# Fair votes in practice

Denis Mollison  Heriot-Watt University, Edinburgh, Scotland
<EMAIL_ADDRESS>
###### Abstract

Criteria for a good voting system have been given particularly careful
scrutiny in recent years (BC Citizens Assembly, 2004; McAllister et al., 2017;
Make Votes Matter, 2019), with general agreement that the core values are fair
results, voter power and choice, and local representation.

Proportionality is a key aspect of achieving fair results. It is a little over
200 years since the first election of a set of representatives under a
proportional system, and 50 since the refinement of that system for the
computer age. This paper reexamines the basic ideas of that system, the
‘single transferable vote’ (STV), which has voter empowerment and choice at
its heart, and uses both core criteria and data from a range of political
elections to evaluate its performance in comparison with alternative
proportional representation (PR) systems in terms of both principles and
practice. The alternatives considered are List-PR and Mixed Member
Proportional (MMP), also known as the Additional Member System (AMS).

I shall look particularly closely at proportionality, examining three aspects:
broad linearity, the threshold for representation and the threshold for
gaining a majority.

As regards local representation, an important question is how to design multi-
member constituencies. It will be argued that using constituencies based on
natural demographic boundaries (such as local government areas) can combine
better local representation with stability over time and better
proportionality,

The broad conclusions are that List-PR is certainly an improvement on single-
member pluraility, alias ‘First Past the Post’ (FPTP), in proportionality, and
has significant advantages on both the other criteria. STV in turn improves on
List-PR in respect of voter choice and local representation, and arguably in
proportionality. MMP, which is essentially a compromise between FPTP and List-
PR, can be similarly proportional depending on how it is implemented, but has
a range of significant potential problems, including non-linearity typically
in the crucial near-majority case, and vulnerability to tactical voting
(Bochsler, 2012).

I shall make some use of the recent public availability of large preferential
voting data sets for STV elections, which make possible new analyses of how
STV functions in practice, and of voter behaviour, particularly examining how
voters’ second preferences relate to their first preferences.

###### keywords:

Voting system, proportional representation, single transferable vote, List PR,
Mixed Member Proportional, Additional Member System, STV, MMP, AMS, Meek,
Thomas Wright Hill, electoral reform, data analysis, large data

As submitted to RSS, 10 Feb 2023

## Contents

  1. Introduction

  2. 1

Choosing a set of representatives

  3. 2

Criteria

  4. 3

Voter power and choice

  5. 4

Local representation

  6. 5

Proportionality and thresholds

  7. 6

Patterns of preference

  8. 7

Some problems and possible solutions

  9. 8

Calculation and presentation of results

  10. 9

Discussion

## Introduction

This paper springs from an unashamedly positive motivation: to combine theory
and data to help design the best possible electoral systems to support
democracy. Electoral reform is often pushed down the agenda because it does
not directly affect social wellbeing; but the system for choosing political
representatives is the foundation, the most basic building block of democracy.
If it is not proportional it gives more power to some parts of the population
at the expense of others; and it has huge influence in determining the culture
of politics (antagonistic or collaborative) and the ways in which it can
change (adaptable or fossilised) (Duverger, 1964; Straw, 2015). That reform is
slow and often reversed should not surprise anyone familiar with elementary
game theory: those who owe their power to an unfair system have a strong
motivation to resist change. The best remedy would seem to lie in better
public understanding of its importance, and of the options for change.

This paper takes a fresh look at the main proportional voting systems,
clarifying their raisons d’etre, assessing them against ‘good system’
criteria, and using election data to evaluate how they perform in practice.
The emphasis is thus on social principles rather than mathematical ones, and
on data - what actually happens - rather than looking for theoretically
possible anomalies. There is a large literature on voting systems, ranging
from politics to mathematical detail, but a good starting point for this study
can be found in three mutually rather different approaches to identifying
principles for a good electoral system: the British Columbia Citizens Assembly
of 2004 (BC Citizens Assembly, 2004), and two recent reviews, one initiated by
the Welsh parliament (McAllister et al., 2017), and one by a grass-roots
cross-party organisation (Make Votes Matter, 2019). These two reviews came up
with broadly similar criteria against which to assess voting systems (see
Figure §1); at their core, these revolve around the principles of fairness,
voter power and choice, and local identifiability that were identified by the
BC Citizens Assembly.

Key topics include primary questions: do all voters have equal power and
sufficient choice? what is proportionality? how should we draw electoral
boundaries? and a range of secondary questions aimed at improving our
understanding of what happens in actual elections, such as what data on
preferential voting can tell us about how voters view different parties.
Sections 3 to 6 attempt to answer these questions, in each case using some new
(though fairly simple) methodology.

I shall use data from a number of countries, primarily within Europe and
particularly the British Isles - Ireland has been using the oldest
proportional voting system, the single transferable vote (STV) for 100 years,
while devolution within the UK has conveniently provided good data on a
variety of systems over the past 25 years. The largest of these data sets, for
Scottish council elections, has provided for the first time a large body of
complete preferential data that throw new light on STV.

In analysing the new riches of preferential data, it is tempting also to
consider how best to make a single political choice, such as when electing a
president or mayor, or in a referendum. But I agree with Hill (1988) that this
is a quite different matter from choosing a set of representatives; it will be
the subject of a separate paper (Mollison, 2023).

Supplementary material is online; also software, in particular a new R package
for the computation, presentation and analysis of STV elections (Mollison,
2022).

##  1 Choosing a set of representatives

The French Revolution led to some fresh thinking on how best to choose
representatives, with Mirabeau in 1789 perhaps the first to state clearly the
ideal of proportionality (Hoag and Hallett, 1926). However, the first known
election under a proportional scheme took place on 17 December 1819, using an
embryonic form of STV to elect a committee for the Birmingham Society for
Literary and Scientific Improvement.

###  1.1 The Single Transferable Vote (STV)

The basic idea of STV (in the US sometimes known by the simpler and perhaps
better name of ranked choice voting or simply choice voting) is that
proportionality should be achieved by matching the voters’ preferences with
the candidates in such a way that each person elected represents the same
number of voters. [We call the number of votes required for election the
quota, $q$ say.] The first scheme to implement this principle is due to Thomas
Wright Hill (1763-1851), who ran a progressive school in Kidderminster, and he
apparently had the original idea when organising the election of a
representative committee from his pupils.

His idea was that the pupils should organise themselves in equal-sized groups,
each of which could choose one representative. This naturally leads to a
process of several stages: initially the pupils form groups, according to
their first preferences; then groups that are larger than necessary shed
surplus pupils, who transfer to other groups, while members of groups that are
too small disband to transfer to larger groups. At the end of this process we
should have a number of groups of exactly the required size $q$, each electing
one representative, and a remainder of at most $q-1$ individuals
unrepresented.

