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Tom Monto

A study of votes wasted under STV, or How much is a quota?

Updated: Aug 19

Under STV, a large proportion of votes are used to elect representatives. But how many are not so used?


However much it is, it is less than the number wasted under First Past The Post where 40 to 66 percent of the vote in each district or more are usually ignored. (In particularly defective elections in Canada's past, only 18 percent of the votes were used to elect the single winner (Toronto municipal election 2014; Alberta Army member, 1944).


Under FPTP, only the votes cast for the winner - whether he or she received a majority of the votes or not - are used to elect someone.


But how many are not used under STV?


Under STV most of the representatives are elected by their vote tally surpassing the quota. Then all but the quota is stripped from them to be transferred to other candidates. (However the last winners are usually left with their full vote totals because there is no reason to transfer the surplus. This makes the vote tallies look deceiving - the least popular successful candidates - the ones elected last - are left with larger vote tallies than the most popular ones who had their surplus taken away from them.)


At the end of the vote count in each election it is possible for one or two or so to win seats by merely being the last surviving candidate when the field thins, but we'll ignore that possibility for the study in hand.


To get quota, first establish the total number of valid votes. if the Hare quota is to be used, divide the number of valid votes by the number of seats being contested in the district and add one (or round up). This quota is simple to explain but compared to another method it reduces the representation of the most-popular party if for example one of its candidates receives much of the party's votes. Hare is large so relatively fewer of the winner's votes are transferred to aid other candidates.


Instead, the Droop quota is usually used nowadays. This is derived by dividing the number of total valid votes by the number of seats plus one and then adding one (or rounding up). The smaller quota under Droop compared to Hare means that more votes are transferred from a winning candidate to other candidates, aiding generally the most popular party who has two or more quotas among its combined candidates' vote tallies. (On the other hand, the Droop makes it possible for a party with just enough votes to make the Droop quota to elect one, while under Hare the larger quota might have made it impossible for the party to win a seat.)


The Droop quota is usually defined as the ultimate minimum that a number of candidates equal to the number of open seats can win. If quota was any lower, it could possibly allow too many candidates to be elected. But this is based on a hypothetical case. Often as the vote count progresses, 10 or 20 percent of the ballots are consigned to the "exhausted " pile - due to having no un-used back-up preference marked for candidates who have not already been either elected or eliminated. So as soon as ballots start to be labelled "exhausted", the quota could be reduced with no fear of electing too many. (The exhausted votes may actually prevent the last seats being won if it was required to get quota to win a seat. The existence of exhausted votes is why in some cases the last seats are won by the last surviving candidates with no requirement that they have to have quota.)


In the STV system used in Canada, the quota set in the first count was not reduced during the vote count, despite the presence of exhausted votes reducing the number of valid votes still in play.


So then, using Droop, we see the following quotas:

say with 120 seats in a legislature elected by count of 12,000 valid votes:

if the land is divided into two or more districts, in each of which there is an equal number of voters, we have the following:


if there are two districts, each electing 60, quota in each district would be 6000/61 = 99

quota of 99 X 120 = 11,880 votes used to elect someone (more or less)

120 votes not used = 1 percent of the overall votes are wasted in each district

2 percent overall


if there are three districts, each electing 40, quota would be 4000/41 = 98

quota of 98 X 120 seats overall = 11,760 votes used to elect someone

240 votes not used = 2 percent of the overall votes


if there are four districts, each electing 30, quota would be 3000/31 = 97

quota of 97 X 120 overall = 11,640 votes used to elect someone

360 votes not used = 3 percent of the overall votes


if there are five districts, each electing 24, quota would be 2400/25 = 97

quota of 97 X 120 overall = 11,640 votes used to elect someone

360 votes not used = 3 percent of the overall votes

(yes, result is the same as in four districts above)


