Sainte-Lague Party-List, the Unbiased Proportional-Representation
Topics in this article:
- Why party-list proportional-representation
- Parliamentary Government
- Sainte-Lague allocation-rule
- Why Sainte-Lague is unbiased
- Open List Sainte-Lague
- Districts or at-large?
- Ranked Party-List, Woodall’s Proposal
- The Sainte-Lague Optional Systematic-Procedure
- Other PR allocation-rules.
- Partition
————-
- Why Party-List Proportional-Representation:
Of course equal-representation for all is its own justification (something Constitutionally-denied now).
Party-list can be, & has been, devised to maximally achieve that goal. It’s Sainte-Lague allocation-rule achieves unbias, not favoring large or small parties (as shown below).
An unbiased proportionally-elected governing-body is a near-perfect scale-model of the population.
STV (Single-Transferable-Vote) can only roughly, sloppily, approximate that ideal. Additionally, STV usually (always?) uses the Droop-quota, which, I’ve read, is as large-biased as d’Hondt, which, under-represents the [1,2) seat-quotas-interval by 33%. That’s really bad.
A big advantage of party-list PR is its rapid immediate set up & implementation. That could matter a lot in uncertain times.
Party-List PR doesn’t need any new balloting equipment or ballots. The only difference is that parties instead of candidates are named next to the voting-bubbles on the ballot.
Party-list doesn’t need any different kind of counting. Exactly the same count-software can & would be used. Just count the votes for each party, as we currently count votes for each candidate.
…& then, given the vote-count numbers, anyone could do the seats-allocation, at their kitchen-table, using a hand-calculator or pencil-&-paper.
Therefore, there’s absolutely no delay in setting-up & implementing party-list PR.
———
Right now, there’s been more emphasis on single-winner voting-systems. That’s because the 1787 Constitution’s requirements, due to “the Great Compromise”, would make a joke of any attempt at national equal-representation. Constitutional-amendments are particularly difficult, & it’s felt by many that the necessary amendments would be quite impossible.
Hence the interest in single-winner voting-systems.
e.g., Ranked-Choice Voting (RCV) has been sweeping the country, in municipalities, & even in 2 states. But look what happened to the 5 state-proposals in November ’24: They all failed by a huge margin. They failed as spectacularly as RCV had previously been spectacularly succeeding everywhere. Let’s just say that it looks like nationwide electoral-reform of any kind isn’t going to happen.
That’s not to say that RCV shouldn’t continue to be tried anyway. Maybe its popularity is so great as to be able to overcome no matter what. But, in any case, it’s necessary to try anyway, even if only for the sake of showing everyone what the situation is.
But, as mentioned above, maybe it’s time to be looking at party-list PR for state legislatures.
—————-
But PR has other advantages too:
Choosing a single winner is a hard-choice, inevitably. No single-winner method escapes that difficulty.
Single-winner methods differ in their problems. Approval, Score & STAR mitigate the spoiler-problem, compared to Plurality.
Ranked-Choice Voting (RCV) has less spoiler-problem than theirs (RCV completely eliminates spoiler-problem for the Mutual-Majority).
The very best winning-votes (wv) Condorcet versions, such as Ranked-Pairs(wv), arguably have the least spoiler-problem of all (with their considerate, though not essential, luxury of eliminating the spoiler problem for the voters not in the Mutual-Majority), but they completely lack RCV’s success & precedent.
But all single-winner methods share the hard-choice of a single-winner. That can affect their ability to avoid the spoiler problem, for example.
Party-lists PR, in comparison, is a simple, direct & unproblematic proportional-allocation, without the problems that inevitably accompany the choice of a single winner.
But it should be added that, even with the best electoral-system, in our pathological society, there’s no such thing as a reliably completely unproblematic democracy, because there always remains the societal problem of the possibility of an antagonistically-divided majority.
- Parliamentary Government:
The public should be in charge. As mentioned above, a proportional governing-body is a near-perfect scale-model of the population.
Therefore, that governing-body should be the absolute, supreme, exclusive power in government. With pure parliamentary-government, the governing-body (Parliament) achieves that. Every other part of government would be appointed by Parliament, & could be over-ruled or replaced by Parliament at any time.
