1885 - Distribution of seats

[islington]

I am asking for comments and criticism from any forum member with a interest in this subject.

I have been working on the redistribution of Parliamentary seats in 1885, which is a landmark because it produced the first really modern-looking boundary map.

I am in the process of preparing a paper on the subject. It is my intention ultimately to share it with a largely academic group with which I am involved. I am well out of practice when it comes to writing academic-style papers so I want it to be as good as it can be before I put it to the academic group and have it shot to pieces.

So I have three requests.

1. If you have knowledge of this subject, please point out any errors or oversights.
   
2. Whether or not you have knowledge of the subject, I welcome feedback on general comprehensibility and cogency or lack thereof.
3. Please ask questions. I'll try to reply to each one.
   

I haven't finished it yet so I'll post it in instalments. This means you get the boring scene-setting stuff first before we get to the meaty stuff later. Sorry 'bout that.

[islington]

The Use of Population in the Redistribution of 1885 - Fit the First

This paper concerns the redistribution of seats in the UK Parliament as part of the collection of measures in 1884 and 1885 that comprise the Third Reform.

In would be fair to say that each of the First and Second Reforms, in 1832 and 1867-68 respectively, took as its starting-point the current system, whatever it happened to be, and proceeded to modify it to take into account other factors of which population was the most important. So this is not to say that the outcome was determined by population: rather, that the principal consideration was the preservation of much of the pre-existing system, but subject to modifications that were influenced by population.

The 1885 redistribution, however, took a markedly different approach that was almost revolutionary in its effect. For the first time, population (as found at the 1881 census) was the predominant consideration, although a strict population-based approach was substantially modified by taking into account elements of the pre-existing system.

Another key element, as a result of a negotiated agreement between the parties, was a dramatic shift towards single-member constituencies. This continued a trend that was already significant in the First and Second Reforms [numbers]; but the Third took matters much further so that the single-member constituency became the norm throughout the UK and only 24 double seats survived, all of them boroughs and 21 of them in England (one each in Wales, Scotland and Ireland). The remaining 613 territorial constituencies were all single member.

For these reasons – the dominant role of population and the preponderance of single-member constituencies – the 1885 distribution is the first that looks really modern.

This paper addresses two issues.

• First, how in practice was the population principle applied?
• Secondly, how was this done in a manner that reflected elements of the old system that a political decision had been taken to preserve (or at least, to take into account)?

To answer this, the paper assumes that a rules-based approach was taken. This seems inevitable. With 670 seats in all, a process of wrangling and horse-trading, on a matter of such political salience, would surely have been impossibly complex and protracted and in all likelihood would have become completely bogged-down in detail. This does not exclude the possibility (as we shall see) that a certain element of political bargaining may have been involved in the final outcome; but it seems certain that this would merely have involved detailed adjustments to a distribution substantially based on the application of agreed rules.

The paper, then, seeks to discern from the final outcome what the effect of these rules will have been. And as an important preliminary caveat, it must be stressed that this is not quite the same as discerning what the rule actually was: the paper should not be interpreted as predicting that there may lie undetected among some politician’s papers a document referring to ‘priority values’ and the like. So the present paper’s aim in setting out the rules outlined below is simply to show, based on the evidence of the 1885 distribution, what a mathematic population system might look like: and to argue from its closeness to the actual outcome that such a system was almost certainly used in 1885 even though it may have been conceived and expressed in different terms.

[islington]

The Use of Population in the Redistribution of 1885 - Fit the Second

Let us start with a few basic ground rules.

Although the agreed size of the House is 670, the distribution does not involve the 9 university seats. These are unaffected. This leaves 661 territorial seats to be distributed.

Although the majority of Parliamentary boroughs are also parts of the relevant Parliamentary county, to avoid double counting of inhabitants this overlap is ignored and the numbers for Parliamentary counties are exclusive of any boroughs within their boundary.

The population figures are from the 1881 census. These cannot be taken directly from the 1881 census report because, although it contains population figures for Parliamentary constituencies, they naturally cover those in existence at the time – in other words, those resulting from the Second Reform of 1867-68 (subject to minor later adjustments). However, the 1891 census provides population data for the constituencies defined in 1885, including comparison data for the same areas in 1881.