In formalising his scheme for the Literary and Scientific Society Hill
introduced the idea of written votes, which needed to be revised in a series
of rounds as transfers progressed (Hill, 1819). The key idea to make the
scheme practical at large scale is preference voting, whereby each voter can
in a single step completely specify how their vote should be treated. The use
of preference voting is due to Carl Andrae (1812-93), who when Prime Minister
of Denmark independently invented a similar scheme to Hill’s for the election
of a new federal parliament, the Rigsraad, in 1855.

Putting together the methodological ideas of quota, transfers and preference
voting, we can give a precise algorithm for STV:

  * (a)

votes are initially assigned to the voter’s first choice;

  * (b)

we calculate the number of votes a candidate needs to be elected (the quota);

  * (c)

if a candidate has more votes than needed, the excess is passed on by
transferring the same proportion of each vote to that voter’s next choice;

  * (d)

if not all seats are filled, the candidate with fewest votes is excluded, with
the whole of each vote being transferred to the voter’s next choice.

Steps (b-d) are repeated, as necessary, until all seats are filled.

If this simple prescription is followed exactly, we have STV as defined by
Meek (1969). The reasons for the prevalence of a variety of untidier variants
of STV will be explained in describing its history (§1.2).

Notes:

(a) To make the system precise we also need to specify the quota $q$; if there
are $v$ votes and $s$ seats to be filled, then $q=v/(s+1)$ (since it is
impossible for more than $s$ candidates to have more than this number of
votes).111Andrae, and Thomas Hare (1806-91) writing two years later in
England, suggested that the quota should be $v/s$; in 1868 Henry Droop
(1832-84) pointed out that $v/(s+1)$ suffices (his precise proposal was the
integer value $[v/(s+1)]+1$). If as the count progresses some votes or parts
of votes drop out (because they are to be transferred, but the voter has given
no further preferences), then $v$ decreases, and $q$ can be reduced
accordingly.

(b) One methodological detail so obvious that it might pass without being
noticed is that each voter has exactly one vote, though it may be shared out
among candidates through transfers. This is where the name single transferable
vote comes from.

(c) With Meek’s definition of STV, the exact state of the count at any stage
can be specified simply through the keep values $k_{i}$ of each candidate $i$,
being the proportion that candidate keeps of each vote that reaches her. The
final keep values at the end of the count allow each voter to calculate
exactly how their vote was used.

(d) Meek also suggested that voters should be allowed to give equal
preferences to sets of candidates: equal preference for $n$ candidates
involves (conceptually) splitting a vote into $m=n!$ micro-votes representing
all possible orderings of those candidates, each with weight $1/m$. This is
impractical for hand-counting, but easy enough on a computer.

###  1.2 History of STV

The main difficulty in implementing STV lies in ‘transferring the same
proportion of each vote’ (step (c) above). The principle is easy to state (the
first to do so was Gregory in 1881) but difficult to follow without a
computer. As a result, over the period in which STV was being adopted for
political elections, several slightly different STV methodologies evolved,
varying in their compromises in fairness, clarity or repeatability made to
facilitate counting votes by hand. In Ireland, the counting method still only
deals in whole votes, so that surplus transfers are by a form of stratified
sampling, meaning that results are not exactly repeatable (Gallagher and
Unwin, 1986).

Meek (1969,1970) saw that the availability of computers allowed one to sweep
away these compromises, and to follow exactly the simple prescription of STV
(as outlined in §1.1). And he saw that this would also achieve more strictly
some of the basic advantages claimed for STV, including (a) treating voters
equally (especially when making transfers of surpluses), (b) minimising wasted
votes (therefore reducing the quota when any votes become non-transferable),
and (c) minimising any incentive for tactical voting (the various hand-
counting methods have unnecessary discontinuities that make possible a degree
of tactical voting). He also argued that voters being allowed to express their
real preferences meant that they should be allowed to put some candidates
equal, and (as mentioned in §1.1) indicated how this should be done.

STV has been used continuously in Tasmania since 1909, in Ireland since its
first independent election in 1921, in Malta also since 1921, in Northern
Ireland off-and-on but continuously for the Northern Ireland Assembly since
1998, and in Scotland for council elections since 2007. It is also used for
the Australian Senate, though in a form that makes it difficult to vote for
individuals so that the system is close to a party list system. With the
conservatism typical of most countries as regards their voting systems, each
has kept its own slightly different variant.

Meek’s method was first used by a number of NGOs, including the Royal
Statistical Society, in the 1980s, and for public elections in New Zealand in
2004; the version allowing equal preferences was first used by the John Muir
Trust (for Trustee elections) in 1998, and by the London Mathematical Society
in 1999.

###  1.3 List Proportional Representation (List-PR)

The second half of the 19th century saw the development of today’s most used
form of proportional representation, List-PR, which is based on parties rather
than individuals. Though later, this is a less sophisticated idea than STV,
corresponding to just the first stage of Hill’s original scheme, i.e. where
pupils form their initial groups according to their first preferences. Then
the $i$th group is allotted a number of representatives $r_{i}$ equal to the
size of the group, $v_{i}$ say, divided by a quota $q$, rounded to an integer.

The pioneers of this development were aware that they were adopting a
simplified alternative to STV; as one of their leaders, Ernest Naville, wrote
in 1895: ‘I do not mean to say that the Swiss reformers considered the list
voting as the best in theory. I, for one, would prefer the method which, like
that of Mr. Hare, gives the elector a chance of preferential vote without the
party official list, for the purpose of realizing better than any other the
idea of representation’ (Hoag and Hallett, 1926).

The commonest forms of List-PR are d’Hondt, using rounding down, so that
$r_{i}$ is the integer part of $v_{i}/q$, and Saint-Lague, where $v_{i}/q$ is
rounded to the nearest integer. In each case, assuming as is almost universal
that the total number of representatives $n$ to be elected is fixed, $q$ is
any number in the range of values for which the total of such allocations
$\sum r_{i}$ is equal to $n$. [For Saint-Lague, the system is sometimes
modified so that the minimum number of votes required to get a representative
is somewhat greater than $0.5q$; for example, Sweden has recently changed from
using $0.7q$ to $0.6q$.] There is no general formula for $q$; the commonest
way of finding an appropriate value is by starting with the value, $v_{max}$,
that would give $n=1$, and reducing it sequentially until the desired value of
$n$ is reached.

The simplest form of List-PR is Closed-list, where the groups standing for
election (usually political parties) put their candidates in an order that
determines which of them will be elected. Many countries (e.g. in
Scandinavia), use Open-list systems where the voter can indicate preference
for a specific individual candidate of the party of their choice.

