent parties would be: given NV=NV02/3, we must also have NV0=NV3/2. So NV0=3.73/2 = 7.12. This number by itself is not so interesting, but it makes all the remaining steps of answering our question possible.

Our expected number of seat-winning parties from a situation in which we know NV=3.7 works out to be 6.12 (which we might as well just round and call 6). We get that as follows. First, NS0=NV0-1: the number of pertinent vote-earning parties, minus one. We already estimated the pertinent vote-earning parties to be 7, so we have an estimated average of 6 parties winning at least one seat. This is realistic for current Canadian politics, as recently five parties have been winning seats (Liberal, Conservative, NDP, BQ, and since 2011, Greens). With PR, the PPC likely would win a few seats on current strength, and the Greens probably would continue to do so, assuming they either recover from their current doldrums (especially once PR were adopted) or that any legal threshold would not be applied nationally and thus even their 2.3% showing in the 2021 election would not lock them out of parliament. (In 2021, Greens still got 9.6% in PEI, 5.3% in BC and 5.2% in New Brunswick, for example (per Elections Canada).)

If we have an expected number of seat-winning parties, based actual mean NV, that is equal to six, what would be the seat product (MS) that would be expected? Once again, the seat product is the mean district magnitude (M), times the assembly size (S). Given M=1 (single-seat districts) and S=338, Canada’s current seat product is 338. Based on one of the formulas comprising the SPM, a seat product of 338 should be expected to result in an effective number of seat-winning parties (NS) of 2.64 and effective number of vote-earning parties (NV) of 3.04. It is working out pretty close to that for seats (average NS=2.8). Yet voters are voting more like they had a PR system given the average over recent elections of NV=3.7.

One of the formulas of the SPM, which like all of those referenced here, is empirically accurate on a worldwide sample of election results, predicts that NS0=(MS)1/4. Thus if we have an expected value of seat-winning parties of around 6, as expected from NV=3.7, we can simply raise it to the power, 4, to get what the seat product is expected to be: MS=64=1296. In other words, based on how Canadian voters are actually voting, it is as if their country had an electoral system whose seat product is around 1300, rather than the actual 338. For a comparative referent, this hypothetical PR system would be quite close to the model of PR used in Norway.1

Any electoral system’s mean district magnitude is M=(MS)/S,so taking a House of 338 seats,2 our hypothetical PR system has M=1300/338=3.85. That is, based on how Canadian voters are actually voting, it is as if their country had an electoral system whose mean district magnitude is around 3.85. Comparatively, this is quite close to the Irish PR system’s mean magnitude (but it should be noted that Ireland has a seat product of closer to 600, due to a much smaller assembly).

So there we have it. The mean district magnitude that would be most consistent with Canada’s current vote fragmentation would be just under 4, given the existing size of the House of Commons.

If Canada adopted a PR system with a seat product of 1300, its expected effective number of seat-winning parties (NS) would rise to 3.30, and its expected largest party would have, on average, 40.8% of the seats, or 138. (This is based on two other predictive formulas within the SPM: NS=(MS)1/6 and s1=(MS)–1/8, where s1 is the seat share of the largest party.)

A largest party with 138 seats (as an average expectation) would then require another party or parties with at least 32 seats to have a majority coalition, or a parliamentary majority supporting a minority government. The NDP would reach this easily under our hypothetical PR system, given it can win around 25 seats on under 18% of the votes under FPTP (and 44 seats on just under 20% as recently as 2015).

The Bloc Quebecois also would be available as a partner, presumably for a minority government, with which to develop budgets and other policy, thereby preventing the NDP from being able to hold the Liberal Party “hostage” to its demands. The BQ won 32 seats in 2019 and 33 in 2021. However, because it is a regionally concentrated party, we should entertain the possibility that it might do worse under PR than under FPTP, which rewards parties with concentrated votes. The only way to estimate how it would do might be to run the SPM within the province.

Estimating Quebec outcomes under PR

Quebec has 78 seats total, such that 33 seats is equivalent to 42% of the province’s seats. On Quebec’s current seat product (78) its largest party should win 45 seats (58%). So it is actually doing worse than expected under FPTP!

If the province had a mean district magnitude of 3.85, its seat product would be 300, for which the expected largest party size would be 49%, or 38 seats. In other words, when the BQ is the largest party in Quebec, it could do a little better on the very moderate form of PR being suggested here than it currently is doing under FPTP. (Suppose the model of PR had a mean magnitude of 9 instead, then we’d expect the largest provincial seat winner to have 44.1%, or 34 seats, or roughly what it has won in the last two elections. Only if the mean M is 16 or higher do we expect the largest party in Quebec—often the BQ—to have fewer than 32 of 78 seats. Obviously, in 2011 when the BQ fell all the way to 23.4% within the province, PR would have saved many of their seats when FPTP resulted in their having only 4 of 75 in that election. In 2015 they did even worse in votes—19.3%, third place—but much better in seats, with 10 of 78. Under the PR model being considered here, it is unlikely they would not have won at least 10 seats, which is 12.8%, on that provincial share of the vote.)

Do Canadians actually ‘want’ a still more proportional system than this?
It is possible we should use a higher NV as reflective of what Canadians would vote for if they really had a PR system. I have been using the actual mean NV of recent elections under FPTP, which has been around 3.7. But in the final CBC polling aggregate prior to the 2021 election, the implied NV was 4.12. It dropped by almost “half a party” from the final aggregate3 to the actual result either because some voters defected late from the NDP, Greens, and PPC, or because the polls simply overestimated the smaller parties. If we use 4.12 as our starting point, and run the above calculations, we’d end up with an estimated average of 7.4 parties winning at least one seat. Maybe this implies that the Maverick Party (western emulators of the BQ’s success as a regional party) might win a seat, and occasionally yet some other party. In any case, this would imply a seat product of 2939, for a mean M of 8.7. The largest party would be expected to have only 36.8% of the seats with such an electoral system, or about 125.

How to use this information when thinking about electoral reform

I would advise, as the way to think about this, that we start with what we’d like the parliamentary party system to look like. I am guessing most Canadians would think a largest party with only around 125 seats would be an overly drastic change, despite the fact that they are currently telling pollsters, in effect, that this is the party system they are voting for as of the weekend before the election!

The expected parliamentary party system from an average M around 4, yielding a largest party averaging just over 40% of the seats (around 138) is thus probably more palatable. Nonetheless, armed with the information in this post, drawn from the Seat Product Model, we could start with a desirable average share of the largest party, and work back to what seat product it implies: MS=s1–8, and then (assuming 338 seats in the House), derive the implied district magnitude from M=(MS)/S. Or one can start with how Canadians are actually voting, as I did above–or from how we think they would (or should) vote, using MS=[(NV3/2)–1]4, and followed by M=(MS)/S.

Whichever value of the seat product, MS, one arrives at based on the assumptions about the end state one is hoping to achieve, remember that we’d then expect the seat share of the largest party to be s1=(MS)–1/8. As we have seen here, that would tend to be around 40% if mean magnitude were just under 4. This implies a typical largest party of around 138 seats.4

But herein lies the rub. If you tell the Liberal Party we have this nifty new electoral system that will cut your seats by about 20 off your recent results, they probably will not jump at the offer. The parties that would benefit the most are the Conservatives (twice in a row having won more votes than the Liberals but fewer seats), NDP, and smaller parties, including apparently (based on above calculations) the BQ. But this isn’t a coalition likely to actually come together in favor of enacting PR. Thus FPTP is likely to stick around a while yet. But that’s no reason not to be thinking of what PR system would best suit Canadian voters, given that they have been voting for a while as if they already had a PR system.

