Overview of ERRE Constrained Simulations

Introduction

I presented my modelling work to the House of Commons Special Committee on Electoral Reform (ERRE) on 19-Oct-2016. The next day they passed a motion asking for more work within three constraints:

  1. Keep the composite Gallagher index as low as possible.
  2. Each projection must maintain the total number of Members for each Province and Territory, unchanged from the number allocated through the 2015 redistribution
  3. If any of the projections require a redistribution, the redistribution must be capable of being executed on an expedited basis, and as a result the only method of redistribution employed should be to merge existing ridings in sets of two, three, or more.

These are significant constraints, cutting to the core of the most common techniques for obtaining proportionality. Pure multi-member systems such as STV meet these constraints easily, but anything that requires a top-up layer such as MMP or RU-PR is hard.

This page summarizes the systems developed for ERRE.

The report submitted to the Committee is here.

Proportionality

erre_mmp5050_LargeRegions-MMP_AV1721662%-0%0%-0%-1%1.4%2.5%46%97%2015
erre_mmp5050_LargeRegions-MMP_FPTP1721661%-0%0%-0%-0%1.1%2.3%47%97%2015
erre_mmp5050_ProvRegions-MMP_AV1721661%0%0%-0%-0%0.8%1.8%46%99%2015
erre_mmp5050_ProvRegions-MMP_FPTP1721661%0%0%-0%-0%0.8%1.8%47%99%2015
erre_mmp5050_SmallRegions-MMP_AV1721662%1%0%-0%-3%2.6%3.5%46%96%2015
erre_mmp5050_SmallRegions-MMP_FPTP1721663%1%-0%-0%-3%2.8%3.7%47%96%2015
erre_ru_multiples_10pct-RcRUPR301372%-0%0%-1%-0%1.3%2.4%88%99%2015
erre_ru_multiples_15pct-RcRUPR287511%0%0%-0%-0%0.9%2.0%88%99%2015
erre_ru_multiples_20pct-RcRUPR268701%0%0%-0%-0%0.8%1.8%85%99%2015
erre_ru_singles-RcRUPR293454%-1%0%-1%-2%3.2%5.6%76%98%2015
erre_ru_singles-STVplus293455%-1%-1%-1%-2%3.8%5.6%77%98%2015
erre_ru_singles-FptpList293451%1%-0%-0%-1%1.4%2.9%79%98%2015
Alternative Vote338024%-11%-4%-4%-3%19.0%24.0%48%94%2015
First-Past-The-Post338015%-3%-7%-2%-3%12.0%17.2%48%94%2015
kingsley-RcRUPR33808%-3%0%-1%-3%6.7%10.7%79%93%2015
lpr_no_topup-SimpleLPR33804%0%0%-1%-3%3.8%5.8%90%97%2015
Local Proportional Representation
(no top-up)
33805%-0%0%-1%-3%4.0%6.0%90%97%2015
lpr_no_topup-STV33807%-1%-2%-1%-3%5.6%6.5%91%97%2015
stv_huge-STV33802%0%0%-0%-1%1.9%2.4%98%98%2015
Single Transferable Vote
(Medium-sized Regions)
33804%-0%0%-0%-3%3.3%4.3%96%97%2015
Single Transferable Vote
(Small Regions)
33807%-1%-2%-1%-3%5.6%6.5%90%93%2015

Footnotes

  1. Number of Local MPs is the total number of MPs representing specific ridings. Those ridings may be either single-member ridings or multi-member.
  2. Number of Regional MPs is the total number of MPs that represent multiple ridings. This happens in systems with top-up seats such as MMP and RU-PR.
  3. Over-Representation by Party is the percentage of MPs in Parliament minus the percentage of the popular vote. For example, in 2015 under FPTP the Liberals received 54.4% of the seats but only 39.5% of the vote for an over-representation of (54.4 - 39.5) = 14.9%. Negative numbers mean the party was under-represented.
  4. Gallagher Index is a measure of disproportionality. It combines both over and under-representation for each party into a single number. Gallagherindicies less than 5 are excellent.
  5. Gallagher Index 2015 is the Gallagher Index for the simulated 2015 election.
  6. Gallagher Index Composite is the average of the Gallagher Indices for each province and territory, weighted by its number of seats. This corrects for a problem in calculating the Gallagher Index for the nation as a whole, which can can hide regional disproportionalities such as the significant over-representation of Conservatives in the Prairies offsetting the over-representation of Liberals in the Maritimes.
  7. % Voters with Preferred Local MP is the percentage of voters who have an MP representing their riding from the same party as their first choice candidate. Systems with multi-member ridings will do better under this measure.
  8. % Voters with Preferred Regional MP is the percentage of voters who have an MP representing their region from the same party as their first choice candidate. Systems with top-up seats will do better under this measure.
  9. Short System Name is a very consise abbreviation of the key parameters for this simulation.