###  1.4 Mixed Member Proportional (MMP/AMS)

A hybrid form of proportional representation was devised for West Germany
during Allied occupation after WW2. This is known as the Mixed Member
Proportional (MMP) or Additional Member System (AMS), and is currently used in
a number of countries, including Germany, New Zealand, Scotland and Wales. It
is usually described as a modification of the UK single-seat plurality system
(FPTP), with ‘compensatory’ List-PR seats added with the intention of making
the overall result party-proportional. But if the MMP system is working as
intended, that is with sufficient compensation, the FPTP part makes no
difference to the number of seats each party wins. It therefore offers more
insight to think of MMP as a modification of List-PR; the distribution of
seats among parties is determined by List-PR, but in deciding which
individuals fill those seats priority is given to the winners of FPTP
contests.

MMP can go ‘wrong’ (meaning fail to achieve the same level of proportionality
as List-PR) when the FPTP contests give a party more seats that it is entitled
to by its list vote: this is known as an overhang. The probability of
overhangs occurring depends on the proportion of compensatory seats, the size
of regions (see next subsection), and the political demography of the country.
In the German parliament additional seats are added to compensate for
overhangs, i.e. to restore overall party proportionality. However, in the
Welsh and Scottish parliaments overhangs are simply accepted as part of the
system, and have a significant effect on proportionality (see §5.3).

###  1.5 Districting and proportionality

With very few exceptions (notably Israel and the Netherlands), nations using
proportional representation divide their country into regions or districts for
electoral purposes. For all proportional systems the size of these regions or
districts has a significant effect on how they work, and indeed on their
proportionality. Districting will be discussed at length in §4, but some
introductory remarks are appropriate here.

Small districts are good for local identifiability and for voters being able
to assess individual candidates, but may give a less proportional result.
Carey and Hix (2011) analysed data from nearly 200 countries and concluded
that using ‘low-magnitude’ districts was associated with best governance.
Their explanation includes that ‘systems with median magnitude between four
and eight seats …offer opportunities for multiple winners, and thus afford
voters an array of viable options, but at the same time do not encourage niche
parties or overwhelm voters with a bewildering menu of alternatives’. For STV,
the slightly lower range of 3 to 7 seats has generally been found optimal,
reflecting STV’s superiority in achieving proportional outcomes in low-
magnitude districts (see §5.1).

Note that the optimal range identified by Carey and Hix does not include the
case of single-member districts. Apart from their inability to provide
proportionality, they have other defects. For example, the phenomenon of safe
seats whereby the same party holds a district for many years not only means
that many voters feel disenfranchised, it also means that many potential
candidates have no hope of being elected where they live: the result of this
is ‘parachuting’: a survey in 2017 by Demos found that less than one-third of
Conservative MPs had a significant link to their constituencies prior to being
adopted as candidates (Morris, 2017). It is then not surprising that a British
Academy research report found that “The ‘constituency link’ seems to be
stronger under Open-List PR in Denmark or STV in Ireland than under the
British FPTP system” (Hix et al., 2010).

There are two ways of districting, which we may call ‘natural’ and ‘equal’:
natural here refers to boundaries determined by demography, equal to where
high priority is given to equality of parity (the number of electors per
representative), and sometimes to having equal numbers of representatives from
each district. Obvious concerns with using natural districting are that the
variations allowed in parity and seat sizes might have an adverse effect on
proportionality; but we shall see in §s 5.4, 5.5 that, for STV at least, the
opposite is more likely to be the case.

##  2 Criteria

STV, List-PR and MMP were all conceived with the aim of empowering and giving
fair representation to voters. Against what criteria should we judge their
success?

Possibly the most thorough evaluation of PR systems from the voter’s point of
view was that conducted by a Citizens Assembly in British Columbia in 2004 (BC
Citizens Assembly, 2004). This came down in favour of STV on the basis of
three basic values, fair results, voter power and choice, and local
representation, together with the additional practical criterion of ease of
use. More recently in the UK, the campaigning group Make Votes Matter came up
with a set of 10 principles embodied in the Good Systems Agreement (Make Votes
Matter, 2019); while in Wales a Commission set up by their parliament, the
Senedd, came up with another set of 10 principles (McAllister et al., 2017).

These latter two sets of principles are in very good agreement; though using
rather different wording, they can be pretty well matched 1-1 as shown in
Figure 1. Most of them also match quite well with the rather simpler
principles of the BC Assembly shown in the left column. However, some of their
wording relates not to fair voting so much as to good government, which is not
covered in the BC criteria. An extreme example of this top-down approach can
be seen in the New Zealand Royal Commission report (Wallace et al., 1986),
which assumes that there is a good government model that the voting system
should serve. This calls to mind Brecht (1953)’s sardonic comment: ‘Would it
not be easier in that case for the government to dissolve the people and elect
another?’

Figure 1: Criteria for a good voting system: comparison of BC 2004, GSA 2019,
McAllister 2017

The simpler BC criteria, in agreement with Thomas Wright Hill, reflect the
alternative voter-centred, or bottom-up, approach, whereby government should
(if necessary) be adapted to the representatives the voters choose, rather
than vice versa. There are persuasive arguments that voter empowerment through
more proportional systems encourages more collaborative and flexible politics,
as well as greater diversity among representatives. Certainly, FPTP has a
strong tendency to result in two-party politics (Duverger, 1964) with the
combativeness and inflexibility that implies (Straw, 2015). We shall look
closely at the threshold vote required to win a majority (§s 5.3, 5.3), but
should bear in mind that constitutional reforms to encourage collaborative
politics may make achieving a simple majority less crucial: the current power-
sharing legislation of the Northern Ireland Peace Agreement (NIPA, 1998)
provides a good example.

If we wish to examine how well specific voting systems satisfy the basic
criteria of fairness, choice, and local representation, we need to see the
system through the eyes of the voters: are they on an equal footing? can they
express their real preference(s)? do they have one or more representatives
they can identify with? This suggests simple practical criteria for evaluating
a system:

  * (a)

fairness: votes are equally effective: this implies that outcomes should be
proportional in the sense that it takes at least approximately the same number
of votes to elect each representative

  * (b)

voters have effective choices: this should include
(i) voters have a wide choice of candidates that have a reasonable chance of
being elected,
(ii) voters can express their real preferences, and there is minimal incentive
to vote tactically, i.e. other than in accord with those preferences,
(iii) voters can effectively influence whether their preferred candidates are
elected,
(iv) voters understand the way in which their votes affects the outcome of the
election,
(v) a minimal proportion of votes is wasted

  * (c)

representatives have local ties - ideally voters should have a local
representative that they voted for

##  3 Voter power and choice

The three most-used proportional systems, as introduced in §1, are STV, List-
PR and MMP. All have good claims to satisfy the fairness criterion expressed
in (a) above (but see discussions of proportionality in §5). All can be
implemented in a way that to some extent represents local ties - and for all
this requirement has some conflict with achieving overall proportionality. The
chief respect in which they differ is voter empowerment and choice.