Government by the people as literally as possible.
- The Sainte-Lague Allocation-Rule:
X% of the votes should win X% of the seats. Therefore, multiply each party’s vote % by the desired house-size, to get that party’s “rightful-seats”.
But of course each party’s rightful-seats will end in a fraction. So round it off to the nearest whole number. Assign to it that whole number of seats.
But, due to the vagaries of rounding, the assigned-seats might not sum to exactly the desired house-size. The might sometimes differ from it by a seat.
That would be fine, as-is. A variation of 1-seat in house-size wouldn’t be problematic.
But, because a constant pre-defined house-size is usually desired, that’s easily achieved:
Just multiply each party’s rightful-seats by some same number (the “multiplier”) before rounding to the nearest whole number.
Choose a multiplier that achieves the desired house-size.
A suitable multiplier can be found by trial-&-error. Or there are systematic-procedures for finding one. It doesn’t matter how the multiplier is found. All that’s necessary is that the allocation-rule specify that a multiplier be used which achieves the desired house-size.
The use of that multiplier is the obvious & natural way to adjust the house-size to its desired value.
When Sainte-Lague was first proposed by Daniel Webster in 1830, for apportionment of the House of Representatives (by the name of “Webster’s method”), it was evidently proposed with no systematic procedure, & so evidently trial-&-error was intended & used for Webster’s method apportionment. It was used for a number of House-apportionments.
Of course now Sainte-Lague is in use for proportional-representation in a number of countries.
- Why Sainte-Lague is Unbiased:
This is about a quotas-interval.
A party’s “quotas” is its rightful-seats, or its rightful-seats multiplied by the multiplier.
A quotas-interval (“interval” for short) is an interval between two consecutive whole numbers of quotas.
The interval between the whole numbers a & b is written [a,b), to denote that it includes a, but not b.
So, [a,b) could be [0,1), or [1,2), or [7,8), etc.
It’s the interval in which are the parties that have between a & b quotas.
For parties between a & b, all of the “divisor-methods”, including Sainte-Lague, round a party’s rightful-seats to a whole number. Sainte-Lague, most reasonably, rounds to the nearest whole number. d’Hondt always just rounds down (to a).
I’ll call the rounding-point “R”. That’s the point above which the party’s rightful-seats is rounded up, from a to b.
So, for Sainte-Lague, R = a + .5
For d’Hondt, R = b.
This is all just a symbolic way of describing the obvious natural Sainte-Lague allocation that I described above, for the purpose of demonstrating Sainte-Lague’s unbias.
Well, in [a,b), what’s the average number of quotas? It ranges from a to b, where (because a & b are consecutive whole numbers) b = a + 1. So of course the average number of quotas for all the possible party-sizes in [a,b) is a + .5
I’ll call that average “Q”
Q = a + .5
Now, what’s the average number of seats for all the possible party-sizes in [a,b)?
Well, half of them have less than R, because R, = a + .5, is the halfway-point. So half get only a seats. The other half, symmetrically, have more than R, & get b seats. So then, isn’t it reasonable to say that the average number of seat awarded to a party in [a,b) is R? …which is equal to a + .5
I’ll let S represent the average number of seats awarded in [a,b).
S = a + .5
…&, from before, that’s what Q is too. S = Q.
i.e. S/Q = 1.
That’s the seats per quota, s/q, in [a,b). …in every [a,b). a & b could be any two consecutive whole numbers, & s/q for that interval is 1.
So s/q is the same for every interval [a,b), no matter which two consecutive whole numbers a & b are. Because s/q is the same for all intervals, no matter how large or small a & b are, that’s why Sainte-Lague can be said to be unbiased.
QED. (quod erat demonstrandum: “…which was to be demonstrated”).
What was demonstrated above is only true because R = a + .5
It wouldn’t be so for any other R.
e.g., for d’Hondt’s R = b, it isn’t so. d’Hondt under-represents [1,2) by 33% (as mentioned in section 1, & demonstrated later below).
- Open-List PR:
On an open-list ballot, one can mark a vote for a party, or, instead, for one of the candidates listed below the party-name. If you do the latter, your vote counts not only as a vote for the party, but also as a vote for that candidate, in the determination of which candidates fill the seats that the party wins.