Model Summary

Proportional electoral systems have many design parameters that can be tweaked. This table has two rows for each model. The bottom row applies to the riding; the top row applies to the region.

The first column of that table gives the name of the riding design (top) and the election algorithm used and the year of the election it's based on (bottom). The riding design specifies a particular mapping from old (e.g. 2015) ridings to new ridings, how the new ridings are gathered into regions, and finally how the regions are gathered by province. Riding designs are described in more detail at the bottom of this page and by following the riding design link.

Region# Tot Seats% SeatsAvg # Seats/RegionAvg #Reg/ProvAvg Adjust Seats / Region
RidingYear# Tot Seats% SeatsAvg # Seats/Riding% Single% MultipleComp. Gallagher
erre_mmp5050_LargeRegions16649%12.92.6
MMP_AV201517251%1.0100%0%2.5%
erre_mmp5050_LargeRegions16649%12.92.6
MMP_FPTP201517251%1.0100%0%2.3%
erre_mmp5050_ProvRegions16649%33.51.0
MMP_AV201517251%1.0100%0%1.8%
erre_mmp5050_ProvRegions16649%33.51.0
MMP_FPTP201517251%1.0100%0%1.8%
erre_mmp5050_SmallRegions16649%7.14.7
MMP_AV201517251%1.0100%0%3.5%
erre_mmp5050_SmallRegions16649%7.14.7
MMP_FPTP201517251%1.0100%0%3.7%
erre_ru_multiples_10pct3711%33.51.03.7
RcRUPR201530189%3.85%95%2.4%
erre_ru_multiples_15pct5115%33.51.05.1
RcRUPR201528785%3.65%95%2.0%
erre_ru_multiples_20pct7021%33.51.07.0
RcRUPR201526879%3.211%89%1.8%
erre_ru_singles4513%33.51.04.5
RcRUPR201529387%2.063%37%5.6%
erre_ru_singles4513%33.51.0
STVplus201529387%2.063%37%5.6%
erre_ru_singles4513%33.51.0
FptpList201529387%2.063%37%2.9%
fptp00%0.01.0
MMP_AV2015338100%1.0100%0%24.0%
fptp00%0.01.0
MMP_FPTP2015338100%1.0100%0%17.2%
kingsley00%0.01.80.0
RcRUPR2015338100%2.362%38%10.7%
lpr_no_topup00%0.01.3
SimpleLPR2015338100%4.25%95%5.8%
lpr_no_topup00%0.01.3
LPR2015338100%4.25%95%6.0%
lpr_no_topup00%0.01.3
STV2015338100%4.25%95%6.5%
stv_huge00%0.01.0
STV2015338100%26.023%77%2.4%
stv_med00%0.01.0
STV2015338100%10.910%90%4.3%
stv_small00%0.02.8
STV2015338100%4.15%95%6.5%

Riding Design Descriptions

erre_mmp5050_LargeRegions

This riding design is based on erre_mmp5050_ProvRegions. This splits a number of provinces into multiple regions where as ProvRegions has only one region per province.

An MMP system where compensatory seats are created by merging pairs of existing ridings wherever possible. The merged riding has 1 constituency seat and creates one compensatory seat. Where it isn't possible to merge a pair of ridings, a large riding with sparse population (e.g. Kenora) or another riding that seems to make sense (e.g. Halifax) is left as a single constituency seat with no compensatory seat produced.