Here STV has an intrinsic advantage in that it is designed to make use of more
information on the voter’s preferences than simply their first choice. Before
comparing with alternative systems, it is worth while critiquing some variants
of STV, flourishing chiefly in Australia, that undermine that intrinsic
advantage. First, filling casual vacancies by ‘countback’ of the previous
election’s votes results in many more candidates, and pressure to allow what
are effectively party lists. If this is compounded by a rule that voters must
give a minimum number of preferences - or even that they must put all
candidates in order - it is not surprising that voters mainly choose the party
list option, even when this is further compounded by parties being allowed to
control their full preference order, which offers wide scope for tactical vote
management by parties (Raue, 2022; Difford, 2022). Such modifications not only
comprehensively undermine the idea of STV and the good system criteria (b)
above; they also (repeating Carey and Hix (2011)’s words) ‘encourage niche
parties and overwhelm voters with a bewildering menu of alternatives’.

###  3.1 List-PR

The advantages of STV over List-PR are perhaps most clearly seen if we
consider how one might modify the latter so as to better meet the criteria set
out in (b) above. First, some versions of List-PR (‘Open-list’) allow voters
to express preference for individual candidates; STV takes this further by
allowing the voter’s preference order to range across all candidates, not just
those of one party. [Allowing equal preferences as can be done in STV should
satisfy those who do wish to simply vote for a party.]

Second, if a candidate has more votes than they need, STV allows the surplus
votes to be transferred; and when excluding a candidate allows their votes to
transfer to a candidate still in the contest. This elimination of wasted votes
has the technical benefit of setting a simple quota for election: if you are
electing $n$ representatives, they are elected if and only if they have more
than a $1/(n+1)$ share of the votes.

But the crucial benefits of allowing transfers are that it avoids wasted
votes, and that it allows voters to express their real preferences (for
example, for a candidate from a minor party or an independent) without fearing
that their vote will be wasted if that candidate is excluded. Without
transfers, it is not only that a party can more easily achieve a majority
because there are wasted votes that are ignored; worse, the vote for a party
that perhaps justifies a majority may be based on tactical voting and thus not
reflect their true support. Finally, a system with transfers such as STV
encourages a more cooperative politics, with parties benefitting from
transfers from others with similar policies.

If we look at the comparison the other way round, it is interesting to note
that STV can be turned into d’Hondt List-PR by imposing restrictions: not
allowing transfers of any kind, only allowing voters a first preference, which
has to be an equal preference for the candidates of one party, and by letting
that party determine how the resulting tied results among their own candidates
shall be decided.

###  3.2 MMP

List-PR in turn has many advantages over MMP. The idea of MMP is that it
embeds FPTP style local representation in a List-PR proportional system.
Unfortunately, in doing so it combines most of the faults of both systems. In
particular, it has safe seats of both types - FPTP seats in localities where
one party dominates, and safe list seats by virtue of being hghly ranked in a
party list. It adds further problems for the voter through the interaction of
the two components. If MMP is working as intended, the winner in a
constituency simply displaces a winner of the same party on the list. This
sets up conflicts both of fact - constituency candidates are in competition
with others of the same party on the list, but it is difficult to assess who
might displace whom, and therefore for voters to vote on the merits of those
candidates - and of perception: although the constituency results should make
no difference to party seat totals, they are seen as the chief battleground
between parties.

Constituency results can matter when there are overhangs; as already mentioned
(§1.4), these disrupt the proportionality of the system. In UK elections using
MMP, such overhangs are allowed to stand, with the other parties getting fewer
seats than their list entitlement; in New Zealand both overhangs and list seat
entitlements stand, so that the total number of representatives is increased
by the size of the overhang. In Germany, overhangs and list entitlements both
stand, but additional ‘compensation’ seats are added to restore
proportionality. We shall look more closely at the occurrence of overhangs,
and the associated disproportionality, in §5.3.

In Germany the number of compensatory seats needed may be large. In 2017 the
number of members had to be increased from 598 to 709, and while that restored
party proportionality at both state and national level, it introduced a
disproportionality in parity, with the number of members per voter varying by
up to 20% between states; to have rectified that disproportionality as well
would have required increasing the parliament to 1001.

Finally, MMP is open to tactical voting. On the list there are incentives not
to vote for a party if it is likely either to be below threshold or to achieve
overhangs. This can achieve innocent distortions, as with the informal
alliance of the pro-independent SNP and Green parties in Scotland, whereby the
SNP win more constituencies, and the Greens more list seats, than they might
otherwise have done, with a mutual benefit arising from the SNP overhangs that
result. More seriously, it is possible to game the system by setting up
‘decoy’ parties to gain similar benefits. In Scotland a new pro-independence
Alba party made this their explicit intention at the 2021 election, but failed
to get above the threshold to win any list seats.

However in several countries that do or did use MMP this ‘decoy’ party tactic
has been very successful (Bochsler, 2012). In Albania in 2005 the two main
alliances each set up decoy parties, with the effect of more than halving the
representation of minor parties. And under a similar system in Italy, the same
tactic by the alliance led by Berlusconi in 2001 was so successful that they
won more list seats than they had candidates to fill.

It is perhaps not surprising that the MMP system is poorly understood: in a
survey of voters with experience of the system in Scotland, over 50% when
surveyed gave the wrong answer to basic binary questions on how it operates
(Bromley et al., 2006). Overall, MMP performs badly on all the criteria of (b)
above, List-PR being better at least for (b)(ii-iv), and is the only one of
the three systems compared to have serious problems with sustainability and
stability (because of natural drift into overhang territory as observed in
Germany (see §5.3), and the possibility of tactical cheating as in Albania and
Italy), or with the McAllister Report’s criterion of ‘Equivalent status’ (see
Figure 1).

##  4 Local representation

As already mentioned, there is evidence from PR systems around the world that
they work best in multi-member constituencies - ‘districts’ for short -
electing a small number (between 3 and 8) of representatives (Carey and Hix,
2011). Some countries (for example Denmark and Norway) have formal
constitutional constraints, requiring administrative districts to have
separate representation. But if we have freedom to choose electoral district
boundaries, how should we choose?