if there are six districts, each electing 20, quota would be 2000/21 = 96

quota of 96 X 120 overall = 11,520 votes used to elect someone

480 votes not used = 4 percent of the overall votes


if there are eight districts, each electing 15, quota would be 1500/16 = 94

quota of 94 X 120 overall = 11,280 votes used to elect someone

720 votes not used = 6 percent of the overall votes


if there are ten districts, each electing 12, quota would be 1200/13 = 93

quota of 93 X 120 overall = 11,160 votes used to elect someone

840 votes not used = 7 percent of the overall votes


if there are twelve districts, each electing 10, quota would be 1000/11 = 91

quota of 91 X 120 overall = 10,920 votes used to elect someone

1080 votes not used = 9 percent of the overall votes


if there are 15 districts, each electing 8, quota would be 800/9 = 89

quota of 89 X 120 overall = 10,680 votes used to elect someone

1320 votes not used = 11 percent of the overall votes


if there are 20 districts, each electing 6, quota would be 600/7 = 86

quota of 86 X 120 overall = 10,320 votes used to elect someone

1680 votes not used = 14 percent of the overall votes


if there are 24 districts, each electing 5, quota would be 500/6 = 84

quota of 84 X 120 overall = 10,800 votes used to elect someone

1920 votes not used = 16 percent of the overall votes


if there are 30 districts, each electing 4, quota would be 400/5 = 81

quota of 81 X 120 overall = 9720 votes used to elect someone

2280 votes not used = 19 percent of the overall votes


if there are 40 districts, each electing 3, quota would be 300/4 = 76

quota of 76 X 120 overall = 9120 votes used to elect someone

2880 votes not used = 24 percent of the overall votes


if there are 60 districts, each electing 2, quota would be 200/3 = 67

quota of 67 X 120 overall = 8040 votes used to elect someone

3960 votes not used = 33 percent of the overall votes


So we can see in each case that, because each district is the same as each of the others each time (in this unusual case!), the percentage of the vote not used to elect someone in each district is the same as the overall percentage ignored.


Thus taking the number of seats in the district and taking the divisor in the fraction used to make the quota and subtracting one percent, we arrive at the percentage ignored in each case.


(In First Past The Post we can have any percentage from 80 or so, to 40 percent or less being used to elect the member. Thus the number of effective votes may be about 4/5ths of the votes cast or about 2/5ths, with as little as 18 percent used in a specific district being a possibility. Thus, FPTP creates a much wider range of effective votes in a district, with members being elected by wildly-varying number of votes.)


Thus, under STV,

in a district electing 3, three-quarters of valid votes are used and one percent less than a quarter - 24 percent - is ignored.

in a district electing 10, 10/11ths are used and one percent less than one-11th - 7 percent - is ignored.

in a district electing 9, 9/10ths are used and one percent less than 1/10th - 9 percent - is ignored.


I have not checked it but I expect this would work for any combination of votes and seats and districts.


There is another thing to think about when deciding on number of districts - the effect on the type of representation.


Bootmakers - A thinly-spread group wanting representation

If only two districts are used, to win a seat a party would have to have only 99 of the 12,000 votes. So with a two-district scheme, a small party with its supporters spread across the land - say a movement for better conditions for bootmakers - would have a good chance of winning a seat - if it had at least 99 supporters in one of the districts. There would be nothing another group could do to stop the bootmakers from getting that one seat out of 120 - if the bootmakers had at least 99 votes overall.


A very-popular party

A party with massive appeal across the land would get many seats under any scheme. Sometimes under FPTP, a single party takes all the seats even with only about 60 percent of the vote. (False majorities where a party takes a majority of seats in the assembly with less than half the votes, as few as 40 percent of the votes are common.)


But there is no way a party can take all the seats in a STV election - unless it had more than something like 90 percent of the vote, hardly possible in real life. (Under STV, each party gets a share of seats proportionate to its vote share in a district. Extended across the jurisdiction. each party gets a share of seats in the assembly that is proportionate to its vote share overall.)


Say a party has 4500 votes spread across the land. (the total vote being 12,000)

In a two-district system that party will likely take at least 45 seats in the 120-seat legislature.

In a 15-district system, it may have 300 votes in each district. This might get 3 of a district's 8 seats. Thus also 45 seats.


In a 40-district system the party may have 100 votes in a district, if its supporters are distributed equally. This would give it one of a district's three seats. Thus again about 40 seats. Some variation could happen due to non-uniform distribution of its voters and the drawing of the district boundaries.


(Under FPTP, such a party might have 35 votes in most of the land's 120 districts, some more and some less. So it might win many seats or it might win only a few, depending on vote-splitting among its opponents and the variables of political geography. There would be much chance in such an election. This kind of reminds me of Canada's federal elections today.)


"Save Our Mountain" - a small local group wanting representation

But in a two-district scheme a party with 80 votes in one corner of the land - for instance a "Save our Mountain" movement - would have no chance to get a seat (unless it got solidarity support from others not directly involved in the local issue).