The seats that the party wins are first filled by its listed candidates getting the most votes. If the candidates voted-for aren’t numerous enough to fill all of the won seats, then the remaining seats are filled from the remaining listed candidates, in order of rank in the list.
Or, optionally, you can just vote for the party instead of for one of its candidates.
It’s probably best to propose open-list.
- Districts Or At-Large?:
There’s a prevailing notion that it’s necessary to elect in districts, to allow local representation. Not so! Any quota-size group, in any locality, has the power to vote for & elect a candidate (an independent, or a member of a local party). …if they want to.
Those are the key words! If you want to elect a local candidate, but others in your locality prefer to vote for other candidates that they like, that’s their right.
It’s one thing, a good thing, to allow people to vote for & elect a local candidate. It’s quite another thing to require & coerce them to. Coercion isn’t freedom. Voters should have the freedom to vote for & elect whomever they want to, local or otherwise.
There are big advantages to at-large PR:
- It avoids the expense, time & trouble of districting. Earlier I mentioned that party-list can be quickly immediately set up & implemented. Much moreso if it isn’t necessary to do the time-consuming & contentious districting.
- It completely eliminates the gerrymandering problem.
- An at-large PR election is a many-seat election, allowing finer proportionality. The average size of a state-legislature is about 150 seats. If it’s unicameral, that’s a 150 seat at-large PR election.
In Sainte-Lague, the percent of the vote needed to get a seat is typically:
70/(the house-size).
So, with 150 seats, that’s 70/150 of a percent = 7/15 of a percent = about .4467 %
i.e. less than half of a percent.
Less than half of a percent of the vote will typically win a seat in Sainte-Lague, in a 150-seat at-large Parliamentary election.
- Ranked Party-List PR:
Woodall proposed (& many others likewise have) that party-list PR allow a voter to indicate a 2nd choice party, or even a 3rd choice party, or, in Woodall’s proposal, even an arbitrarily-long ranking of parties.
That would enable you to vote for your favorite party without concern that your vote will be wasted if it doesn’t win a seat. If that party wins a seats, then that completes your vote & its effect. But, if it doesn’t win a seat, it’s eliminated. If any 1 or more parties fail to win a seat, then a 2nd allocation is done, in which votes for an unseated (now eliminated) party go to the voter’s next choice.
That could just consist of allowing a 2nd choice party, & a 2nd allocation. Or 2nd & 3rd choice party & 3 allocations.
…or, at the extreme, it could be an arbitrary number of choices, with as many allocations as thereby needed. That would be something feasible for an automated poll, or for a public electoral system to maybe consider some time after the simpler & more modest 1, 2 or 3 choice election has been adopted & in use..
It’s most often proposed to only allow a 2nd choice party, with 2 allocations. I like 3 choices & allocations.
With only 2 or 3 choices allowed, it would be easy to implement, by letting the voter make-out 2 or 3 ballot-papers instead of just one. Thereby no new ballot or balloting-equipment is needed.
That would suit the need for an electoral-system that could be immediately quickly be set up & implemented, as I spoke of above.
- Optional Systematic Procedures for Sainte-Lague:
If you look up Sainte-Lague, you’ll find it defined as a systematic-procedure. That’s how European countries have evidently chosen to legally define it. It needn’t be defined that way.
For House-apportionment, When Daniel Webster proposed the same allocation-rule in 1830, there evidently wasn’t any systematic-procedure specified. It’s completely sufficient to simply specify that a multiplier be used that adjusts the house-size to the desired value.
Nonetheless, as I said, European countries evidently define Sainte-Lague by a systematic-procedure. Let me describe it:
- A) The European Wording:
Suppose we start with a very small multiplier, much smaller than 1. …a multiplier so small that no party gets a seat.
Then we start increasing the size of the multiplier, until we start seating parties.
Initially, when no one has a seat, & we start increasing the multiplier: Which party will be the first one to get a seat? Well, divide each party’s rightful-seats by the first round-up point. The 1st round-up point, for a party’s 1st seat, is .5
So divide each party’s rightful-seats by .5 & award the first seat to the party with the largest division-result. …because it will be the first party to qualify for a seat as the multiplier is raised.