The traditional approach to districting has been to give primacy to parity
(i.e. the number of electors per representative), but to allow some
flexibility to improve the fit to natural communities. The Venice Commission,
without giving any specific reason, recommends that parity should not vary
beyond $\pm$10%, or $\pm$15% in ‘special geographical circumstances’; some
countries, notably the US, insist on much tighter parity (though at the same
time allowing party-controlled gerrymandering that has far greater distorting
effects!).

Another traditional strand, especially for STV, has been to insist on ‘equal-
size’, i.e. that each district has the same number of representatives; some go
further in insisting that this must be an odd number (e.g. the Proportional
Representation Society of Australia, following Howatt (1958)). Tasmania, Malta
and Northern Ireland all use STV with 5-member districts.

Here I shall follow through the implications of instead giving priority to
fitting natural communities, and not worrying about parity or equal-size. What
variability in parity does this lead to? and more importantly, what
variability from overall proportionality? The reason for taking this approach
is that we have already identified that voters having a link to particular
‘local’ representatives is a key feature of a good electoral system; and in
practice voters’ concerns over districting show much more concern over
boundaries than over parity. Inherent in the attraction of localism is that
the localities involved should be meaningful, preferably defined in terms of
natural communities, and that their role as electoral districts should be
stable over time.

I shall concentrate here on districting for STV because while it is generally
agreed that it needs a districting process, the details, especially the
question of variable or fixed size, have for long been a key topic of debate.
List-PR and MMP have often been used within already defined regions of widely
varying size, so that the districting question hardly arises; but with
increasing interest in good local representation it has recently received more
attention, at least within the UK (McAllister et al., 2017). Much of what I
shall say here applies equally to List-PR and MMP, though the sizes of their
districts may need to be rather larger to attain reasonable proportionality.

Since natural communities vary unsystematically in their numbers of electors,
a scheme that fits them well must allow variability in size, the number of
elected members for a district, as well as in parity. Since they also vary in
topology, there are no certain rules for joining up or splitting communities
to form contiguous districts. However it is found in practice that a twofold
range, i.e. ‘any size between $x$ and $2x$’, where $x$ is chosen to give a
number of seats in a desired range, makes the process of districting feasible,
but that a smaller range does not. The intuitive reason for this is that a
twofold range guarantees that combining two districts that are too small
cannot give one that is too big, nor can halving one that is too big give two
that are too small.

The requirement for a twofold range gives two options within the 3-8 seat
‘sweet spot’, either 3-5 seats or 4-7: the former gives more emphasis to
localism, the latter to proportionality. Districting then becomes a matter of
choosing areas solely on the basis of their best fit to communities within an
appropriate twofold range; for the 3-5 seat case this could be $2.7<e<5.4$,
where $e$ is the area’s entitlement calculated as $e=E\times S/N$ ($E=$
district electorate, $N=$ total electorate, $S=$ total seats).

A major advantage of giving natural boundaries priority in this way is that
they do not need to change over time. Changes in electoral numbers can instead
usually be accommodated by changing the allocation of seats between districts;
if an allocation rule is agreed (e.g. the US’s Huntington-Hill proportionally-
nearest integer) this updating can be done instantaneously whenever electoral
numbers are updated. The price paid for local identity and stability is some
variability in parity: the worst case is where a district’s entitlement falls
half-way between two possible integer sizes, which can mean a variation from
parity around the Venice Commission maximum of $\pm 15\%$. We shall see in
§5.5 that the effect of such variability on overall proportionality is smaller
than one might expect.

That the natural communities method is feasible and achieves stability in
practice can be confirmed from examples. McAllister (2017) includes a scheme
for the Senedd (the Welsh parliament) of 90 members in 4-7 member
constituencies based on local authority boundaries. If this had been adopted
at the creation of the Senedd in 1999, it could have been used with no change
of boundaries since then, just a small number of $\pm$1 changes in seat
numbers, and with standard deviation in parity steady at about 6%.

A similar scheme for the UK is illustrated in Figure 2 \- for full details see
the Supplementary material.

Figure 2: Part of the map of an STV scheme for the UK parliament based on
Local Authority areas. Each coloured area represents a constituency.

###  4.1 Isolated and sparse communities

A common problem in territories with diverse geography is whether and how to
accommodate isolated or sparse communities. The BC Citizens Assembly (2004),
while recommending up to 7-seat districts for populous areas, proposed a small
number of 2-seat districts for the remoter rural regions. This is a
flexibility to be used with restraint, as wards with fewer than 3 seats have
an adverse effect on overall proportionality. More recently, the Electoral
Reform Act (Scotland) 2020 introduced greater flexibility for council wards,
extending the range from 3-4 to 2-5, with the possibility of 1-seat wards in
the case of islands. The first review under this Act, in 2020-21, is
interesting both in demonstrating how the flexibility of allowing 2-seat wards
can give a much better fit in sparse and island areas (particularly in this
case in Eileanan–an-Iar), and also how achieving a good fit to communities
naturally leads to an agnostic view of parity (see Supplementary material).

Although the Islands Review was carried out with the overt aim of demarcating
wards with parity appropriate to 2, 3, 4 or 5 seats, the outcome is a
distribution of seat entitlements that is not significantly different from
uniform over a range from below 2 to above 5, just as though the ‘natural
communities’ method proposed here had been used. The review also made
allowance for special geographic circumstances for sparse areas; while no
numeric amount is specified, their recommendations correspond closely to
adding 10% to such areas’ entitlements. It is interesting to see that there
have been no public objections to the larger than usual variations in parity,
nor to the extra allowance for sparse areas in the recommendations. To the
contrary, the review’s proposals for Highland Council were rejected by the
Scottish Parliament because of strong feeling throughout that council that not
enough allowance was made for sparse areas.

More radically, there is a good argument for departing far from parity when a
parliament is representing a confederation or a collection of independent
territories of very different sizes. Penrose (1946), thinking of the newly
founded United Nations, showed that if such components vote independently for
or against each issue, then for them to have influence on decisions
proportional to their populations they should have representation proportional
to the square root of those populations (see also Kurz et al. (2017)). In this
context it is interesting to note that the representation of countries in the
European Parliament is closely fitted by a 2/3 power law222See Supplementary
Material.

##  5 Proportionality and thresholds

Proportionality, the idea that $x$% of votes should give $x$% of seats, seems
a very simple idea, but is much less so when considered carefully.
Numerically, proportionality is often assessed using a single statistic, such
as the standard deviation of seats from votes (in %s) (Gallagher, 1991). We
shall assess proportionality in more detail, in three aspects: broad linearity
and two key thresholds:

(a) Dependence of seats on votes over the main range of votes, from any
threshold up to the level at which a party can expect an overall majority: for
proportionality, this relationship should be linear, with slope close to 1 and
little variability from the expected value.