But if 40 districts were used - and one district encompassed the 80-strong "Save Our Mountain" movement, the movement could easily take one of the three district seats and there would be nothing any other group could do to stop them.


Unfortunately it could happen accidentally or on purpose that the 80-strong Save Our Mountain-ers might be split by the drawing of the district boundary, although this is more likely under FPTP where districts are only one third as large as in a three-seat/district STV scheme.


Under FPTP, the "Save our Mountain"-ers would likely have no chance to be elected. Say with their numbers split in third by the many district boundaries, in a single district they would have only 27 votes in a hundred-vote district. Likely at least one other group would have more votes and take the district's one seat. And this would happen in each of the three districts that would divide the area's electorate under DPTP.


Under the 40-district STV scheme, one district would likely encompass all the SOM-eers.



Bootmakers - A thinly-spread group wanting representation

Meanwhile the "Better Conditions for Bootmakers" movement has a poor chance under the 40-district STV scheme. Unless there are at least 76 bootmakers and their supporters in a single district they would not have any representation. Even if they had 1500 votes overall spread thinly in every part of the land, without 76 in a single district they would have no representation.


But if election officials adopted a system of 15 districts, if the bootmakers had 90 supporters in any district covering one-15th of the land, they would have a seat. And if that happened in many districts, their say 1500 votes might give them 15 seats in the 120-seat legislature.


Conclusions

Few districts means larger districts.

So having few districts is better for thinly-spread parties. This would include the large umbrella parties - the Liberals and the Conservatives - and also direct representation of populous but poorly organized sections of the population - workers, farmers, women and other "minority" groups. But it would be proportional - the large parties will take many seats and the less-popular parties/groups will take fewer seats.


Having many districts drops the quota necessary to win a seat in any one district and so gives a locally-strong movement a good chance of being elected to a seat.


A combination of medium-size districts, say of 5 to 10 seats each, with an overall "at-large" scheme works to provide fairness for all types of parties and groups, while still ensuring proportional representation -

A more-popular party will take more seats than a less-popular party.

All parties will take their due share of the overall vote.


These are generalities that are subject to exceptions. For example, under STV if a small party has its support in only three districts it may take six seats (five percent of the overall seats) even if it has less than one percent of the overall vote. It is unlikely but it could happen.


But a very small party has its best chance under FPTP

Far more likely is the 1982 FPTP election of a Western Canada Concept candidate to the Olds-Didsbury seat. The candidate took a seat (1.1 percent of the province's seats), with only 4000 of the province's 945,000 votes, a percentage of the overall vote so small you need a microscope to see it. It is only .4 percentage of the votes cast.

(See my blog "Local representation -- under FPTP WCC, under STV Save Our Mountain" for information on the 1982 WCC victory.)


A Fair and simple MMP system

Here's an example of an MMP system that combines single-member districts and medium-size districts of 5 to 9 seats each, with an overall "at-large" scheme --

-each city with currently 5 to nine seats would be made into a city-wide district electing the same number of reps. through STV

- each city with 10 to 18 seats would be made into two districts, each district electing roughly half the reps. through STV

(and so on for larger cities)

- two or more districts outside the cities, up to 7 districts together, would be combined so their new total land area would be large but not too large to be practical. And the new district would be given the same number of seats as the districts together used to have. Electing its reps. through STV.


A maximum practical size would be 150,000 square kilometres or seven seats, whichever comes first. (Nine members could be taken as maximum seat count, if preferred.) Thus, a new rural multi-member district could have 200,000 voters (seven times an average of 30,000 voters per old district).


A somewhat typical rural riding, Banff-Airdrie riding in Alberta contains 12,000 square kilometres. This riding takes about an hour's drive from one corner of the riding to another, as a bird flies.


As large as that seems, there are many federal ridings much larger. Each is represented by only one MP.


Even with many new districts being established with the maximum size (150,000 square kilometres), there are some current ridings that cover even larger land areas. Admittedly, these ridings are in remote areas (ironically where transportation is very difficult).


These existing large ridings are:

Quebec's Manicouagan riding contains 264,000 square kilometres.

Quebec's Abitibi—Baie-James—Nunavik—Eeyou is crazy large containing more than 800,000 square kilometres.

Ontario: Churchill-Keewatinook Aski riding contains 494,701 square kilometres.

Ontario: Desnethe-Missinippi-Churchill River riding contains 342,903 square kilometres.