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Notes

General note: At the time of writing, a few ridings remained uncalled. Thus the seat numbers mentioned above, based on who is leading these close ridings, could change slightly. Any such changes would not alter the overall conclusions.

1. More precisely, it would be almost identical in seat product to the Norwegian system from 1977 to 1985, after which point a small national compensation tier was added to make it more proportional.

2. I will assume electoral reform does not come with a change in the already almost perfect S for the population, based on the cube root law of assembly size, S=P1/3, where P is population, which for Canada is currently around 38 million. This suggests an “optimal” number of seats of about 336.

3. This is based on the Poll Tracker final aggregate having vote shares of 0.315, 0.310, 0.191, 0.070, 0.0680, 0.035 for the six main parties (and 0.011 for “other”).

4. I am deliberately not going into specific electoral system designs in this post. I am stopping at the seat product, implicitly assuming a simple (single-tier) districted PR system, meaning one with no regional or national compensation (“top up” seats). Arriving at a seat product to produce the desired party system should be the first step. Then one can get into the important finer details. If it is a two-tier system–including the possibility of mixed-member proportional (MMP)–one can generate its parameters by using the result of the calculations as the system’s “effective seat product,” and take it from there.

## 81 thoughts on “What electoral system should Canada have?”

1. Okay, let’s presume we must get buy-in from LP and use that as a constraint for incorporating PR into Canada’s system…

Then why not consider ridings of size 4, and one at-large seat, and use 3-seat LR Hare for 3 of the seats, and 1-seat RCV/AV for the other seat in every riding, as well as for the at-large seat?

Let’s then treat it as 2 different systems and average their results(ignoring the at-large seat for now).

I’ll presume there are 320 seats, so there’d be 80 ridings. So, for the 80 RCV seats, there’d be 80^(1/6)=2.076 =Ns and s1=.5782.

For the 240 seats, adjusting for how the Hare quota makes it very hard for one party to win all 3 seats, at one extreme the best the biggest party would get 2/3rds of the seats.
As for the Ns0, let’s assume it’s impossible for one party to win all 3 seats, beating the 2nd place party by more than 2/3rds of the vote. So the geometric mean of the number of parties winning a seat per riding would be Sqrt(6). Then, at one extreme, there could be 2 parties who win one or two seats in all of the districts. At the other extreme, let’s say two different parties winning at least one seat in each riding/district, but the 3rd seat is held by other parties in (Sqrt(6)-2)S/3 ridings/districts. Thus, the geometric mean for the number of parties winning seats, Ns0, would be Sqrt(2(2+S/3(sqrt(6)-2))), instead of (3S)^(1/4). If Ns0^(-1/2) is still the lower limit of s1 but 2/3 is now the upper limit, then s1 can be calculated as their geometric average and Ns=s1^(-4/3). So, if S=240, then Ns0=8.71, s1=.475 and Ns=2.69.

The number of seats won by the top party would be 80.5782 + 240.475 + 1 or 161.3 out of 321. Thus, there would most likely still be a majority in place. The Number of effective parties seems harder to calculate so for a short-cut, I’ll take a weighted geometric mean.

2.076^(80/321)2.69^(240/321)=2.51.

So, even though, the biggest party would be expected to get half the seats, there’d be plenty of pluralism in representation and the biggest party would need to maintain its majority of support in the country. In practice, the LP wd not have a problem getting at least 1 out of the 3 seats used for PR in every riding/district, ‘n it’d get at least half of the other seats and the at-large seat. So that’d be a floor of 121 out of 321 seats for the most popular party, not too far from what the LP has now, got now but practically it would more likely be closer to 161, a bare majority, consisting of MPs from all over the country.

It’d also would make who won the 3rd seat more likely to be competitive so that outsiders’ ideas come up, ‘n if they also moved the hearts of people who tended to vote for the biggest party, that party would be more prone to adopt them than otherwise…

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2. I think you should probably correct for the Bloc (seeing as it is a regional party) by using the average district-level Nv.

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• I’d be very resistant to the specific approach JD suggests. See some previous posts in this series: mean district Nv itself can be modeled from nationwide Ns. So using it to work “backwards” towards what electoral system would most match how Canadians are currently voting would be inappropriate. Besides, the SPM already implicitly assumes that there may be regional parties, by the fact that the expected numbers all increase as S goes up, even M=1.

There may be ways one should adjust for Quebec’s “distinct” role, perhaps by disaggregating the national numbers into regional ones. This would be very tricky and complex–the entire federal party system exists in Quebec even though the entire Quebec party system does not exist in the rest of the country–and I prefer to keep it simple. If this post were ever to generate sufficient interest to do more subtle analysis in the service of electoral system design in Canada, I’d be happy to take it up!

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• One thing to keep in mind here (maybe it is obvious) is that the mean district Nv will be tend to be somewhat lower than the national. So if one plugged that average into the process above, it would imply a less proportional system as the one that Canadians have been voting as if they had. I don’t think that would make a lot of sense, honestly.

Here are the figures for elections I had readily available. Pardon the Stata-to-WordPress formatting challenge.

The variables are the election year, the mean district-level N’v, the nationwide Nv, and the ratio of the latter two. The mean ratio for these elections, 1949-2011 is 0.820. It appears to be clearly lower in recent years (though for now, this is going to have to end with 2011).

``````  yr   Nv_dist_mn       Nv_nat   ratio
1949   2.438549   2.8399999   .8586441
1953   2.514313   2.8599999   .8791304
1957   2.605045        2.99   .8712525
1958   2.319531   2.4400001   .9506275
1962   2.754879        3.23   .8529038
1963   2.695099         3.2   .8422185
1965   2.756055   3.3099999   .8326451
1968   2.539013        2.97   .8548866
1974   2.526652        2.96   .8535985
1979   2.462016   3.0899999   .7967688
1980   2.382806   2.9300001   .8132443
1984   2.551373        2.75   .9277719
1988   2.753291        3.04   .9056878
1993   2.827454   3.9300001   .7194539
1997    2.91317   4.0900002   .7122664
2000   2.706572        3.77   .7179236
2004   2.727972         3.8   .7178875
2006   2.832149        3.77   .7512332
2008   2.875112   3.8699999   .7429231
2011   2.736154   3.4300001   .7977126
``````

The above is just the national mean district N’v, with the ratio being this national mean divided by the nationwide aggregate Nv. But in the spirit of the question, we want to know it by province.

Here is what we get for mean district N’v in Quebec and the rest of Canada, focusing on the years since the BQ emerged.

Quebec

``````     yr |        Mean N'v
``````

————+————
1993 | 2.3869064
1997 | 2.7389243
2000 | 2.5525925
2004 | 2.5373216
2006 | 3.0146058
2008 | 3.2396953
2011 | 3.1216285

``````     yr |        Mean N'v
``````

————+————
1993 | 2.9503586
1997 | 2.9601791
2000 | 2.7613697
2004 | 2.7989885
2006 | 2.7673314
2008 | 2.7489428
2011 | 2.5978699

In the three years of Conservative wins (2006-11)–but not before then–we see a much higher district mean in Quebec than in the rest of the country. This was the part of the secret to their success–they were performing well enough in Quebec to win some pluralities in divided contests (in 2011, the NDP won the most seats in the province even as the Conservatives won a House majority). When I get a chance I will check this for 2015 and beyond, when Liberals recovered.