However, there is a problem. It is hard to reconcile some of the figures in the 1891 census with the approach known to have been taken in 1885: for instance, some boroughs are shown as having populations in 1881 below the figure of 15000 that is known to have been the politically agreed minimum below which boroughs would be abolished.

The population figure for each seat given in the Debrett Parliamentary guide for 1886 [check proper name] does not involve any such irregularities; and while one would normally, without hesitation, prefer an official source such as the 1891 census over a private production like Debrett, the figures in the latter appear to have been compiled with a high degree of diligence and must surely be derived from an official source. Therefore, although with considerable hesitation, the Debrett figures for individual counties and boroughs have been used where they differ from the 1891 census, although the reasons for the discrepancies clearly require further investigation.

However, the global figures for counties and boroughs taken together, and for each constituent nation of the UK, are taken from the 1881 comparison numbers in the 1891 census. Any resulting discrepancy will be minor: the discrepancies between Debrett and the census are mostly small and the Debrett numbers, where different, do not appear to be generally larger or smaller than the census numbers so it has been assumed that the variations will broadly balance out.

‘Seat’ and ‘constituency’ mean territorial seats and constituencies unless otherwise stated.

‘Borough’, ‘burgh’ and ‘burghal’ refer to Parliamentary boroughs (burghs in Scotland) or, in the case of burghal districts, to the district as a whole rather than to the individual boroughs (or burghs) within it.

‘County’ and ‘comital’ refer to Parliamentary counties, which are the same as applied after the First Reform in 1832 with the following exceptions:

• The areas defined as being within boroughs (or burghs) are excluded
  
• Each of the divisions of Yorkshire (5) and Lancashire (4) existing after the Second Reform is treated as a separate county, subject to the inclusion of the whole Municipal Borough of Bacup, as created in 1882, in North East Lancashire (previously divided between NE Lancs and SE Lancs).
• The Isle of Wight is treated as a separate county, not as part of Hampshire.
• In Scotland, each of the joint county constituencies is treated as a single county: Orkney and Shetland; Ross and Cromarty; Moray and Nairn; Clackmannan and Kinross; Peebles and Selkirk. The boundary between Moray & Nairn and Invernesshire is as defined in the [check name and date (1871?) of this Act].
  
• In Ireland, the boundaries of counties and counties corporate reflect the [check name and date (1836?) of Act], which removed detached parts.
  

[finsobruce]

May 15, 2022 11:56:37 GMT islington said:

I am asking for comments and criticism from any forum member with a interest in this subject.

I have been working on the redistribution of Parliamentary seats in 1885, which is a landmark because it produced the first really modern-looking boundary map.

I am in the process of preparing a paper on the subject. It is my intention ultimately to share it with a largely academic group with which I am involved. I am well out of practice when it comes to writing academic-style papers so I want it to be as good as it can be before I put it to the academic group and have it shot to pieces.

So I have three requests.

1. If you have knowledge of this subject, please point out any errors or oversights.
   
2. Whether or not you have knowledge of the subject, I welcome feedback on general comprehensibility and cogency or lack thereof.
3. Please ask questions. I'll try to reply to each one.
   

I haven't finished it yet so I'll post it in instalments. This means you get the boring scene-setting stuff first before we get to the meaty stuff later. Sorry 'bout that.

Sounds interesting.

[islington]

The Use of Population in the Redistribution of 1885 - Fit the Third

After the Second Reform, the House had 649 territorial seats: that is, the traditional total of 658 less the 9 university seats.

In 1870 four boroughs, with a total of 6 seats, were disfranchised for corruption. This left 643 seats: England (incl Monmouthshire) 454; Wales 30, Scotland 58, Ireland 101. A political decision was taken to increase the size of the House to 670, i.e. 661 territorial and 9 university.

It seems a further decision was taken that each of the four nations of the UK should receive at least its current number, totalling 643. This meant that 18 seats were available for distribution to take the total to 661. How might this most equitably be done, given the population of each nation as found in 1881?