Now, what about the next seat to be awarded?
Well, just as before, for each party, divide its rightful-seats by the roundoff-point for a next seat (one more than it currently has). Again, the party with the highest division-result will be the one that gets a next seat as we raise the multiplier. Award the next seat to that party.
Continue that procedure until the desired house-size is reached.
That’s the European Sainte-Lague systematic-procedure.
Two things to mention:
- a) To deter & thwart splitting-strategy, the 1st roundoff-point is raised from .5 to .7
- b) Because when Sainte-Lague was first adopted, there were no computers or electronic-calculators, it was desirable to avoid the fractional numbers that are the round-up points, such as 1.5, 2.5, 3.5m 4.5 etc. That was done by doubling all of those numbers to 3, 5, 7,9 etc.
(of course the doubling changes the .7 round-up point to 1.4)
The result of those 2 changes is the following sequence of numbers by which to divide the party’s rightful-seats, to determine who wins the next seat:
1.4, 3, 5, 7,9 …etc.
That’s the Sainte-Lague rule defined in Europe. It’s often called the Odd-Numbers Rule.
I should add that of course, instead of the partys’ rightful-seats, their raw vote-totals could be used just as well, & that’s how it’s said in the European systematic-procedure.
- B) The Briefest Systematic Procedure:
The European wording starts with everyone having 0 seats, so that it will consist only of adding seats. That simplifies the wording. But of course, with 150 seats, that unidirectional systematic-procedure us going to take a long time. It would be a lot faster to just use trial-&-error to find a multiplier that gives 150 seats.
But there’s a way to do a systematic-procedure that’s quicker than both trial-&-error & the European rule:
Start as I described in my initial definition: Determine each party’s rightful-seats, by multiplying each party’s vote % by the desired house-size. Then round off each party’s rightful-seats to the nearest whole number.
What if that awards a seat too few? Then you do exactly what the European rule says: Divide each party’s rightful-seats (or its votes) by the next round-up point, to determine which party will be the first to get a seat, as the multiplier is raised from 1 to larger values.
Of course, just as in the European rule, if it’s necessary to add more than one seat, the procedure is repeated.
But what if the number of seats initially awarded is more than the desired house-size? Then just do the opposite!
Divide each party’s rightful-seats by the next rounding-point below its current number of seats. The party with the lowest division-result is the one that will round-down first, as the multiplier is lowered.
So reduce its number of seats by one.
…&, as before, repeat the procedure if it’s necessary to lose more than one seat.
That’s the quickest briefest way to do the allocation.
9, Other Allocation-Rules:
The other allocation-rules in use are d’Hondt & Largest-Remainder.
- a) d’Hondt:
The only difference is that, instead of rounding to the nearest whole number, everyone is rounded down. So, effectively, in [a,b), the round-up point is b.
The resulting European d’Hondt systematic procedure therefore, for each next seat, divides each party’s seats by that next number of seats: 1,2,3,4…etc. for each of its successive next seats.
d’Hondt under-represents the [1,2) interval by 33%. Here’s why:
Average quotas for the possible numbers of quotas in [a,b): Same as in Sainte-Lague: a + .5
That’s of course independent of what the allocation-rule is.
So Q = a + .5 just as with Sainte-Lague.
What about the average number of seats awarded, over all the possible number of quotas for parties in [a,b)? Well, every party in [a,b) gets a seats, so the average is a.
S = a.
So, S/Q = a/(a+.5)
So, in the [1,2) interval:
S/Q = 1/1.5 = 2/3.
Thus, d’Hondt under-represents [1,2) by 33%
QED.
- b) Largest-Remainder:
Largest-Remainder awards to each party the whole-number part of its rightful-seats. Then it awards the remaining seats to the parties in order of their remainders (fractional-parts of their rightful-seats). That’s a bastardized patchwork of two allocation-procedures, & its proportionality is as sloppy & inaccurate as that of STV.
- Partition:
I just have to ask: Does anyone think that it makes any sense for people who want opposite kinds of government to have to live under the same government???
That’s unfair to everyone. Everyone is dissatisfied.
Why not let the different-preferring sets of people to have the governments that they want? That’s democracy. Otherwise isn’t.
Leave a Reply