I am not sure what one would want to do with this information in terms of the question posed in the title of the post. But it would be fair to say that voters in Quebec are voting as if the system was PR even more than is the case elsewhere, at least in 2006-11. Yet it should be kept in mind that many parts of BC and Atlantic Canada are also quite multiparty. In fact, BC’s mean district N’v for the entire 1949-2011 period is over 3.0, and its value of 2.68 in 2011 was the lowest it had been since 1984; it has likely risen again since). So the trend in “too high” Nv is certainly driven by more than Quebec.

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• (It actually formatted okay, at least in the view I get. This is a pleasant surprise.)

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3. Ryan Matthew Campbell says:

How likely do you think it is that Canada would continue to trend above the seat product model’s predictions even after electoral reform? A concern I often hear raised in is that because we’re a diverse country, adopting a more proportional system would cause a proliferation of parties. It would be good to know how likely that is.

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• It is certainly possible. But with a moderate PR system like what I am proposing based on current (i.e., above expectation) Nv, it is not so likely that Ns would go well above the new SPM prediction for the system.

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4. Ryan Matthew Campbell says:

Also I am still quit mystified even after reading the provided link when it comes to how to convert this to a region size for AMS.

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• You mean MMP? (We really should dispense with this “AMS” business. I did not know that terrible British term had entered into Canadian discourse.)

You’d need to run the predictive formula for two-tier PR backwards to generate the needed size of the upper tier to reach the same expected impact on Ns as a simple (single-tier) system of the same ‘effective’ seat product. Then, assuming basic-tier M=1 (for MMP), you’d be able to generate an appropriate seats in each province to reach that tier ratio.

I have not tried to do it, and I am unsure whether I will get around to it. If there was sufficient interest north of the border in this post so as to need me to work out a design for MMP for Canada, I’d be delighted!

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• Ryan Matthew Campbell says:

I guess the distinction I’m making is that MMP in the Canadian context can have considerably smaller regions than provinces. I believe the “moderate” model Byron Weber Becker (half of the pair the modelled for the ERRE committee) had 8 seat regions for example. It’d be nice to have some sort of equivalence between what our default parameters should be for both STV and MMP (the two dominant models in FVC circles – though I am an anomoly on Fair Vote’s board in being more of a Finnish-style quasi-list PR guy).

Unrelated question, but have you tried adding your index for candidate/party centred electoral systems as a parameter for your model?

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5. Rob says:

Justin Trudeau wants a preferential voting system, why not STV in multiple member districts a la Ireland, 3, 4, 5 members would be optimal?

If not possible for fear that rural areas will elect People’s Party representatives, then use Single member or 2 member districts for rural areas, and urban and suburban areas elected 4 and 5 members.

MMP can only be used for Federal elections within a province, I don’t know if districts can span the territories in the North. Any at large district across the provinces is unconstitutional, perhaps the only way this would work is let each providence have an additional 2 more seats and the seats are allocated to parties nationally.

It is sad that no Providence in Canada has jumped on the PR bandwagon to see if such experiments work. I am surprised when the NDP formed government in Alberta, that they didn’t change to a PR system, in Japan, it took a non LDP government to change the electoral system, it doesn’t have to be all that proportionate, the Spanish and Portuguese systems are such examples.

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• Any MMP for Canada would surely need to have compensation only within provinces, and most likely at sub-regions within the larger provinces. This is almost enough complexity for me to recommend not using MMP in Canada. However, the online appendix to Votes from Seats (which you can find linked within the page with that title linked at the top of this blog) shows that the modelling mostly still works for Scotland’s regional compensation model. However, the caveats about the parts that do not work–concerning Nv and deviation from proportionality–make me somewhat skeptical that voters really know how regional MMP works.

As for Justin Trudeau’s “preference” of course, he wanted STV in M=1 (i.e., AV).

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• So, what of my counter-offer, use STV, M=1, for 25%+ of elections, ‘n 3-seat LR w/ Hare quota for the rest!

LP wd still likely be in power, but they’d face different incentives, have a more regionally diverse set of MPs, and become a different party because of such.

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• Ryan Matthew Campbell says:

My understanding is that on the committee studying the issue (ERRE), the Liberals refused to make any sort of offer or counter offer to the pro-reform parties (Greens, NDP).

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• Rob says:

Maybe the Conservatives and the NDP should agree with the Liberals to implement preferential voting, so that if the Liberal Party ever suffers a wipe out that they would get their vote transfers.

Now that Canada has had two reverse pluralities in a row like New Zealand, is the electoral reform going to come, or is there something else that will cause electoral reform?

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• Rob, for the NZ scenario to play out in Canada you’d need the Conservatives to be the party prepared to study the issue upon coming back to power some day. This seems very unlikely, as they surely expect PR to help the broad center-left and hurt the center-right.

In NZ, it was Labour that was shut out twice in a row, with plurality reversals that led to majority government by National, not minority governments as in these last two Canadian elections. Even then, it still took a lot of steps–and a fair amount of luck– for MMP to end up being adopted.

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• dlw says:

If an at-large seat across provinces in Canada is currently unconstitutional, then have a 4-seat ridings ‘n an at-large seat per province.

The point is to make the LP likely to be in power, but needing to work harder to be in power.

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6. The upper limit for Ns0 with 3-seat LR Hare cd be sqrt(S/3), if the 2nd or 3rd seat were always held by parties who’d been in 1st place somewhere, or LTPs who aren’t competing w/ major parties…
Then, the bounds wd be 2 and sqrt(S/3) so the geometric mean guesstimate of Ns0 wd be sqrt(4/3*S) instead of sqrt(S).

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• Alan says:

I’m afraid my counteroffer is that Rule 1 for any electoral system is not to ask people to use different voting systems at the same election. Australia devotes considerable effort to electoral education and quite small differences between voting for MHRs and senators generate huge confusion in the public mind.

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• In this case, the 3-seat election wd work almost just like FPTP. The single seat cd stay FPTP, but that wd be stupid. Methinks, using both approval voting and rank voting or two different types of PR wd be problematic but not something new(RCV) and something old(but still new cuz of how there’d be 3 seats instead of 1).

It’d give the LP what they want: they’d still tend to be in power, but with different mix of MPs and different incentives than before…s.t the two biggest parties in Canada wd tend to evolve in the right direction so outsiders like us cd be less upset with them on average.

dlw

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• Alan says:

Whether two systems are new or old is irrelevant to the fact that they are fundamentally different and would create a serious electoral education problem.

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• Why would it create a serious electoral education problem?

The complexity of campaigning for seats determined by two different rules wd apply to the parties and the candidates, not the voters. Voters can compartmentalize. If one of the 2 rules is a 3-seat LR Hare then it is almost like 1-seat LR Hare, or FPP, the rule that people already are familiar with. So, you’d only need to teach 1-seat RCV and that’d be not too hard to teach…(I like the version that gives people up to 3 rankings ‘n treats the rankings as approval votes in a first round that wd narrow the number of candidates down to 4 (or 5) options and then apply the remaining relevant rankings to determine the winner. This wd avoid complicated equipment, allow tallying to be done locally in a decentralized manner, and avoid the use of “recursion”.