Here we introduce the concept of the ‘priority value’. This is a number assigned to each potential additional seat for each nation. Where an additional seat falls to be assigned, it is awarded to the nation with the highest PV for an extra seat.

Acknowledgment is made here of the system for assigning between states seats in the House of Representatives in the US. However, although the US system uses PVs in the same way as described here, it calculates them in a slightly different manner that has the effect of generating a very slight bias in favour of smaller states. This bias is so slight that it makes no difference in most apportionments, but there does not seem to be any justification for it as a feature so in the present exercise the PV is calculated in a manner that generates no such bias between larger and smaller areas.

With 643 seats already assigned, how do we allot the 644th?

England has 454 and its PV for a 455th is found by dividing its population of 24615015 by the mid-point between its current allocation, 454, and its proposed allocation, 455 – that is, by 454.5. 24615015 / 454.5 = 54158. This is England’s PV for a 455th seat.

Ireland has 101 and its PV for 102 is found by dividing 101.5 into its population. 5174836 / 101.5 = 50984.

Scotland’s PV for a 59th seat is 3728124 / 58.5 = 63729.

Wales’s PV for a 31st seat is 1359424 / 30.5 = 44571.

It will be seen that Scotland’s PV for an additional seat is comfortably the highest and therefore the 644th seat is a 59th for Scotland.

For the 645th seat, we repeat the exercise with each nation’s PV remaining as above except that Scotland’s PV for a 60th seat is 3728124 / 59.5 = 62658.

This is still the highest so Scotland also receives the 645th seat, but note that with each additional seat the divisor is increased and the PV therefore reduced. As additional seats are awarded, all to Scotland, we reach the 654th seat, Scotland’s 69th, by which point the PV has fallen to 54425.

When we come to the 655th seat we see that Scotland’s PV for a 70th seat is 53642 and this figure is below England’s PV for a 455th, namely 54158. Seat 655 therefore goes to England, whose new PV, for a 456th seat, is now 54040. This is greater than Scotland’s PV for an additional seat so the 656th seat also goes to England. In fact from this point England, because of its much larger population, dominates the process and it also receives seats 657, 658 and 659, its PV for a further seat declining slightly each time. Seat 660, however, goes to Scotland (its 70th), and seat 661, the final one, to England again for a total of 460.

England’s PV for its final seat is 53569, and this compares with the PVs of Ireland and Wales for an additional seat of 50984 and 44571 respectively. This shows that these two countries are overrepresented compared with England and Scotland, although in the case of Ireland the overrepresentation is not by all that much.

So, using this method, the national totals for 661 seats are: England 460; Wales 30; Scotland 70; Ireland 101.

These are the exact numbers that were actually assigned in 1885.

Is this significant?

Well, it could be argued that 643 of the 661 seats were preassigned by the political decision that no nation should lose representation; and with only 18 seats in play, the exact coincidence of the numerically calculated outturn and the actual result is no more than suggestive of the method adopted. But a more stringent test may be applied by applying the same method to the allocation of seats within each country, and in particular within England as it has far more seats than the other nations.

[islington]

May 15, 2022 12:10:31 GMT finsobruce said:

Sounds interesting.

But apparently isn't, judging by the complete lack of other responses.

Never mind. 

Undeterred, I'll plough on later with a discussion of the treatment of boroughs and counties, before going on to look at the actual distribution within England.

Also, I'm making a note here to remind myself that I need to add something about the rules adopted in 1885 regarding multi-member constituencies (the effect of which was that they almost all bit the dust). 

[Deleted]

May 16, 2022 9:38:57 GMT islington said:

May 15, 2022 12:10:31 GMT finsobruce said:

Sounds interesting.

But apparently isn't, judging by the complete lack of other responses.

Never mind. 

Undeterred, I'll plough on later with a discussion of the treatment of boroughs and counties, before going on to look at the actual distribution within England.

Also, I'm making a note here to remind myself that I need to add something about the rules adopted in 1885 regarding multi-member constituencies (the effect of which was that they almost all bit the dust). 