Ontario: Timmins-James Bay contains 251,599 square kilometres.

BC: Prince George - Peace River -Northern Rockies riding contains 243,276 square kilometres.

BC: Skeena-Bulkley Valley riding contains 327,275 square kilometres.

The Yukon riding contains 482,443 square kilometres.

The NWT riding contains 1,346,106 square kilometres.

The Nunavut riding contains 2,093,190 square kilometres.

(info from Wikipedia -- Population of Canadian federal ridings)


Each of these are represented by one single MP. Under STV a large rural district would be represented by 2 to 7 MPs, so it seems perfectly do-able. (Concern about the loss of local representation under such a large district is discussed in the footnote.)


The current riding of Fort McMurray-Cold Lake, with 147,412 sq. kms., is about the size of the maximum size of a "grouped riding."


Thus, under this new system, Banff-Airdrie could be combined with something like six other of our current ridings to create a single multi-member riding, covering the south and west corner of Alberta with a single riding.


- Those districts too large to be made larger (larger than say 100,000 square kilometres) would be left as single-member districts, each electing its rep. through Alternative Voting perhaps. The ridings left as is would include the ten largest ridings listed above. They would not be made any larger, although the voters present in each are but a fraction of that of populous urban ridings like Victoria, which has 90,000 voters.


The attached at-large scheme could be as simple as allocating a supplemental hundred seats to the political parties as per their vote tallies, or

allocating an additional supplemental 50 seats to the political parties as per their vote tallies.

The largest remainder system used to allocate seats as fairly as possible.


If a party had one or two percent of the vote, it would get a seat or two in the at-large scheme, even if its support was so spread out it could get no seats in district elections even with the fair representation achieved through the district-level STV used in a large part of the country.


That would be fairness. That would be democratic representation.

------------------


* Those who fear loss of local representation if rural STV grouped districts are brought in, those skeptical should consider that each and every group within the new larger district that has quota will elect a rep.

We saw that in the text above, where the "Save Our Mountain" group with less than one percent of the votes was pretty much guaranteed to take one seat (if overall chamber has 120 members). That applies to any other local group as well.

Under FPTP, the only way for a small group to win a seat is if it has more votes than any other group in a district, for example more than the local Liberal or Conservative candidate.


On the other hand, under STV in a three-seat district, any group or party just needs a quarter of the vote in the district to take a seat. And in a five-seat district only one-sixth of the vote. And so on.


This gives a much better chance to most parties, except for a hyper-localized party that has only sizable support in a small FPTP district. That kind of accidental victory warps and does not improve the effectiveness of our election system.


Thanks for reading.

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More on Quota

(from Wiki "Proportional Representation")


In STV elections, there is no explicit electoral threshold, but there is a natural threshold based on District Magnitude.


Winning the [[Droop quota|quota]] (ballots/(seats+1), plus 1) of first preference votes assures election.


As well, the need to attract second preferences tends to promote consensus and disadvantage extremes. A candidate who attracts good second (and third, etc.) preference support may be elected while an initially more popular candidate falls behind and is eliminated.


Due to vote transfers, a candidate who attracts good second (and third, etc.) preference support may pass quota and win a seat even if they start with only half the quota of first preference votes. In those cases, it may be said that in a six-seat district the effective threshold would be 7.14% of first preference votes (100/(6+1)/2).<ref name="DMstvPdf" />


For a district magnitude of 3, the STV electoral threshold is 12.5 %, significantly higher than typical party-list PR. (Of course, in such a district, the number of votes involved is less than in an at-large contest, so quota in a MMD is lower than the natural threshold in an at-large contest.


A city election electing 8 members at-large is regulated by natural electoral threshold of 11 percent (Droop quota); if the same city is divided into four districts, each electing three members, the natural threshold would be 25 percent in the district, which is only 7 percent of the total vote.)


As well, in STV it is possible to win without passing the quota, by being neither elected nor eliminated when the field of candidates is thinned to the number of remaining empty seats.


The electoral threshold has different effects on STV than on Party-list PR. For STV many of the votes for candidates below natural threshold are not wasted but transferred to the next-indicated choice, thus many of the voters assist in electing someone even if not the voter's first choice.


In party-list PR, a vote cast for a party below electoral threshold is an unrepresented vote, in almost all cases.


Only a small portion of list PR systems allow votes to cross party lines.


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