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• I must insist that 3-seat LR Hare is the exception to that rule, since it is the simplest 3rd party friendly election rule.

I call this system the Alternative Mixed Proportional system, or AMP. If one only lets people rank up to 3 candidates, ‘n uses the short-cut I described above, it wouldn’t too hard, even if some people only voted for one candidate in each of the 3 votes they would get.

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• Then let’s let voters vote the fptp way for all 3 votes if they wish, or vote using RCV for all 3 votes with up to 3 rankings, if they wish…

The real no. 1 rule is you need an elite-mass interaction, so you need an alternative to pure single-seat RCV that would still likely keep the LP in power.

Would it be worth it? Methinks, 3-seat LR Hare PR wd change both the LP and CP, or their next incarnations, by changing their incentives and geographical make-up…

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• dlw says:

It may give way to another election reform eventually, but letting the LP remain in power, is I think something that has to be woven in to whatever is proposed. The LP wd become a different party as a result of the changes.

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• Presuming that Alan’s rule is based on experience, has there ever been a case where a FPTP-like rule of picking just one candidate/party and a simplified ranking (of up to 3) has been tried and caused far more voter confusion/ education than if just the ranking rule had been used?

3-seat LR Hare is very similar to FPTP; there’d likely be only one closed party list of length 2, so people can still vote for the candidate.

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7. The kicker is to sell it as a mixed system that would tend to keep the LP in power, akin to pure single-seat RCV, but change its incentives better, ‘n make it have MPs from every riding in Canada and have to campaign country-wide for the at-large seat.

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8. We could even give people the option of asking for the FPTP election ballot that would only have the FPP options for those who don’t want to learn a new system.

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9. BTW, my mom grew up in Canada and we vacationed there a lot, so I oft feel Canadian in spirit if not fact.

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10. So, I adapted my idea to your “rule”…, if one had a 4-seat riding(and had fractional MPs so that it were used consistently throughout Canada) + 1 at-large seat, and in each 4-seat riding used 3-seat PR w/ a Hare quota and 1 seat then one cd let each voter decide if they want to use FPTP or RCV(w/ up to 3 rankings that wd be used in a 1st stage as approval votes to reduce the # of candidates down to 5) for all 3 votes, So, no one wd have to learn RCV if they didn’t want to, ‘n one could explain the process of RCV w/o using recursion.

What say you? I stand by my math and my political-economic reasoning that this wd be a better elite-mass interaction since it would favor the biggest party(ies) and small, but local, parties rather than small ideological-based parties.

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• Alan says:

I would need to know a great deal more about what you think RCV is. I say everyone has the right to a fully preferential vote. That has to mean voting as many or as few preferences as they wish. An otherwise unexplained ‘simplified RCV’ that also works as an approval vote sounds like a an electoral chimera I have not encountered before.

Moreover I just do not see the rationale for two tiers of districts where some have magnitude 4 and some have magnitude 1. No parliament elected by RCV uses the Hare quota. There are solid mathematical reasons why a multimember RCV electorate should have an odd number magnitude so I would find M=4 districts questionable.

I am trying to understand what you’re proposing but I’m afraid I have very little idea of your actual proposal.

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• Alan wrote: I would need to know a great deal more about what you think RCV is. I say everyone has the right to a fully preferential vote. That has to mean voting as many or as few preferences as they wish. An otherwise unexplained ‘simplified RCV’ that also works as an approval vote sounds like a an electoral chimera I have not encountered before.

dlw: But w/ RCV, typically most of the lower rankings don’t get used, esp. w/ a single-seat election. If only a few want to express their prefs so thoroughly, they need not be accommodated at the cost of complicating the voting process for everyone else.

My hybrid, which I thought I had explained, is to treat the top 3 rankings as approval votes in a first round and use that information to eliminate all but 5 candidates. Then, the remaining rankings of the 5 candidates would be used to eliminate candidates. If there’s only 1 seat that would only need to require seeing if the top candidate has 50% or eliminating the candidate w/ the fewest top votes among the remaining candidates until there is only left.

Alan: Moreover I just do not see the rationale for two tiers of districts where some have magnitude 4 and some have magnitude 1. No parliament elected by RCV uses the Hare quota. There are solid mathematical reasons why a multimember RCV electorate should have an odd number magnitude so I would find M=4 districts questionable.

dlw: You misunderstood me, I want all districts to have 4 seats, but 2 different voting rules used. 3 of the 4 seats would be elected using RCV(or LR) with the Hare quota, and the other 1 would use RCV. I also want there to be an at-large seat for each province to make the number of seats odd and to sweeten the deal for the LP, and to be a stopgap against extremist parties.

I am trying to understand what you’re proposing but I’m afraid I have very little idea of your actual proposal.

My apologies, I am a dilettante electoral analyst and my language reflects that sometimes.

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• Earlier, I engaged in a thought-experiment comparison of the complex system I proffer w/ a 4 seat w/ Droop quota to illustrate the diff using a Hare quota and then a single-seat election in a 4-seat riding/district would make. For a 3-seat w/ Hare quota, I tried to follow the methodology illustrated in Votes from Seats. First, I presume the Hare quota wd make it effectively impossible for a party to win all 3 seats, not unlike how a Backgammon is effectively impossible though it could happen and does change the strategies of parties/players. Thus, the predicted number of parties per 3-seat district is root(6) or about 2.45. This implies the top party would tend to win the 3rd seat about 55% of the time. That’s likely to stir up excitement locally so not “all politics is national”. Then I presume the number of parties overall taking 1st or 2nd place w/ S seats using 3-seat w/ Hare quota election rules wd tend to be the same as the number of parties taking 1st place, or bounded by root(S/3) and 2. So Ns0’=(4/3 s)^(1/4), somewhat less than (3 s)^(1/4). The s1 for the 2/3rds of the seats would be bounded by 1/3, since no party can take both the first and 2nd seat, and (2/3)1/Ns0, or if all the parties who win 1st or 2nd seats split the 2/3rds of the seats equally. So the geometric mean of 1/3 and (2/3)/Ns0 would be the sqrt((1/81)^(1/4) (16/81)^1/4 * (3/4)^(1/4)s^(-1/4))) or ((2187/4) s)^(-1/8), smaller than (3 s)^(-1/8). But the top party likely would also win a sizeable fraction of the 3rd seats by beating the 3rd place party by more than 33.3% of the vote or by being the 3rd place party. The percent of seats that wd be won by the top vote-getting party with s/3 seats/districts according to the SDPM is (s/3)^(-1/8). I use this as the probability that the top party wins the 1st seat. The probability they would win the 1st and 3rd seat then cd be (this is a work in progress.) (s/3)^(-1/8)(3-Sqrt(6)). So, the proper s1 would be

(108 s)^(-1/8)(1 + (s/3)^(-1/8)(3-Sqrt(6)))+percentage of seats won when the top party took 3rd place and was within 33.3% of the top party. If S=240 then the first part of this would be: .37. I’m still thinking about how to model the top party winning the 3rd seat… If they took 1/3rd of the seats won by the 3rd place party it would add .05 so s1=.42.