You might get more interest when everything is typed up. I expect a lot of people are also interested enough to read but not knowledgeable enough (or not confident enough in their knowledge) to comment 

[yellowperil]

May 16, 2022 9:46:15 GMT @europeanlefty said:

May 16, 2022 9:38:57 GMT islington said:

But apparently isn't, judging by the complete lack of other responses.

Never mind. 

Undeterred, I'll plough on later with a discussion of the treatment of boroughs and counties, before going on to look at the actual distribution within England.

Also, I'm making a note here to remind myself that I need to add something about the rules adopted in 1885 regarding multi-member constituencies (the effect of which was that they almost all bit the dust). 

You might get more interest when everything is typed up. I expect a lot of people are also interested enough to read but not knowledgeable enough (or not confident enough in their knowledge) to comment 

count me in that category

[islington]

The Use of Population in the Redistribution of 1885 - Fit the Fourth

Before turning to individual nations, however, we should note that this was very much a UK-wide distribution, effected by a single Act of Parliament extending to all four nations, and the evidence is that a political decision was taken that a similar approach should be taken in each nation.

One way in which this was evident was in the treatment of multi-member constituencies. These were a long-established feature of the UK system. Traditionally, prior to the First Reform, in England all counties and almost all boroughs operated as undivided double seats (although in the other nations the picture was more mixed, with a substantial number of single seats). The First and Second Reforms tended to increase the number of single seats in response to pressure to make representation more granular; but even so, after the second reform every English county constituency elected two MPs, except that seven undivided counties returned three and the Isle of Wight only one. As for boroughs, [check number] out of [check number] elected two MPs and a handful of very large industrial towns had three. Meanwhile the City of London elected four as it had for centuries.

The 1885 redistribution swept all this away. Throughout the UK, each county with more than one seat would be split into the appropriate number of divisions returning one MP each. The treatment of boroughs with three or more members was the same; but, in a limited concession to supporters of the two-member tradition, a borough assigned two MPs would remain undivided, returning them both at large, provided that it currently had two or more MPs. (A newly-created two member borough, however, or an existing single-member borough assigned an extra seat, would be divided. In other words, while some existing double seats would be suffered to continue, no new ones would be created.)

The effect was that only 24 double seats, all boroughs, survived after 1885. Of these, 21 were in England; Wales, Scotland and Ireland had one each (respectively, Merthyr Tydfil, Dundee, Cork City). It is this dramatic shift to a preponderance of single-member seats that is the biggest single factor in giving the 1885 distribution its much more modern appearance.

Perhaps even more significant, however, although less obvious from a glance at a map, was the decision that the ratio of population to each MP should nowhere be less than 15000 or greater than 90000.

This six-fold variation may seem unreasonable and excessive by modern standards, but in its historical context it was a huge step forward in terms of equalizing representation. Under the arrangements applying prior to 1885, the ratio (as found at the 1881 census) varied from 301655 (N Lanarkshire) to 2426 (Portarlington), meaning the inhabitants of the latter seat were 124 times more generously represented than those of N Lanarks.

To give effect to the minimum, in all parts of the UK any existing borough with a population below 15000 was abolished. The only exception was Warwick, pop 11800, which survived because it was linked with the immediately adjoining and larger town of Leamington Spa to form a seat with a population of 37879. Even with the salvaging of Warwick, the resulting cull of [check number] boroughs was easily the biggest in UK history.

Note that no Parliamentary county had fewer than 15000. Bute with 17489 was the smallest.

Some boroughs, all in England, were abolished for reasons other than population. Two, Macclesfield and Sandwich, were abolished for corruption. Six, which for historic reasons included very extensive rural areas, were deemed insufficiently urban in general character to continue as boroughs: Aylesbury, Cricklade, East Retford, Shoreham, Stroud, Wenlock. And nine boroughs (seven in the London area, plus Stoke on Trent and Wednesbury) were split into two or more smaller boroughs all of which, in this situation, were regarded as new.

To give effect to the 90000 maximum, any borough or county with a population exceeding that number is automatically preassigned a second seat.