This division of current ridings into pairs is from Wilf Day. Byron Weber Beckergropued those pairs into regions.

erre_mmp5050_ProvRegions

An MMP system where compensatory seats are created by merging pairs of existing ridings wherever possible. The merged riding has 1 constituency seat and creates one compensatory seat. Where it isn't possible to merge a pair of ridings, a large riding with sparse population (e.g. Kenora) or another riding that seems to make sense (e.g. Halifax) is left as a single constituency seat with no compensatory seat produced.

This design places all of the compensatory seats into a single province-wide region. Other designs are very similar to this except that they use smaller regions.

This division of current ridings into pairs is from Wilf Day.

erre_mmp5050_SmallRegions

An MMP system where compensatory seats are created by merging pairs of existing ridings wherever possible. The merged riding has 1 constituency seat and creates one compensatory seat. Where it isn't possible to merge a pair of ridings, a large riding with sparse population (e.g. Kenora) or another riding that seems to make sense (e.g. Halifax) is left as a single constituency seat with no compensatory seat produced.

This riding design is based on erre_mmp5050_LargeRegions (which was based on erre_mmp5050_ProvRegions). This riding design simply splits any region larger than 5 into two regions. The split is very quick and dirty and probably results in regions that are NOT contiguous. It's just to get a feel for the results of using smaller regions.

erre_ru_multiples_10pct

338 ridings total. Topups taken out of larger multi-member ridings. Emphasis on multiples.

erre_ru_multiples_15pct

A Riding-Centric Rural-Urban design, similar to erre_ru_multiples but with fewer adjustment MPs. That's accomplished by being less aggressive in assigning seats from multi-member ridings as adjustment seats.

erre_ru_multiples_20pct

338 ridings total. Topups taken out of larger multi-member ridings. Emphasis on multiples.

erre_ru_singles

Current 338 ridings grouped into a mix of single- and multi-member ridings allowing more single-member ridings. Compensatory seats are gained by reducing the number of seats in the largest multi-member ridings.

fptp

Canada's current riding design: 338 single-member ridings with no compensatory seats. Thisdesign can be used with either FPTP or Alternative Vote.

kingsley

Single member ridings in sparsely populated areas; multi-member ridings in densely populated areas; no top-ups.

lpr_no_topup

Local Proportional Representation: cluster ridings into regions of (ideally) 4-8 ridings. Run an STV-like algorithm to elect as many MPs as ridings in the region. The difference is that a candidate who is the last one left in his or her riding can't be eliminated. This guarantees that each existing riding will have an MP to represent it.

For comparison, the same mapping of ridings to regions was also simulated using STV to see the effects of the special rule described above.

stv_huge

STV (Single Transferable Vote) with ridings that are as large as possible to explore the question of how proportional the system can be in a Canadian context that places constitutional limits on riding sizes. that average 10.9 seats each. The three territories are left as single-member ridings and PEI is of necessity only 4 seats. All other ridings are 7 seats or larger -- ranging up to one with 18 seats.

This particular grouping of ridings is based on work by Antony Hodgson, President of Fair Voting BC.

stv_med

STV (Single Transferable Vote) with medium-sized ridings that average 10.9 seats each. The three territories are left as single-member ridings and PEI is of necessity only 4 seats. All other ridings are 7 seats or larger -- ranging up to one with 18 seats.

This particular grouping of ridings is based on work by Antony Hodgson, President of Fair Voting BC.

stv_small

STV (Single Transferable Vote) with smaller ridings that average 4.1 seats each. The three territories and Labrador are left as single-member ridings. There are 5 two seat ridings in large, sparsely populated areas. Most ridings have four seats and the largest is eight.

This particular grouping of ridings is based on work by Antony Hodgson, President of Fair Voting BC.

Election Strategy Descriptions

Election strategies are the specifics of how ballots are counted to determine which candidate fills a seat. Each strategy has three parts: how single-member ridings are handled, how multi-member ridings are handled, and finally how top-up or adjustment seats are handled.

AV

Single-Member Ridings:

During the election in each riding, votes were transferred in two steps. First, if a member of party X is eliminated and there are other members of party X still in the race, ALL of the votes are split equally between the remaining members of party X.

When the last member of a party is eliminated, the votes are transferred according to the following table.