Then, since (80)^(-1/8)=.58, the s1 overall wd be 3/4.42+1/4.58=.46. This wd be greater than (4*320)^(-1/8) or .41. The at-large seats would raise the likely s1 as well, so that it becomes uncertain, and thereby interesting, whether the top party, like LP, would get a majority or have to rule with a coalition gov’t.

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• Alan says:

The definition of RCV you offer is not RCV. It is as if you argued that the Canadian house of commons is elected by MMP without a compensatory tier, or that the Danish Folketing is elected by FPTP with a strong element of proportionality. It simply makes no sense.

A district that has 4 members where 1 is elected by 1 rule and 3 are elected by another is two districts where M=1 in one and where M=3 in the other.

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• It is a hybrid of RCV and Approval Voting and some RCV rules do limit the number of candidates one can rank. This takes that a step further and uses those up to 3 rankings at least once initially to reduce the number of candidates down dramatically at first so there is no need to explain the method with the term “recursion”.

It does make sense. It is an alternative way to combine single-winner and multi-winner/PR relative to MMP. I call it Alternative Mixed Proportionalityish (AMP); it offsets the impact of a Hare quota by using it simultaneously with a single-winner election. This tends to help smaller parties relative to the 2nd place party, rather than the top party, so the net effect is more to change the incentives of the top party, making it become effectively a different party, than to make the system no longer tend to favor the top party.

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11. dlw says:

Or just use the same simplifed RCV for all the elections, with one at-large seat per province, and show the LPs that they’d most likely stay in power due to the continued use of 1st seat elections and the reduced number of ridings and how 3-seat RCV w/ a Hare quota wd reward the top party w/ 2 seats more often than the 2nd place party relative to 3-seat RCV w/ a Droop quota.

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• Alan says:

I’m sorry but I still have no idea what simplified RCV is except that it is clearly not RCV.

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• Why would using some of the rankings akin to approval votes to winnow down the field of candidates to 5 and then using the relevant rankings to determine the 1 or 3 winners not be RCV? It is a rank choice vote. Is it becuz only 3 rankings were feasible? That’s not important, one can use the top 3 rankings as “approval votes” regardless. Is it becuz so many candidates would be eliminated at once?

Besides, what of my bigger point of focusing on pushing PR in ways that doesn’t change the tendency for the top party, like the LP, to win, but rather changes their incentives for winning and the geographical composition of their party so that they become effectively a very different party?

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12. I’m sorry for the repeated posts, but I fear my math may have been a lot off before as it was late.

If the bound for number of parties who win 1st or 2nd place is between 2 and (4/3* S)^(1/4) (I’m using 4/3rds since that’s the predicted number of parties winning a seat with M=2 and 2*S/3 seats) then their geometric mean is: (64/3 *S)^1/8, which is hard to compare with (3 S)^1/4.

If the bounds for the percent of the total seats won by the top party among the seats awarded as the 1st or 2nd seat are 1/3 and 2/3 / (64/3 S)^(1/8) then its geometric mean will be Sqrt( 2/9 / (64/3 S)^(1/8)). I can move the denominator to the numerator if I make it to the -1/8ths power instead. Then, since 2/9=(9^8/2^8)^(-1/8), it will be (3^15/4 *S)^(-1/16). If S=240 then this would be .2764. This is the predicted percent of the time the overall first place party takes either 1st or 2nd place in a 3-seat election. Above, I used the percent of the top party with S/3 seats and M=1, (S/3)^(-1/8), as a proxy for how often the top party would win 1st place. If S=240 then this would be .578. That would imply that .192 of the overall seats would be won by the top party getting first place, and .2764-.192=.0844 were due to the top party getting 2nd place, which would happen in about 1/4th of the districts. What is left is the seats that would be won if the top party beat the 3rd place party by more than .33333 of the total vote or if the top party came in 3rd place within .3333 of the top place in one of the districts where it placed neither in 1st or 2nd place.

The probability of the top party in a district winning a seat was (3-Sqrt(6)) since the expected number of parties in a district was the geometric mean of 2 and 3 or the sqrt(6). If we multiply the probability of the top party being in first place, or .192, by 3-sqrt(6), then we get the percent of seats they won as their 2nd seat to be .192.55=.106. .2764+.106=.382. Then, what remains is the percent of seats won as the 3rd place party who is within .3333 of the 1st place party in a district. The percent of districts they won 1st or 2nd place is implied to be .27642=.5528, so that leaves .4472 of the remaining districts. If they got a seat from 1/6th of those districts then they would win .075 more, which would bring the s1 up to .457. If a regular 3-seat PR were used for 240 seats then the predicted s1 would be (3*240)^(-1/8) or .44 so not a big difference, but that would change if the percent of times the top party took 3rd place and was within 1/3rd of 1st place in the remaining 4/9ths of districts was higher than 1/6th.

But a 1.7% edge can make a difference in the probability of winning a majority instead of needing to be in a coalition. If the top party wins .55 of the 80 seats that were single-seat and .457 of the 3-seat elections then the weighted average would be: .55/4 + .4573/4=.48. If we added that each province had a single-seat election and the top party wins 10^(-1/8) =.75 of them then it would .5580/330+.457240/330+.7510/330=.488. So the top party would easily be within reach of getting a majority; it would simply need to work harder at winning a second seat in more elections or getting out the vote in areas where they are relatively unpopular. This could entail merging with existing third parties who are strong in areas of the country where the LP is weak.

The bottom line is: the LP has the power, so why not focus on incorporating pr in a way that lets them tend to stay in power. This would still work if the LP (and the CP) became changed as a result of the changed incentives given to them by the electoral reform.

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• Woops, .2764*3, or .8292 is the probability of the top party winning 1st or 2nd place. So, there’s only .1708 of the elections left and if they got 3rd place and were within 1/3 of the top place in 1/6th of them that would be only .029, making the final percent .411. So, the percent of 3rd places that would win another seat for the top party is still important to discern…

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• I think it would be safe to say that when the % of seats for the top party is between .45 and .5, which is an exciting range for an election since a majority would be feasible but not guaranteed.

As for the effective party size, the extremes would be one party getting 2/3rds of the seats and another getting 1/3rd of the seats with a Ns of 1.8 and the other extreme is the NsO, or (64/3 S)^1/8, for those coming in 1st or 2nd place in an election splitting 2/3rds equally and other small parties getting the other 1/3rd. This would be 1/(NsO(2/(3NsO))^2+ S/3(1/S)^2)=6.49. The geometric mean of 1.8 and 6.49 is: 3.42. The SPDM would predict (3*240)^(1/6)=2.99. The use of a Hare quota for a 3-seat election likely increases the effective number of parties and may increase the s1 as well.

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• Well, with respect to the remaining elections where the top party did not win 1st or 2nd place, there are two extremes: they win none of them, or they win all of them. If they win none of them then s1=.382. If they win all of them then s1=.382+.1708/3=.4389. The geometric mean of those two is .41 which is about the same as if they got the 3rd seat in 1/6th of the elections where they did not win 1st of 2nd place. So the predicted overall s1 would be .5580/330+.41240/330+.7510/330=.45. If they kept the 338 seats in the parliament and still used single seat elections then the predicted s1 would be: 338^(-1/8)=.48. So, they’d be down 10 seats on average, but if they switched to 3-seat droop quota with 339 seats then the s1 would be predicted to be (3339)^(-1/8)=.42. So, the reduction in seats expected to be held by the top party would be halved by using a mix of 3-seat PR and single seat elections along with at-large province seats.