In addition, as a nod to the current system, any existing borough or county with a population between 50000 and 90000 is preassigned a second seat, but only if it already has at least two seats. [Note no special treatment for City]

Any borough or county with fewer than 50000 receives only one seat regardless of the number it currently has.

These arrangements preallocate, in England, 276 seats. To bring it up to its total of 460, therefore, a further 184 need to be assigned by population.

In Ireland, 75 seats are preallocated with 26 more to be assigned by population.

In Scotland, 59 are preallocated with 11 more to be assigned.

In Wales, 27 are preallocated with only 3 more to be assigned.

Before we assign seats to specific counties or boroughs, however, there is a further factor to take into account. This is the key historic distinction between boroughs and counties, and in particular, especially in England, the long-established overrepresentation of the former compared with the latter. The First and Second Reforms had somewhat tilted representation away from boroughs and toward counties, but despite this the imbalance was still severe. A central aim of the 1885 redistribution, above all in England, was to rectify this long-standing anomaly by ensuring that boroughs collectively received no more representation, based on their population, than counties collectively.

(Regarding other nations, Wales was treated the same as England in this respect although with only 3 seats to be assigned the effect was marginal. In Ireland the treatment of boroughs and counties was ambiguous as noted farther on. In Scotland it seems no distinction was made in terms of distribution between counties and burghs.)

[J.G.Harston]

May 15, 2022 12:08:16 GMT islington said:

• In Scotland, each of the joint county constituencies is treated as a single county: Orkney and Shetland; Ross and Cromarty; Moray and Nairn; Clackmannan and Kinross; Peebles and Selkirk. The boundary between Moray & Nairn and Invernesshire is as defined in the [check name and date (1871?) of this Act].

When refering to combined entities, join them with "&" not with "and" (as in your Moray & Nairn example), otherwise long lists get incomprehensible.

Shiregreen and Brightside and Stocksbridge and Upper Don and Dore and Totley.

Shiregreen & Brightside and Stocksbridge & Upper Don and Dore & Totley.

Fit the Forth:

In addition, as a nod to the current system, any existing borough or county with a population between 50000 and 90000 is preassigned a second seat, but only if it already has at least two seats.

Should that be an additional seat? You can't be "given a second seat as long as you already have a second seat".

[YL]

May 16, 2022 21:52:38 GMT J.G.Harston said:

In addition, as a nod to the current system, any existing borough or county with a population between 50000 and 90000 is preassigned a second seat, but only if it already has at least two seats.
Should that be an additional seat? You can't be "given a second seat as long as you already have a second seat".

My interpretation was that if it
- has a population between 50000 and 90000
- currently has two (or more) seats
then it gets preassigned those two seats to keep (or is preassigned two if it currently has more than two).

If it has a population over 90000 then it is preassigned two seats regardless of what it currently has, and if it has a population under 50000 (but above the minimum of 15000) then it is preassigned one seat regardless of what it currently has. And if it has a population between 50000 and 90000 but currently has only one seat it is preassigned one, not two.

[J.G.Harston]

May 17, 2022 6:40:37 GMT YL said:

May 16, 2022 21:52:38 GMT J.G.Harston said:

In addition, as a nod to the current system, any existing borough or county with a population between 50000 and 90000 is preassigned a second seat, but only if it already has at least two seats.
Should that be an additional seat? You can't be "given a second seat as long as you already have a second seat".

My interpretation was that if it
- has a population between 50000 and 90000
- currently has two (or more) seats
then it gets preassigned those two seats to keep (or is preassigned two if it currently has more than two).

If it has a population over 90000 then it is preassigned two seats regardless of what it currently has, and if it has a population under 50000 (but above the minimum of 15000) then it is preassigned one seat regardless of what it currently has. And if it has a population between 50000 and 90000 but currently has only one seat it is preassigned one, not two.

Might be worth re-writing that sentence then.

I hope we're not writing your paper by comiittee, issy. Get that too much at campaign meetings.

[islington]

YL's interpretation is correct, but JG's post was very helpful because it made me realize that I hadn't properly explained the concept of preallocation, and how it acts to modify the application of a strict population rule in such a way as to protect some elements of the previous system that it was clearly thought politically important to maintain.