(b) Threshold for representation, or more broadly the relation between votes
and seats for minor parties. It is widely thought that a threshold level of
votes to win representation is a good thing, discouraging too much political
fragmentation. Relevant questions are: how large is the (explicit or implicit)
threshold? Is it sharp or smooth? what happens to the votes cast for below-
threshold parties or candidates? and how does it affect proportionality?

A basic problem with thresholds, shared by List-PR and MMP, is that votes for
minor parties may be depressed by tactical voting, so that the % of votes for
each party does not truly reflect the voters’ preferences. As noted in §1.4,
under MMP there can be additional distortion where supporters of major parties
vote tactically for another party on the list because they expect their
preferred party to win all or more than its share of seats in constituencies
(a potentially self-fulfilling prophecy).

(c) Threshold for a majority: a crucial question is what % of votes can be
expected to give a party or coalition 50% of seats? This should be close to
50%, and again, there should be low variability about this threshold value.

To illustrate these aspects, let us look first at just how disproportional is
the FPTP system (plurality in single-member seats). Figure 3 shows the
relation between votes and seats for the 27 UK General Elections of the past
100 years.

Figure 3: Relation between national vote and seat %s for UK general elections
1922-2019

On average, over the range of votes from 15 to 60%, a party has a seat:vote
ratio of 1.82, i.e. it gets 1.82% seats for each extra 1% of votes (solid line
in figure). If this disproportionality were consistent, there would be an
argument in its favour as allowing a clear winner with a mandate to govern,
but FPTP is not consistent: there is huge variability about the average
($\pm$8.5%). Two examples of this variability are the 1983 election, when two
parties with 27.6 and 25.4% of votes got respectively 32.3 and 3.5% of seats;
and that of 2015, when parties with 12.9 and 4.7% of votes got respectively
0.2 and 8.6% of seats.

As to threshold, only the geographically concentrated Nationalist and Northern
Irish parties have ever won significant numbers of seats with less than 15% of
the national vote. And as to majority, the linear fit estimates that 42.7% of
votes is required to give a majority of seats, but because of the great
variability a majority has been achieved with 35.2% of votes (in 2005), and
not achieved with 48.8% (in 1951).

Before looking at relevant data for proportional systems, we should note that
STV does not fit easily into the straightjacket of most proportional voting
analysis, which looks only at (implicitly first choice) votes for parties. It
is therefore important to stress that the idea of STV, matching
representatives to equal numbers of voters by allowing transfers, is in a
sense (within each district) exact proportionality. And, because preferences
are expressed for individual candidates rather than simply parties, STV gives
proportionality in terms of whatever is most important to the voter, which
might be cross-party issues. [This is why it is especially good for non-
political elections, where the outcome should reflect the proportions of
voters who support different priorities of an organisation (e.g. conservation,
campaigning or education for an environmental charity).]

A key advantage of STV is that it avoids wasted votes (whether surplus or
excluded) through its use of transfers. This has qualitative benefits, for
voters who can express their real preferences, knowing that if their first
preference does not have sufficient support their vote will not be wasted but
passed on; and for parties in encouraging more collaborative politics and
campaigning. But it also has quantitative benefits, in that it comes closer to
requiring 50% support to win 50% of seats (see §5.4).

###  5.1 Linearity

Figure 5 shows generalized linear model (GLM) fits for data from a range of
parliamentary and assembly elections. These examples are chosen from systems
having a strong element of local representation. Thus List-PR is represented
by the GB elections for the European Parliament (1999-2019), and estimated
outcomes from Northern Ireland Assembly elections to 6-seat constituencies
(1998-2016) - the latter allows an interesting comparison with the actual STV
results for those elections.

Figure 4: Proportionality: comparison of systems used in the UK and Ireland

It is perhaps surprising that STV achieves the best proportionality among
these examples, despite the data used being first preferences rather than
simply votes for a party.

It is further interesting to note that the smaller variable-sized (3-5) seat
system of Ireland has better proportionality than the 6-seat system of
Northern Ireland (1998-2016) - a point I shall return to in §5.4. The
variability of the Irish system is somewhat higher (standard deviation of
residuals 2.0% as against 1.2%), which is at least partly explained by
transfers being relatively more important in smaller size districts. Perhaps
surprisingly, the List-PR examples have similar variability (1.4 and 1.8 %
respectively).

The fitted slope parameters (‘seat:vote ratios’) for the two STV examples are
1.10 (Ireland 1951-2019) and 1.14 (Northern Ireland 1998-2016), and for the
two List-PR examples are 1.41 (GB EU elections, 3-11 seat districts) and 1.29
(estimates using NI Assembly 6-seat districts).

Can we explain the linear relationships we have found, particularly for List-
PR? For a single region or district, vote data can be well-fitted using the
simple approximation that the total vote for parties not gaining seats
(‘wasted votes’) is a fraction $w$, and each party gaining seats, by achieving
some multiple of a quota $q$, has a surplus $x$ uniformly distributed on [0,q)
for d’Hondt, and on [-q/2,q/2) for Saint-Lague; it also needs to be noted that
the probability of being the party that wins the last seat, and thus sets the
quota, is roughly proportional to that party’s vote. This leads to an estimate
for Saint-Lague, $s_{i}=v_{i}/(1-w)$, i.e. exact proportionality if wasted
votes are ignored.

In contrast, the d’Hondt rounding down method favours larger parties, and a
similar approximate calculation gives $s_{i}=b(v_{i}-v_{0})$, where
$b=(1+m/2n)/(1-w)$, where $m$ is the number of above-threshold parties and $n$
the number of seats in the region; and $v_{0}=100/2nb$. Thus for a region with
16 seats, 4 main parties, and with 10% wasted votes, we expect
$s_{i}=1.25(v_{i}-3.9)\%$, compared with $s_{i}=1.11v_{i}$ for Saint-Lague.
If, as is likely, the proportion of wasted votes falls under St-Lague, its
proportionality will be even better. For example, using Scottish Parliament
list vote data (1999-2021) we find overall national fits of
$s_{i}=1.22(v_{i}-2.0)\%$ for d’Hondt (with 8.9% wasted),
$s_{i}=1.09(v_{i}-0.4)\%$ for St-Lague (with 6.8% wasted) (see Figure 5).

Figure 5: Effect of varying both voting system (MMP or List) and allocation
system (St-Lague, modified St-Lague or d’Hondt), using Scottish Parliament
data (1999-2021).