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• I think that I shd have used (8/3S)^(1/4), the geometric mean of 2 and Sqrt(2/3S), as the number of parties predicted to be winning at least one seat by coming in 1st or 2nd place, instead of (4/3*S)^(1/4) it as the upper bound above. That lowered the number of parties which boosted the lower bound of s1 due to winning the first or 2nd seat in a 3-seat election.

If that s1 is bounded between 2/3/Ns0, or 2/3(1/(8/3S)^1/4), and 1/3rd then when S=240 the geometric mean would be: .2102. This implies that the 1st place party gets 1st or 2nd place in 63.06% of the elections. The other way they can win seats is by coming in first and beating the 3rd place candidate by more than 1/3rd of the total vote or by coming in 3rd place and being within 1/3rd of the total vote of the 1st place candidate.

The probability of coming in 1st used has been (S/3)^(-1/8). That times the probability of the top party winning the 3rd seat, or 3-sqrt(6), divided by 3 is the percent of seats won by the top party by also winning a 2nd seat. I think that if one is the top party overall, then the probability of winning a 2nd seat should be higher than the average probability so this is a lower estimate. If S=240 this would be .192 as the probability of coming in first place divided by 3, and .192(3-Sqrt(6))=.192.55=.106.

The percent of seats not won by winning 1st or 2nd place is 36.94%. In the best case scenario, the top party could win 3rd place and get a seat in all of these election. In the worse case scenario, the top party would never get a seat by winning 3rd place. So, the extremes are (.2102+.106) and (.2102+.106+.3694/3). The geometric mean of these two extremes is: .373. It would be higher if

That is fair amount lower than what I calculated before. I think I need to adjust the .106 to the assumptions of the best and worse case scenario assuming that the overall percent of districts where the first place party wins a 2nd seat is still .55, or 3-Sqrt(6). But if you don’t think this is helpful, I can stop sharing my thought experiments here.

For me, Canada starting to use PR will help the US to use it as well, so I’m invested in Canada electoral reformers playing political jujitsu with the LP or they can first push RCV w/ 1 seat and then work with the next CP to get PR…

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• Alan says:

Approval voting does not have a quota. RCV does. RCV does not arbitrarily treat the first 3 preferences as more important than later preferences. I still have absolutely no idea what it means when you say your simplified RCV (which is clearly not RCV) treats the first 3 preferences as approval votes. It’s not for me to take stabs int he dark by what you mean when you take terms from incompatible systems and say they apply in the same system. It’s for your yo explain what you actually mean.

I have a horrible suspicion, and of course I may be wrong, that what you’ve done is re-invent the block preferential system used in the Australias senate before 1948 where the first 3 preferences all had equal value. The sad result was that the winning party, usually the government party, captured wildly unrepresentative majorities. At the last block preferential election in 1946 Labor took 16 of the 19 seats available and held a senate majority of 33 out of 36.

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• No single-winner rule has quotas. I wd treat the rankings as akin to approval votes initially so as to reduce the number of candidates potentially dramatically w/o in most cases changing the ultimate outcome. RCV does disregard lower rank preferences for voters until their higher rank preferences have been eliminated. This makes the 4th, 5th, etc… less consequential than the 1st/2nd/3rd preferences. So, the top 3 rankings, while somewhat arbitrary, is less so if one is choosing only 1 or 3 winners. The implication is that RCV with truncated rankings deters strategic voting somewhat less than a full RCV. But for many low-info voters its consequence is not so great and it does simplify the ballot.

AS for being more concrete, you’d need to tabulate votes 2 x with my approach. First time, you tabulate how many times each candidate is ranked among the top 3 candidates by voters. After 5 candidates are identified by this then you tabulate the 5 candidates into the 5P3 or 5P2 or 5P1 or 5P0 or 86 possible rankings. Then, you can use EXCEL to determine the number of top rankings among the 5 candidates. If there is one seat then you exclude the candidate with the fewest top rankings among voters and transfer the votes as possible. If there is 3 seats but a Hare quota is used then there would be fewer extra votes to transfer from a candidate( or possibly 2) that might reach their quota. Here, I believe there would have to be a restriction of candidate per party and a vice-candidate would be prespecified to win a 2nd seat if the top candidate/party gets enough votes relative to the 3rd place candidate after votes are transferred from the 5th and then 4th place candidates.

I’ve interacted w/ experts and other dilettantes about this idea. It performs very much like RCV; it would speed up the winnowing of candidates if there are quite a few of them on the ballot.

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• Alan says:

No single winner system has quotas.

That is simply untrue. The quota in an RCV election where M=1 is the total number of votes divided by 1 more than the number of vacancies to be filled and then increased to the next highest whole number. Some electoral legislation describes that number as a majority of first preferences or an absolute majority. If no candidate reaches the quota trailing candidates are eliminated until either a candidate reaches the quota or a candidates has more than half of the remaining preferences.

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• Alan says:

You have still not defined what you mean by treating the first three preferences as approval votes. If you mean that the first three preferences are to be treated as of equal value then what you are proposing is the block preferential system The result would be that a party winning a plurality in a district where M=3 would win all seats in that district. Not only would that proposal not be RCV, it would also not be proportional representation.

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• Alan wrote: You have still not defined what you mean by treating the first three preferences as approval votes.

dlw: I thought I had. Voters wd not be forced to rank candidates, but there would be one candidate per party(whose leadership wd determine their candidate ‘n vice-candidate who’d hold the 2nd seat if their candidate won 2 seats) and voters could just vote for one candidate. The (up to) top 3 prefs wd be treated as of equal value in determining which 5 candidates were the front-runners. Then the ranking information on those five would be used to eliminate one candidate at a time until there is 1 or 3 remaining and the top candidate cd still beat out the 3rd place candidate if her/his remaining votes after a hare quota is removed is greater than the number of votes received or transferred to the 3rd place candidate.

Alan: If you mean that the first three preferences are to be treated as of equal value then what you are proposing is the block preferential system . The result would be that a party winning a plurality in a district where M=3 would win all seats in that district. Not only would that proposal not be RCV, it would also not be proportional representation.

dlw: Only if there were just one stage. I am aware that that would simulate the results of FPP that is why I described the 2 stage process. The 1st stage mainly speeds things up and lets the tabulations be done in a more decentralized, transparent manner since the number of possible rankings rises very fast with the number of candidates and there are often good things that can come down the road from letting marginal candidates get their ideas out into the public square.

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• Alan says:

Quotes do not give way to remainders. It may be better for you to get a better understanding of RCV before setting out to invent a new system. As but one example, it is not possible for candidate to win twos eats under RCV because they an no longer receive preferences once they are elected.

I bring a slightly unusual perspective to F&V because I been a returning officer, I have done electoral education and I have drafted RCV rules for some quite large organisations. You’re not proposing simplified RCV first because it is not RCV, second because your system would be an absolute nightmare to explain to voters and third, because it would be a worse nightmare to count. It’s also absolutely obvious that a two count procedure where votes have different values in the two counts reduces the whole thing to a random walk, although one where the plurality party in a district would be massively advantaged.

You have not made clear how the second count is to be conducted. If you keep three equal values for the first preferences in that count the result is still exactly the same as block preferential. If on the other hand the first three preferences do not have equal values in the second count then second and third preferences have different values in the first and second counts.