I'll post something on this when I've had time to write it.

[islington]

For instance, to allocate 661 territorial seats between the UK nations, you could theoretically have started by preallocating only a single seat to each of the four nations and then assigning all the rest strictly by population using the PV system. I've just worked it out. It gives England 466, Ireland 98, Scotland 71, Wales 26. This represents the best and most equal apportionment based totally on population. But on the other hand, it doesn't deliver the political objective that no nation should lose representation.

To square this circle, preallocation is my suggested mechanism. You preallocate seats to guarantee an outcome that meets agreed political requirements; then you apportion however many seats are left strictly by numbers using the PV system. In this example, you have to preallocate the great majority of seats - 643 out of 661 - to protect the nations' current representation (E 454, I 101, S 58, W 30). This leaves only 18 seats to be apportioned by population, in practice all going either to Scotland (12 of them) or to England (the other 6) since these were the least well represented nations prior to the redistribution. The result - England 460, Ireland 101, Scotland 70, Wales 30 - coincides exactly with what happened in 1885.

To explore whether this identity of outcome is significant, or merely a matter of happenstance, I'll be looking in more detail at what happens when you apply the same approach to the distribution of seats within, as opposed to between, the various nations. I'll look at all parts of the UK but in particular detail at England since that's where most of the seats are located. 

[johnloony]

May 15, 2022 13:27:22 GMT islington said:

Here we introduce the concept of the ‘priority value’. This is a number assigned to each potential additional seat for each nation. Where an additional seat falls to be assigned, it is awarded to the nation with the highest PV for an extra seat.

Acknowledgment is made here of the system for assigning between states seats in the House of Representatives in the US. However, although the US system uses PVs in the same way as described here, it calculates them in a slightly different manner that has the effect of generating a very slight bias in favour of smaller states. This bias is so slight that it makes no difference in most apportionments, but there does not seem to be any justification for it as a feature so in the present exercise the PV is calculated in a manner that generates no such bias between larger and smaller areas.

With 643 seats already assigned, how do we allot the 644th?

England has 454 and its PV for a 455th is found by dividing its population of 24615015 by the mid-point between its current allocation, 454, and its proposed allocation, 455 – that is, by 454.5. 24615015 / 454.5 = 54158. This is England’s PV for a 455th seat.

Ireland has 101 and its PV for 102 is found by dividing 101.5 into its population. 5174836 / 101.5 = 50984.

Scotland’s PV for a 59th seat is 3728124 / 58.5 = 63729.

Wales’s PV for a 31st seat is 1359424 / 30.5 = 44571.

It will be seen that Scotland’s PV for an additional seat is comfortably the highest and therefore the 644th seat is a 59th for Scotland.

For the 645th seat, we repeat the exercise with each nation’s PV remaining as above except that Scotland’s PV for a 60th seat is 3728124 / 59.5 = 62658.

You are thus describing Sainte-Laguë divisors, except that you call them 0.5, 1.5, 2.5, 3.5 etc whereas S-L calls them 1,3,5,7 etc. Which means that they do in fact provide a bias in favour of smaller units than bigger ones. It is therefore also a different method from the one used for allocating congressional seats to the states in the USA, which uses the geometric mean rather than your use of the arithmetic mean (except that in the USA the first divisor is 0, in order to guarantee at least 1 seat for each state).

In other words (if I have understood your description correctly), your statement about bias relating to smaller states is the wrong way round.

en.m.wikipedia.org/wiki/United_States_congressional_apportionment

[islington]

May 17, 2022 13:27:37 GMT johnloony said:

May 15, 2022 13:27:22 GMT islington said:

Here we introduce the concept of the ‘priority value’. This is a number assigned to each potential additional seat for each nation. Where an additional seat falls to be assigned, it is awarded to the nation with the highest PV for an extra seat.