In regions with fewer seats (and thus closer to the Carey-Hix ideal) the
seat:vote ratio will be significantly higher, as for the UK’s EU Parliament
elections (3-11 seats) or in estimates for 6-seat districts using the first-
preference data for Northern Ireland Assembly elections. For example, for the
latter at district level the seat:vote ratio is around 1.5, though as we have
seen it is only 1.29 at the national level, owing to demographic variability
between districts. This reduction can be less marked for larger districts, for
example if we compare the list part of Scottish or Welsh parliamentary votes
at regional and national levels, so it is not predictable whether having a
smaller number of larger districts will necessarily give better overall
proportionality than a large number of small ones.

In contrast to both List-PR and STV, the MMP examples show significant
deviations from linearity (see Figure LABEL:wales_prop). Because the overhangs
responsible for this occur principally when one party is close to a majority,
this will be discussed in the majority-threshold section below (§5.3). If we
ignore the non-linearity, linear fits to these data are comparable to the
List-PR examples (1.45 for Wales, 1.27 for Scotland).

###  5.2 Threshold for representation

For any electoral system based on multi-member constituencies, there is an
implicit threshold for the vote percentage required to win seats. The quota
for election under STV is $q=1/(s+1)$ (e.g. 20% in a 4-member constituency),
but this of course does not have to be all in first-preference votes. The
relation between first preference votes and probability of election is shown
in Figure 6 for the main parties in Northern Ireland Assembly elections: the
vote level required to achieve 50% probability of election varies from 55 to
69% of a quota, with the Alliance Party and Sinn Fein at opposite extremes,
reflecting their different abilities to attract transfers. Figure 6 also shows
the thresholds for election if we apply List-PR rules to the STV first
preference data. D’Hondt gives a higher threshold, unsurprisingly as in party
terms it resembles STV without transfers (see §3). Saint-Lague gives a lower
threshold, similar to the best-performing party under STV, but does not make
that discrimination between parties that attract transfers and those that
don’t. Modified Saint-Lague, with threshold 0.7 instead of 0.5 of a quota
(i.e. first divisor 1.4 rather than 1), is closer to d’Hondt.

Similar figures for the Scottish Council elections of 2012, 2017 and 2022 show
similar numerical results. Their larger data sets allow us to look at how the
various party thresholds have changed over these three elections. The main
changes have been that Labour and the Liberal Democrats now have rather lower
thresholds, probably due to exchanging more preferences in a realignment
following the Scottish independence referendum of 2014; while the Greens now
have a higher threshold, having aligned themselves as a pro-independence
party333See Supplementary material.

Figure 6: Threshold comparison for List-PR (d’Hondt, St-Lague and modified St-
Lague) with STV (with, in colour, fits for individual parties for STV), using
Northern Ireland Assembly data (1998-2016).

###  5.3 Threshold for a majority

In both examples of small-district List-PR we find that the threshold for a
majority is close to 41%. This could be improved, i.e. made closer to 50%, by
either increasing district size, or, as in countries such as Denmark and
Sweden allocating a number of ‘top-up’ seats at national level to make the
result as close as possible to exact proportionality. However, both of these
modifications diminish the quality of local representation, and because they
generally give nothing to parties below some threshold (in Sweden 4%), a party
or coalition can still win a majority on less than 50% of the national vote.
In Germany in 2013 the compensatory system allocated the CDU/CSU alliance
49.7% of seats, exactly matching their 49.7% of above-threshold list votes,
but their actual overall vote was only 41.5%.

For MMP, if the system is working as intended, the numbers of seats won are
determined by the list votes, so that proportionality is as above. However,
MMP can have overhangs, where a party wins so many constituency FPTP contests
that it gets more seats than the proportional list system would entitle it to.
This has been discussed in general terms in §3. In numerical terms, where
overhangs are accepted as for the Scottish and Welsh parliaments, the effect
is significant in the vote range that determines whether a party can win a
majority. Figure 7 shows the effect of overhangs in Welsh Senedd elections:
the dependence of seats on votes is significantly quadratic rather than
linear, and the estimated threshold vote required to win 50% of seats is
reduced from 40.8% to 36.4% - as is confirmed by the three elections (2003,
’11, ’21) in which one party has indeed won exactly 50% of the seats on
between 36 and 37% of the votes.

Figure 7: MMP: Welsh Senedd 1999-2021, showing how overhangs introduce
nonlinearity

The frequent occurrence of overhangs in Wales is not surprising, as the
List:FPTP seat ratio is only 50%. However, even in Germany with its List:FPTP
seat ratio of 1:1, substantial overhangs can occur: in their most recent
elections (2017, 2021) their top-up to achieve proportionality at national
level has required adding over 100 seats to their 598-seat parliament. This
plethora of overhangs can be explained by Germany’s mostly politically
homogeneous regions, and to there being a larger number of parties with
substantial support so that constituencies can be won on a low vote. Indeed,
the number of effective parties in Germany, as measured by Laakso and
Taagepera (1979)’s index, has increased from 3.2 in 2002 to 5.8 in 2021. The
number of overhangs has much increased in Germany in recent years, supporting
the reasonable conjecture that under a proportional system the effective
number of parties tends to increase, which can in turn be expected to increase
the probability and size of overhangs.

Figure 8 illustrates the association between the larger effective number of
parties, as calculated by the Laakso-Taagepera index, and the occurrence of
overhangs.

Figure 8: The relation between the number of parties as calculated from votes
with the number calculated from constituency seats, for parliaments using MMP
with regions. The area of each symbol is proportional to the number of seats
in the region. The same relation for UK Parliament elections using FPTP is
shown for comparison.

###  5.4 Winning a majority under STV

How does STV perform when a party or coalition are close to 50% of the vote?
In what is arguably the most important case, where voters are divided between
two clear alternatives (Left vs. Right or Leave vs. Remain, for example), a
simple argument suggests that it is highly accurate. Here each side may be
represented by a mixture of parties (and possibly independent candidates), but
we assume all transfers are within-side. Then the outcome in each district
(provided there are sufficient candidates from each side) will be determined
simply by the number of quotas of first preferences on each side: a side with
proportion $v$ of the vote in a district with $s$ seats will win $[(s+1)v]$
seats. More generally, the outcome in each district depends on what we may
call the final preference distribution, i.e. the distribution of votes at the
final stage of the count, not that of first preferences; but if the first and
final preference distributions differ this indicates that there are transfers
between the sides, so that thinking of the election as simply ‘L v R’ is not
entirely correct.