The only advantage you propose for all this strangeness is that it would be simpler. On the contrary I say that your system approaches the complexity of the modified d’Hondt system used in the first two ACT elections.

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• Alan:Quotes do not give way to remainders. It may be better for you to get a better understanding of RCV before setting out to invent a new system. As but one example, it is not possible for candidate to win twos eats under RCV because they an no longer receive preferences once they are elected.

dlw: I think we’re arguing about semantics on those fronts. More importantly, If there were only 3 seats and 1 candidate per party then there wd have to be the possibility of winning a 2nd seat that would be held by the appointed vice-candidate. This would make RCV a closed list rather than an open list form of PR but that would be less important if there are only 3 seats(and a Hare quota made it harder to win 2 seats)?

Alan: I bring a slightly unusual perspective to F&V because I been a returning officer, I have done electoral education and I have drafted RCV rules for some quite large organisations. You’re not proposing simplified RCV first because it is not RCV,

dlw: Okay, it’s not RCV by the standard def’n of RCV then. It works like RCV in the 2nd round.

Alan: second because your system would be an absolute nightmare to explain to voters

dlw: I disagree. It cd be a challenge, but it would not be a nightmare. I wanted to use 3-seat LR Hare for the 3seat election but you claimed it was not right to have two different types of rules, even if one of them is basically the same as fptp.

Alan: and third, because it would be a worse nightmare to count.

dlw: It hasn’t been used yet. Why concretely would it be a “nightmare” to first tally the number of times each candidate is ranked among the top 3 and then to sort votes based on the relative rankings of the 5 remaining candidates?

Alan: It’s also absolutely obvious that a two count procedure where votes have different values in the two counts reduces the whole thing to a random walk, although one where the plurality party in a district would be massively advantaged.

dlw: You use the method of authority often, afaict, to make strong claims. Jameson Quinn came up with an example that was meant to prove that this voting rule would be dysfunctional and found that it was very difficult for the inclusion of the 1st round to be problematic. This is because the first round mainly potentially affects who is in the 4th or 5th or 6th place, i.e. candidates/parties that normally are not likely to be among the final 1 or 3 winners. IOW, it gets results more or less the same as RCV w/ M=1 and 3, just faster and more transparently, except that it would use a Hare quota and be a closed list form of PR when M=3. That doesn’t massively advantage the 1st place party in the district, because a higher quota makes it easier for the 3rd place candidate/party to win a seat there. I believe it does help the 1st place overall relative to 2nd place overall party, because the latter wd be less likely to win the 3rd seat as often as the former.

Alan:You have not made clear how the second count is to be conducted. If you keep three equal values for the first preferences in that count the result is still exactly the same as block preferential. If on the other hand the first three preferences do not have equal values in the second count then second and third preferences have different values in the first and second counts.

dlw: the 2nd count goes back to how RCV is done. It sorts the votes based on the 86 possible up to 3 rankings of 5 candidates. I specified this step of sorting votes by possible rankings and that implied that the equal values were only used in the first stage.

Alan: The only advantage you propose for all this strangeness is that it would be simpler. On the contrary I say that your system approaches the complexity of the modified d’Hondt system used in the first two ACT elections.

dlw: But that is based on a misunderstanding of what I wrote. I will admit I am not the best communicator for a wide audience, so that is a different problem, not to be judged by my own shortcomings in that task.

I proposed this system based on the simple logic that the best criteria for rallying around pushing for the use of a PR system is to play political jujitsu against the LP. The system I described still favors strongly the top party. It also favors small, local parties rather than the main rivals of the LP. What it wd do is to change the LP, making them have MPs from all over Canada and making them need to work harder to win a 2nd (out of 3) seat. The use of a rule similar to RCV wd also end the ability of the CP to take advantage of FPP. Radicals could run for the 3rd seat and maybe win it sometimes, but they’d have a hard time gaining power since voters in other parties could exit the LP/CP/NDP/GP by forming new small local third parties that wd potentially win a 3rd seat or be key swing-voters in the single-seat elections. For a small parties to be influential, they would have to change the views of people in the bigger parties so that it becomes easier for the party to accommodate them than to resist them.

I don’t see my imperfect understanding of quotas or what exactly is RCV as mattering in terms of the basic intution that one cd replace the use of a droop quota with the use of a hare quota + single-seat elections so that the resulting system wd tend to favor the biggest party(the one in a position to make electoral reform) and small, local parties.

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13. But the concept of a quota does not matter as much for a single-winner election typically.

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• Alan says:

Again, simply untrue. The rules for dignifying* the set of winning candidates are identical where M=1 and M=any other number.

*Obvious predictive typo but I rather like the idea of dignifying winning candidates

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• it depends on whether the rankings selected get exhausted or not, right? The quotas give way to remainders, I presume?
But the possible permutations of candidates grows very fast w/ number of candidates and forcing people to rank all the candidates like they do in Australia is also problematic so exhaustion of rankings happens. But this is a side point that is not too important when M is small and a M of 3 is coupled with the use of a Hare quota, imo!

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14. Fair Vote Canada suggested to the Electoral Reform Committee in 2016 that, for MMP, “Parliament might decide that the average region should have 14 MPs, eight MPs, or some other number” and provided an example of how MMP regions could be configured with an average size of eight MPs per region. (We hopes this would be moderate enough to appeal to the Liberals. Few said so.)
I have used that size to simulate how Moderate MMP would have worked on the votes cast in 2021:
As my post states, this small size (plus a 5% provincial threshold) would elect only eight Peoples Party MPs, not 17. But it also has positive advantages: the region is small enough that the regional MPs are accountable. And it identifies that 14 of those 42 regions are one-party kingdoms, swept by a single party this year.
My guess is that this might lead to five or six parties outside Quebec (a party needs 12 MPs for recognition).

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• So, how might you get the LP/Trudeau to go along w/ this idea this time around?

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15. This is a continuation of a previous line of thought above.

“If that s1, portion of all seats won by taking 1st or 2nd place in a 3-seat LR Hare election, is bounded between 2/3/Ns0, or 2/3(1/(8/3S)^1/4), and 1/3rd then when S=240 the geometric mean would be: .2102. This implies that the 1st place party gets 1st or 2nd place in 63.06% of the elections. The other ways they could win seats is by coming in first and beating the 3rd place candidate by more than 1/3rd of the total vote or by coming in 3rd place and being within 1/3rd of the total vote of the 1st place candidate.

The probability of coming in 1st used has been (S/3)^(-1/8). That times the probability of the top party winning the 3rd seat, or 3-sqrt(6), divided by 3 is a lower bound on the percent of seats won by the top party by also winning a 2nd seat. An upper bound is to presume that the only time the 2nd seat is won by the first place candidate, predicted to overall be 3-Sqrt(6), is when the overall top party’s candidate wins 1st place, which is predicted to be (80)^(-1/8) or .578. This upper bound for the chance that the top party wd win another seat would be .55/.578. If S=240 the geometric mean between the two extremes would be (80)^(-1/8)/3* Sqrt( (3-Sqrt(6))* (3-Sqrt(6))/(80^(-1/8))) or (3-Sqrt(6))*80^(-1/16)/3 or .1395.