Acknowledgment is made here of the system for assigning between states seats in the House of Representatives in the US. However, although the US system uses PVs in the same way as described here, it calculates them in a slightly different manner that has the effect of generating a very slight bias in favour of smaller states. This bias is so slight that it makes no difference in most apportionments, but there does not seem to be any justification for it as a feature so in the present exercise the PV is calculated in a manner that generates no such bias between larger and smaller areas.

With 643 seats already assigned, how do we allot the 644th?

England has 454 and its PV for a 455th is found by dividing its population of 24615015 by the mid-point between its current allocation, 454, and its proposed allocation, 455 – that is, by 454.5. 24615015 / 454.5 = 54158. This is England’s PV for a 455th seat.

Ireland has 101 and its PV for 102 is found by dividing 101.5 into its population. 5174836 / 101.5 = 50984.

Scotland’s PV for a 59th seat is 3728124 / 58.5 = 63729.

Wales’s PV for a 31st seat is 1359424 / 30.5 = 44571.

It will be seen that Scotland’s PV for an additional seat is comfortably the highest and therefore the 644th seat is a 59th for Scotland.

For the 645th seat, we repeat the exercise with each nation’s PV remaining as above except that Scotland’s PV for a 60th seat is 3728124 / 59.5 = 62658.

You are thus describing Sainte-Laguë divisors, except that you call them 0.5, 1.5, 2.5, 3.5 etc whereas S-L calls them 1,3,5,7 etc. Which means that they do in fact provide a bias in favour of smaller units than bigger ones. It is therefore also a different method from the one used for allocating congressional seats to the states in the USA, which uses the geometric mean rather than your use of the arithmetic mean (except that in the USA the first divisor is 0, in order to guarantee at least 1 seat for each state).

In other words (if I have understood your description correctly), your statement about bias relating to smaller states is the wrong way round.

en.m.wikipedia.org/wiki/United_States_congressional_apportionment

Well, I hesitate to cross swords with Mr Loony on this subject, but ...

Since the approach here is to derive a series of numerical values that can be ranked in order, so long as the divisors are in the same ratios to each other the system will still work and the results, in terms of apportionment, will be the same. So it could be 1.5, 2.5, 3.5, 4.5 ... (as I have it); or 3, 5, 7, 9 ... ;  or any other set of divisors you like so long as you keep the same ratios. The reason I prefer 1.5, 2.5 and so on, is that it makes more intuitive sense (to me anyway) and the PVs that result are similar to the size of individual single-member seats.

I've deliberately omitted 0.5 from the list because when it comes to the first seat for a given area, I think it works best if it is assigned automatically; or in other words, as in the US, if the divisor for the first seat is 0 (and the PV therefore is infinity).

Now, on the subject of small state bias: the US uses the Huntington-Hill (H-H) method. In H-H, when a state has n representatives, its PV for an extra seat is calculated by dividing its population by the geometric mean of (n) and (n+1). The geometric mean of two values is the square root of their product, so if n=1, say, the divisor is the square root of 1x2=2, or 1.4142 ... . This is a smaller divisor, and therefore a larger PV, than you get by using the arithmetic mean, i.e. half of their sum, namely 1.5. Under H-H, if n=2, the divisor (to four places) to calculate the PV for a third seat is 2.4495 and for successive numbers it is 3.4641, 4.4721, 5.4772, 6.4807 and so on. We can see what is happening here: as a state's number of seats increases, the geometric mean tends closer and closer to the arithmetic mean, and when we get to larger numbers the difference is trifling (the divisor for a 50th seat, for instance, is 49.4975). This is what I mean by a bias in favour of smaller states.

But this should not be exaggerated. The bias is slight and in terms of the number of seats actually awarded to each state in most US apportionments it makes no difference whether you use the arithmetic or the geometric mean. On the odd occasion where using the geometric mean does make a difference, however, it will always favour a smaller state at the expense of a larger one. 

[islington]

Coming back to preallocation - 

In the distribution of seats within each country, preallocation was used to achieve certain outcomes that were clearly thought to be politically desirable, and that might not have been achieved if a purist population-based apportionment had taken place.

The preallocation of a first seat to any borough or county is assumed to be uncontroversial and is little more than a practical way of ensuring that each borough or county receives a whole number of seats.