Xfer from↓ to→BlocCHPComConGrnIndLbtLibM-LNDPOth
Bloc1961628
CHP
Com
Con81710
Grn1581622
Ind
Lbt
Lib3121045
M-L
NDP661353
Oth

This table is based on Ekos polling performed just before the 2015 election which asked for voters' second choice party. As Wilf Day has pointed out,

On Oct. 14 it had Liberals at 33.5%, Conservatives 32.6%, NDP 22.9%, Greens 5.6%, Bloc 3.4%. However, the E-day figures were Liberal 39.5%, Conservatives 31.9%, NDP 19.7%, Green 3.4%, and Bloc 4.7%. Obviously a lot of NDP and Green second-choices for Liberals had switched by E-day

However, it appears to be the best data we have.

Source: http://www.ekospolitics.com/wp-content/uploads/full_report_october_15_2015.pdf

Multi-Member Ridings:

An election strategy for where none are applicable. For example, for multi-member ridings in a FPTP simulation.

Top-up or Adjustments:

A placeholder election strategy for where no top-up strategy is applicable.

FPTP

Single-Member Ridings:

After collapsing all candidates running for the same party into one virtual candiate, choose the virtual candidate with the most votes.

Multi-Member Ridings:

An election strategy for where none are applicable. For example, for multi-member ridings in a FPTP simulation.

Top-up or Adjustments:

A placeholder election strategy for where no top-up strategy is applicable.

FptpList

Single-Member Ridings:

After collapsing all candidates running for the same party into one virtual candiate, choose the virtual candidate with the most votes.

Multi-Member Ridings:

Calculate the number of votes for each party and from that the determine the number of seats won by each party using a highest averages method -- specifically as described in "Quota system". After calculating the number of seats for each party, make a list of all the candidates for the party, ordered by number of votes in the 2015 election, and choose the first n candidates as the winners.

Top-up or Adjustments:

Iteratively choose the most disadvantaged party.

LPR

Single-Member Ridings:

During the election in each riding, votes were transferred in two steps. First, if a member of party X is eliminated and there are other members of party X still in the race, ALL of the votes are split equally between the remaining members of party X.

When the last member of a party is eliminated, the votes are transferred according to the following table.

Xfer from↓ to→BlocCHPComConGrnIndLbtLibM-LNDPOth
Bloc1961628
CHP
Com
Con81710
Grn1581622
Ind
Lbt
Lib3121045
M-L
NDP661353
Oth

This table is based on Ekos polling performed just before the 2015 election which asked for voters' second choice party. As Wilf Day has pointed out,

On Oct. 14 it had Liberals at 33.5%, Conservatives 32.6%, NDP 22.9%, Greens 5.6%, Bloc 3.4%. However, the E-day figures were Liberal 39.5%, Conservatives 31.9%, NDP 19.7%, Green 3.4%, and Bloc 4.7%. Obviously a lot of NDP and Green second-choices for Liberals had switched by E-day

However, it appears to be the best data we have.

Source: http://www.ekospolitics.com/wp-content/uploads/full_report_october_15_2015.pdf

Multi-Member Ridings:

An STV-like strategy that guarantees electing a candidate in each old riding that makes up a multi-member riding. It does so by prohibiting the elimination of the last candidate in the old riding. Surplus votes are transferred.

Top-up or Adjustments:

A placeholder election strategy for where no top-up strategy is applicable.

MMP_AV

Single-Member Ridings:

During the election in each riding, votes were transferred in two steps. First, if a member of party X is eliminated and there are other members of party X still in the race, ALL of the votes are split equally between the remaining members of party X.

When the last member of a party is eliminated, the votes are transferred according to the following table.

Xfer from↓ to→BlocCHPComConGrnIndLbtLibM-LNDPOth
Bloc1961628
CHP
Com
Con81710
Grn1581622
Ind
Lbt
Lib3121045
M-L
NDP661353
Oth

This table is based on Ekos polling performed just before the 2015 election which asked for voters' second choice party. As Wilf Day has pointed out,

On Oct. 14 it had Liberals at 33.5%, Conservatives 32.6%, NDP 22.9%, Greens 5.6%, Bloc 3.4%. However, the E-day figures were Liberal 39.5%, Conservatives 31.9%, NDP 19.7%, Green 3.4%, and Bloc 4.7%. Obviously a lot of NDP and Green second-choices for Liberals had switched by E-day

However, it appears to be the best data we have.