The percent of seats not won by winning 1st or 2nd place is 100-63.06=36.94%. In the best case scenario, the top party could win 3rd place and get a seat in all of these election. In the worse case scenario, the top party would never get a seat by winning 3rd place within 1/3rd of the top placing party. So, the extremes are (.2102+.1395) and (.2102+.1395+.3694/3). The geometric mean of these two extremes is: .40667, or just about 1.33% lower than the .42 what would be predicted for 3-seat PR w 339 seats by the SPM.

So the predicted overall s1 would be .5580/330+.40667240/330+.75*10/330=.45. This is 3% less than what would be predicted if Canada kept single-seat elections only, but it is higher than the .42 predicted by the SPM if the country switched over to 3-seat PR. So, the LP wd very likely either have a majority or be the dominant party in a coalition gov’t, as they have been as of late. But they would have input from legislators from all over the country and need to work harder in districts where they are popular to win a 2nd seat, or in districts where they are not popular to win the 3rd seat by taking 3rd place within 1/3rd of the total vote of the 1st place party.

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16. furthermore, the CP, as the 2nd place party, wd face similar incentives to work harder to win a 2nd seat or the 3rd seat, and would be more geographically diverse in its leadership so any moderation by it would spill over into further changes by the LP, or the LP cd split between part of the CP and the NDP or other parties. So, Canada wd find itself w/ 2 rather different dominant parties whose duopoly status would be rather contested.

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17. Alan says:

In view of discussions elsewhere, I thought I would include a short extract from Wikipedia on the former electoral district of Sturt in the NSW legislative assembly:

In 1920 Sturt was combined with Cobar, Willyama and much of the area of Murray to create a three-member electorate, elected by proportional representation. At the time it was one of the largest electoral districts in the world, said to be 121,000 square miles, with votes having to be brought sometimes 500 miles by the primitive transportation of the time to the central counting place. But proportional representation pundits as far as away as Canada revelled in STV’s successful implementation under such conditions.[20]

In 1927, it became a single member electorate, but was renamed Broken Hill in 1968.

Sturt is, to my knowledge, the only electorate ever to send a Wobbly to parliament.

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18. Dave Hutcheon says:

Reference 20, to the “proportional representative pundits as far away as Canada,” reads:
“Is proportional representation suitable for all elections in provincial constituencies?”, The UFA United Farmers of Alberta, 15 January 1929, p.9.”

Page 9 reveals the pundits to have been secretaries of the Proportional Representation League, Philadelphia.

The word “all” and the question mark of the article’s title are significant. Alberta already had a hybrid system in which provincial MLAs were elected by STV in Edmonton and Calgary and by AV in rural areas. It was replaced by FPTP in 1955.

The UFA has gone from being a lobby group, the surprise winner of the Alberta provincial election in 1921, to a farm services retail chain 100 years later.

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• Alan says:

Percy Brookfield, the Industrial Socialist Labor MLA for Sturt, was killed in South Australia in 1921 on his way to a meeting of the NSW parliament. Why was he travelling to Sydney via Adelaide and Melbourne? The NSW rail network would not reach Broken Hill until 1927. SA had built a somewhat illegal rail link to Broken Hill (the link was theoretically operated by a private company but the company was actually owned by SA) and the fastest way to Sydney from Sturt was a 4 day rail trip through two other states with train changes for breaks of gauge at the Victorian and NSW borders. Now that’s isolation.

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19. Rob says:

The smallest providence Prince Edward Island is guaranteed 4 seats. How large would Canada’s parliament would have to grow to have something like what is equivalent to the US Wyoming Rule, the rule would be called the Prime Edward Island rule?

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• Mark Roth says:

Back of the envelope: PEI has a population of 157,000, Canada has 38 million.

Canada would need 950 seats with an average of 40,000 residents each.

I could live with that. Not sure if Canadians or Canadian politicians would, Especially if the scores of new backbenchers see that getting onto the front bench is a longshot and decide to act more independently of the party leader.

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20. Dave Hutcheon says:

You get almost the same number with a Nunavut Rule.

PEI and the 3 territories have 7 MPs but their combined population would call for 2+ MPs under rep-by-pop. Invoking a “PEI rule” would be taking a sledge hammer to a peanut-sized problem. As a resident of the ROC, I’m happy to allow over-representation in a few sparsely populated regions and to stick with the deal made when PEI joined the confederation.

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• I’m not Canadian, but I’d rather see each of the 3 Territories get 4 quarter-MPs each and have one at-large MP for the 3 of them together, as well as one at-large MP for all of the provinces, which would entail PEI getting either 12 quarter-MPs, from 3 ridings, or 4 3-quarter-MPs, from one riding, along with their at-large provincial MP.

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• Alan says:

Tasmania has 5 MHRs because, as an original state, they have a constitutional minimum. On a strict population rule they would have 3. Like you, I’m happy to accept a historical anomaly to maintain commitments at the time of Federation. There may be good reasons to have much larger legislative bodies, but a PEI/Tasmania rule is not one of them.

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• Yes, allowing some minimal malapportionment for the sake of federal bargains seems reasonable to avoid what would be a rather mindless increase in assembly size. Canada has a current house size that is pretty close to the cube root law expectation; Australia’s is rather severely undersized.

Note also that neither of these conditions holds in the US: Wyoming is not an original member of the federation but rather a later creature of it, and the House is undersized.

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21. Didn’t Wyoming get made a US state as part of how the Republican party was trying to remove Andrew Johnson from President?
JFK, in his Profiles of Courage, describes how close they came and how just 1 Senator stood in their way. Both increasing the seat size and keeping the president being powerful, making other later presidents able to move the US out of its funk.

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• Wyoming became a state in 1890, so that seems unlikely.

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• Alan says:

In 1889 and 1890, Congress added North Dakota, South Dakota, Montana, Washington, Idaho, and Wyoming—the largest admission of states since the original 13. This addition of 12 new senators and 18 new electors to the Electoral College was a deliberate strategy of late-19th-century Republicans to stay in power after their swing toward Big Business cost them a popular majority. The strategy paid dividends deep into the future; indeed, the admission of so many rural states back then helps to explain GOP control of the Senate today, 130 years later.

It’s obviously wrong this was done to effect the impeachment of Andrew Jackson.

If North Dakota, South Dakota, Montana, Idaho, and Wyoming were a single state, Supermontana would presumably return 2 Republican senators.

The Democrats would be down 1 and the Republicans would be down 7.

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• I stand corrected, but the idea of gaining an advantage by adding a whole bunch of new smaller states immediately came from the addition of new states before, when the GOP wanted to impeach Johnson. And, the increase in states also abetted the number of seats, which by the Seat-Product model was a net positive, until they put the kibbutz on that in the 30s of the last century.

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• Alan says:

No new states were admitted immediately before the Johnson impeachment.

Kansas—29 January 1861
West Virginia—20 June 1863

Johnson was impeached on 24 February 1868.

1 March 1867—24 February 1868 is just under a year.

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• I guess my memory must’ve played tricks on me, woops…

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• The more important point(I remembered it wrongly from a chapter from JFK’s Profiles in Courage) is that the expansion of states helped to increase the number of seats in the US and by the SPM, the predicted effective number of parties in the US. I guess the better precedent in US history is in how during the ante-bellum period in the US, the slave-owning states increased the number of slave-owning states to keep pace with the expansion in number of non-slave owning states in the North. And, the existence of more states would give states an incentive to push for an increase in the number of congressional seats.

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