Threshold -- how do they boil down?

Thresholds (the number of votes needed to take a seat) have some unexpected features.

I mean natural thresholds, not electoral threshold that stays some parties will not get any representation)

Here is proof for the supposition that because list pr and STV have high percentage of effective votes (80 to 90 percent of valid votes cast are used to elect members), the same number of seats (100) is spread over more votes 

and thus in a scientific system (list PR or STV), the threshold is actually higher than in FPTP.

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effective threshold is basic amount that ensures election to a seat either for a party or a candidate depending on system.

assuming fair voting where each successful candidate receive same or about same number of votes.

say where 1M votes cast for 100 seats

districting makes little difference (bear with me)

math shows that if districting is at-large, to take one seat needs 10,000 votes (1/100 X 1M)

if province is divided into ten districts, 100,000 votes in each,

to take one seat takes 1/10 X 100,000 = the same 10,000.

that is using the Hare quota.

Droop quota, the usual used in STV, means there is some difference when districts multiply

if districting is at-large, to take one seat needs 9901 votes (1/101 X 1M)

if province is divided into ten districts, 100,000 votes in each,

to take one seat takes 1/11 of district votes = only 9090.

if province is divided into 20 districts, each electing 5,           50,000 votes in each,

to take one seat takes 1/6th of district votes = only    8333.

if province is divided into 50 districts, each electing 2,           20,000 votes in each,

to take one seat takes 1/3 of district votes = only 6667.

meanwhile as effective threshold

 we say that it is approx. one percent at-large, 

10 percent in DM-10 district.

17 pervent in DM-5 district

33 percent in DM-2 district.

so despite higher precentage (in district), the actual threshold is fewer votes.

and then we look at FPTP

one member in a district     100 districts       so about 10,000 votes in each district 

(district turnout can vary but let's assume unrealistically uniform turnout.)

depending how vote is split and how many candidates are in running.

candidate is certain to win with 5001 votes.

but often a candidate wins with 3300 votes in district with 10,000 vote cast,

some with just 2500 votes.

(and of course some with 8,000 votes cast)

so threshold under FPTP is something like as few as 3000 votes

while under fair voting (STV or list PR threshold is like 8333 to 10,000.

(6667 in two-member district which is too low to be truly PR in overall effect)

so point is that basic effective threshold under STV or list PR is higher than under FPTP so already you prevent small parties from being elected.

while under FPTP any candidate with about 3000 votes may win in a local contest (sometimes has happened historically) and take that one seat, but I would not bank on it - it is just an accidental result that happens when one uses a unscientific method.

but of course a scientific system is more fair and dependable, 

but why is threshold under list pr or STV so much higher than FPTP can be?

the difference is the waste of votes

under FPTP as much as two-thirds of votes in a district may be wasted.

while under List pr or STV the waste is perhaps half or so of quota for each major party,   and about one quota (however much that is) is wasted under STV.

the difference is that

under list PR  1/100th of seats goes to 1/100th of effective votes (10,000);

under STV 1/100th of seats goes to 1/101th of effective votes (9900)

or with districting, perhaps 1/6th of 5/100ths of effective votes (8333)

under FPTP 1/100th of seats goes to perhaps 1/3rd of 1/100th of votes,

or as little as 1/2 of 1/100th of votes, if district majority (Instant-Runoff Voting) is used.

because list PR and STV have high percentage of effective votes (80 to 90 percent of valid votes cast are used to elect members),

the same number of seats (100) is spread over more votes 

and thus in a scientific system, the threshold is actually higher than in FPTP.

more fair but higher threshold.

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Footnote:

came across PR thesis available online      Glashan's "PR in Canada" (1951)

a bit dated  but looks to have interesting info