Source: http://www.ekospolitics.com/wp-content/uploads/full_report_october_15_2015.pdf

Multi-Member Ridings:

An election strategy for where none are applicable. For example, for multi-member ridings in a FPTP simulation.

Top-up or Adjustments:

Iteratively choose the most disadvantaged party.

MMP_FPTP

Single-Member Ridings:

After collapsing all candidates running for the same party into one virtual candiate, choose the virtual candidate with the most votes.

Multi-Member Ridings:

An election strategy for where none are applicable. For example, for multi-member ridings in a FPTP simulation.

Top-up or Adjustments:

Iteratively choose the most disadvantaged party.

RcRUPR

Single-Member Ridings:

During the election in each riding, votes were transferred in two steps. First, if a member of party X is eliminated and there are other members of party X still in the race, ALL of the votes are split equally between the remaining members of party X.

When the last member of a party is eliminated, the votes are transferred according to the following table.

Xfer from↓ to→BlocCHPComConGrnIndLbtLibM-LNDPOth
Bloc1961628
CHP
Com
Con81710
Grn1581622
Ind
Lbt
Lib3121045
M-L
NDP661353
Oth

This table is based on Ekos polling performed just before the 2015 election which asked for voters' second choice party. As Wilf Day has pointed out,

On Oct. 14 it had Liberals at 33.5%, Conservatives 32.6%, NDP 22.9%, Greens 5.6%, Bloc 3.4%. However, the E-day figures were Liberal 39.5%, Conservatives 31.9%, NDP 19.7%, Green 3.4%, and Bloc 4.7%. Obviously a lot of NDP and Green second-choices for Liberals had switched by E-day

However, it appears to be the best data we have.

Source: http://www.ekospolitics.com/wp-content/uploads/full_report_october_15_2015.pdf

Multi-Member Ridings:

An STV-like strategy that guarantees electing a candidate in each old riding that makes up a multi-member riding. It does so by prohibiting the elimination of the last candidate in the old riding.

The multi-member riding typically has at least one of its seats designated as an adjustment seat. The topup algorithm selects a candidate from an empty old riding (ie the adjustment seat).

Top-up or Adjustments:

A placeholder election strategy for where no top-up strategy is applicable.

STV

Single-Member Ridings:

During the election in each riding, votes were transferred in two steps. First, if a member of party X is eliminated and there are other members of party X still in the race, ALL of the votes are split equally between the remaining members of party X.

When the last member of a party is eliminated, the votes are transferred according to the following table.

Xfer from↓ to→BlocCHPComConGrnIndLbtLibM-LNDPOth
Bloc1961628
CHP
Com
Con81710
Grn1581622
Ind
Lbt
Lib3121045
M-L
NDP661353
Oth

This table is based on Ekos polling performed just before the 2015 election which asked for voters' second choice party. As Wilf Day has pointed out,

On Oct. 14 it had Liberals at 33.5%, Conservatives 32.6%, NDP 22.9%, Greens 5.6%, Bloc 3.4%. However, the E-day figures were Liberal 39.5%, Conservatives 31.9%, NDP 19.7%, Green 3.4%, and Bloc 4.7%. Obviously a lot of NDP and Green second-choices for Liberals had switched by E-day

However, it appears to be the best data we have.

Source: http://www.ekospolitics.com/wp-content/uploads/full_report_october_15_2015.pdf

Multi-Member Ridings:

During the election in each riding, votes were transferred in two steps. First, if a member of party X is eliminated and there are other members of party X still in the race, ALL of the votes are split equally between the remaining members of party X.

When the last member of a party is eliminated, the votes are transferred according to the following table.

Xfer from↓ to→BlocCHPComConGrnIndLbtLibM-LNDPOth
Bloc1961628
CHP
Com
Con81710
Grn1581622
Ind
Lbt
Lib3121045
M-L
NDP661353
Oth

This table is based on Ekos polling performed just before the 2015 election which asked for voters' second choice party. As Wilf Day has pointed out,

On Oct. 14 it had Liberals at 33.5%, Conservatives 32.6%, NDP 22.9%, Greens 5.6%, Bloc 3.4%. However, the E-day figures were Liberal 39.5%, Conservatives 31.9%, NDP 19.7%, Green 3.4%, and Bloc 4.7%. Obviously a lot of NDP and Green second-choices for Liberals had switched by E-day

However, it appears to be the best data we have.