The preallocation of a second seat to boroughs or counties with a population over 90000 furthers the intention that the ratio of inhabitants to MP should never exceed 90000. (This is not quite the same as saying that no constituency should exceed 90000 because some constituencies returned two MPs.)

It may be asked why, if we wish to avoid breaching the 90000 limit, further seats are not preallocated to boroughs or counties with very large populations such as, for instance, Liverpool with 601050. The answer is that there are enough seats available for the population-based apportionment to ensure that these areas will be assigned enough seats to allow them to be split into divisions that do not exceed 90000. So further preallocation is not necessary; and moreover, it would make no difference to the outcome because the preallocation of an extra seat to such an area would simply diminish its entitlement to an extra seat on grounds of population. But it cannot be assumed that the same will apply to boroughs or counties whose population exceeds 90000 by only a relatively small account - thus they require the protection of preallocation to ensure that they are not left with only one seat.

This applies even more to boroughs or counties with populations below 90000. If population alone is used, it is likely that many of these will fail to gain a second seat. So, since a political decision has been taken that a multi-member borough or county should retain two seats unless its population is below 50000, preallocation of a second seat is essential.

When we turn to apportionment within the various nations, it will be seen that preallocation of a second seat has a significant impact on the eventual outcome - it results in a second seat for a number of boroughs and counties that would not have received one had population alone been used.
    

[J.G.Harston]

An interesting point - in light of various representations to the current review (and previous zombie reviews) is the use of the census population rather than the electorate. Is there any information why population was used and not electorate? Was there an assessment (or an assumption) that the electorate was a sufficiently consistant fraction of the population that population was a near-equal proxy?

Of relavance to the current reviews, the population in 1881 will have had a much lower proportion of foreigners than today's population, so the census electorate could fairly easily have been asserted to be a count of British Citizens. Non-citizens would have been highly concentrated in possibly no more than a few dozen enumeration districts.

[maxque]

May 17, 2022 21:13:54 GMT J.G.Harston said:

An interesting point - in light of various representations to the current review (and previous zombie reviews) is the use of the census population rather than the electorate. Is there any information why population was used and not electorate? Was there an assessment (or an assumption) that the electorate was a sufficiently consistant fraction of the population that population was a near-equal proxy?

Of relavance to the current reviews, the population in 1881 will have had a much lower proportion of foreigners than today's population, so the census electorate could fairly easily have been asserted to be a count of British Citizens. Non-citizens would have been highly concentrated in possibly no more than a few dozen enumeration districts.

Wasn't there a logistical issue caused by some people qualifying for both the borough and county franchise and, so, being twice on the rolls?

[johnloony]

May 17, 2022 14:26:03 GMT islington said:

Now, on the subject of small state bias: the US uses the Huntington-Hill (H-H) method. In H-H, when a state has n representatives, its PV for an extra seat is calculated by dividing its population by the geometric mean of (n) and (n+1). The geometric mean of two values is the square root of their product, so if n=1, say, the divisor is the square root of 1x2=2, or 1.4142 ... . This is a smaller divisor, and therefore a larger PV, than you get by using the arithmetic mean, i.e. half of their sum, namely 1.5. Under H-H, if n=2, the divisor (to four places) to calculate the PV for a third seat is 2.4495 and for successive numbers it is 3.4641, 4.4721, 5.4772, 6.4807 and so on. We can see what is happening here: as a state's number of seats increases, the geometric mean tends closer and closer to the arithmetic mean, and when we get to larger numbers the difference is trifling (the divisor for a 50th seat, for instance, is 49.4975). This is what I mean by a bias in favour of smaller states.

But this should not be exaggerated. The bias is slight and in terms of the number of seats actually awarded to each state in most US apportionments it makes no difference whether you use the arithmetic or the geometric mean. On the odd occasion where using the geometric mean does make a difference, however, it will always favour a smaller state at the expense of a larger one. 

The point is that the H-H method (geometric mean) as used in the USA is less likely to create a bias towards smaller states than the S-L divisors (arithmetic mean) in your example.