Source: http://www.ekospolitics.com/wp-content/uploads/full_report_october_15_2015.pdf

Top-up or Adjustments:

A placeholder election strategy for where no top-up strategy is applicable.

STVplus

Single-Member Ridings:

During the election in each riding, votes were transferred in two steps. First, if a member of party X is eliminated and there are other members of party X still in the race, ALL of the votes are split equally between the remaining members of party X.

When the last member of a party is eliminated, the votes are transferred according to the following table.

Xfer from↓ to→BlocCHPComConGrnIndLbtLibM-LNDPOth
Bloc1961628
CHP
Com
Con81710
Grn1581622
Ind
Lbt
Lib3121045
M-L
NDP661353
Oth

This table is based on Ekos polling performed just before the 2015 election which asked for voters' second choice party. As Wilf Day has pointed out,

On Oct. 14 it had Liberals at 33.5%, Conservatives 32.6%, NDP 22.9%, Greens 5.6%, Bloc 3.4%. However, the E-day figures were Liberal 39.5%, Conservatives 31.9%, NDP 19.7%, Green 3.4%, and Bloc 4.7%. Obviously a lot of NDP and Green second-choices for Liberals had switched by E-day

However, it appears to be the best data we have.

Source: http://www.ekospolitics.com/wp-content/uploads/full_report_october_15_2015.pdf

Multi-Member Ridings:

During the election in each riding, votes were transferred in two steps. First, if a member of party X is eliminated and there are other members of party X still in the race, ALL of the votes are split equally between the remaining members of party X.

When the last member of a party is eliminated, the votes are transferred according to the following table.

Xfer from↓ to→BlocCHPComConGrnIndLbtLibM-LNDPOth
Bloc1961628
CHP
Com
Con81710
Grn1581622
Ind
Lbt
Lib3121045
M-L
NDP661353
Oth

This table is based on Ekos polling performed just before the 2015 election which asked for voters' second choice party. As Wilf Day has pointed out,

On Oct. 14 it had Liberals at 33.5%, Conservatives 32.6%, NDP 22.9%, Greens 5.6%, Bloc 3.4%. However, the E-day figures were Liberal 39.5%, Conservatives 31.9%, NDP 19.7%, Green 3.4%, and Bloc 4.7%. Obviously a lot of NDP and Green second-choices for Liberals had switched by E-day

However, it appears to be the best data we have.

Source: http://www.ekospolitics.com/wp-content/uploads/full_report_october_15_2015.pdf

Top-up or Adjustments:

Iteratively choose the most disadvantaged party.

SimpleLPR

Single-Member Ridings:

During the election in each riding, votes were transferred in two steps. First, if a member of party X is eliminated and there are other members of party X still in the race, ALL of the votes are split equally between the remaining members of party X.

When the last member of a party is eliminated, the votes are transferred according to the following table.

Xfer from↓ to→BlocCHPComConGrnIndLbtLibM-LNDPOth
Bloc1961628
CHP
Com
Con81710
Grn1581622
Ind
Lbt
Lib3121045
M-L
NDP661353
Oth

This table is based on Ekos polling performed just before the 2015 election which asked for voters' second choice party. As Wilf Day has pointed out,

On Oct. 14 it had Liberals at 33.5%, Conservatives 32.6%, NDP 22.9%, Greens 5.6%, Bloc 3.4%. However, the E-day figures were Liberal 39.5%, Conservatives 31.9%, NDP 19.7%, Green 3.4%, and Bloc 4.7%. Obviously a lot of NDP and Green second-choices for Liberals had switched by E-day

However, it appears to be the best data we have.

Source: http://www.ekospolitics.com/wp-content/uploads/full_report_october_15_2015.pdf

Multi-Member Ridings:

An STV-like strategy that guarantees electing a candidate in each old riding that makes up a multi-member riding. It does so by prohibiting the elimination of the last candidate in the old riding. Surplus votes are not transferred.

Top-up or Adjustments:

A placeholder election strategy for where no top-up strategy is applicable.