For an example of how this might work out in practice, possibly the best
available set of data is from the UK 2016 referendum, where voting figures for
Remain vs Leave are available for each council area, and thus for the STV
districting scheme of Figure 2. This is an interesting empirical distribution,
in that it is significantly skew. We adjust the votes very slightly (under
logit transformation) to give a distribution where each side has 50%. We can
not only calculate how many seats each side would have won (49.8:50.2%) but
can also, by randomising the sequence of district sizes relative to the
empirical vote distribution, calculate the mean and standard deviation for
such a scheme (a roughly equal mix of mainly 3-6 seat districts) for the kind
of demographical vote variation represented by the referendum data: we find
49.9 $\pm$ 0.5 %. Thus the conclusion is that the majority-threshold for STV
with this kind of demographic vote distribution is very close to exact
proportionality, certainly much closer than any List or MMP system is likely
to achieve because of their omission of wasted votes.

This data analysis can be taken further to look at the difference between
variable and equal-size districts. We can use the same randomising idea to
estimate the majority threshold if we used all 5-seat districts, as in
Northern Ireland, Tasmania and Malta. This gives a seat total estimate with
the same variability ($\pm$ 0.5 %) but with a mean (49.2%) that is
significantly different from 50%. The reason variable district sizes do better
is apparent if we plot the empirical vote distribution together with the cut-
offs for the main district sizes used - see Figure 9. What’s going on is that
we’re trying to estimate a continuous distribution using either one or several
rather broad discretisations: it’s not surprising that the ‘several’
alternative does better at capturing the skewness of the vote distribution.
Indeed, under the 5-seat option almost all districts return results of either
3-2 or 2-3, with the dividing line therefore drawn at the median (here 48.7%)
rather than the mean (50%) of the distribution, as it should be for
proportionality of votes to seats.

Figure 9: Distribution of Remain vote in 2016 UK referendum, showing STV quota
cutoffs for 3 (…..), 4 (—), 5 (green) and 6 (-.-.-.) member seats

###  5.5 Equalisation: the effects of variation in parity

The question of how variations in parity affect overall proportionality was
raised in §1.5. However, ‘equalisation’, i..e. a strong emphasis on minimising
variation in parity (as in the current UK Parliament boundary revision),
requires a very detailed process, and is therefore typically updated only at
long intervals (typically 10 years or more). Over such a period, the pattern
of demographic change can introduce significant bias between party totals, for
example if there are significant population movements between urban and rural
areas (Baston, 2010).

In contrast, the natural districts approach described in §4 allows seat
allocations to be kept up to date, thus not giving time for such bias to
develop. We can use the technique of §5.4 to estimate the effect of
variability in parity for this approach. The effect of such variability is to
give a party $n_{i}$ seats in a district, where its fair entitlement would be
the non-integer value $m_{i}=n_{i}e_{i}/s_{i}$, where $e_{i}/s_{i}$ is the
parity ratio for the district. If we again use the 2016 referendum data as
typical of demographic variability in voting, and the STV scheme of Figure 2,
we find that the effect of parity variability is $\pm 0.2\%$ ($\pm$ 1.3 seats
in a 650 member parliament). The explanation for this perhaps surprisingly
small effect in relation to the $\pm 6\%$ variation in parity at district
level is that it is the difference in seats between the two political sides in
each district that counts, not the total number of seats.

An important note is that we are assuming in this subsection, both for
equalisation and for natural districts, that boundaries are drawn up by an
independent body, i.e. that there is no gerrymandering.

##  6 Patterns of preference

The uniquely large set of full preference data available for the Scottish
Council elections (two councils for 2007, all councils for 2012-2022, a total
of nearly 1100 data sets) provides a rich seam for exploration of patterns of
political preference. In this section I shall just scratch the surface with
three examples.

First, it is common to assume that transfers between parties can be
represented by a simple matrix, whose entries $t_{ij}$ represent the
probability that a supporter of party $i$ will have next preference party $j$.
An alternative - a sort of null hypothesis - would be that second preferences
are independent of first preferences $v_{j}$, and have the same distribution.
Linear regression shows that the truth lies between these two extremes. As
illustrated in Figure 10, the transfer frequencies show a roughly linear
dependence on first preferences of the transferee,
$t_{ij}=a_{ij}+b_{ij}v_{j}$, where $0<b{ij}<1$.

Figure 10: Scottish Council election 2017: How the % of transfers going to
Green candidates from other main parties depends on the Green first preference
vote. Note that the green points and line show the % of transfers from Green
going to non-transferable.

One simple way of using transfer preferences is to define a measure of
distance between parties (as perceived by voters). These can then be viewed in
two dimensions using non-metric multi-dimensional scaling (Jardine and Sibson,
1971): Figure 11 shows how such distances between the main parties have
changed over the three Scottish Council elections from 2012 to 2022. Starting
from approximate equidistance from the SNP and Conservatives, Labour moved to
being perceived as closer to the SNP during the latter’s first period in
national government, but then to being closer to the Conservatives after the
2014 Independence referendum. The Greens and Liberal Democrats both moved
closer to the Conservatives in the first period, but then diverged as Scottish
politics became more polarised between pro-independence and unionist parties.

Figure 11: Scottish Council elections 2012-22: MDS plot showing changes in
closeness of voters’ perceptions of main parties.

Lastly, the detailed Council election data can be used to explore the patterns
of preference in each election, for example to examine the frequency of
occurrence of Condorcet cycles, where, if ‘$>$’ means ‘is preferred to’,
$A>B>C>A$. This is of theoretical interest because most if not all examples of
pathological behaviour in STV rely on data with cyclic preferences. We find
that Condorcet cycles occur in less than 5% of elections, and almost all
involve 2 or more candidates of one party; this is explored further in
Mollison (2023), which also goes into the dependence of second preferences on
the distribution of first preferences in greater depth.

##  7 Some problems and possible solutions

One of the advantages of STV is that where a party stands more than one
candidate it is the voters’ preferences that decide which of them (if any) is
elected. Three problems have arisen in this context, which give rise to
varying degrees of unfairness as between candidates, voters and parties.

First, there is ‘donkey voting’, where voters through laziness or indifference
rank the candidates in the order that they appear on the ballot paper. Voting
analysis suggest that around 15-20% of voters behave like this, leading to
substantial unfairness between candidates of the same party. There is also,
surprisingly, a significant ballot-order effect between candidates of
different parties, even between parties as different as the Conservatives and
the Scottish National Party: that is, in wards where each of these parties has
one candidate, there is a significant tendency to give first preference to
whichever party appears first on the ballot paper.

The logically best answer to this problem is to vary the order of candidates
on the ballot paper. A full solution would involve randomly permuting the
order on each paper. However, the partial solution of using just one order,
preferably random, and alternating it with its reverse, would be much simpler
to implement and should greatly reduce the problem.

Second, a significant number (around 1%) of ballot papers have to be
disallowed because the voter has placed the same number, or an ‘x’, against
each of the party’s candidates.

Lastly, both in order to mitigate donkey voting, and to promote less well-
known candidates, parties often go in for a form of vote management in which,
if they have 2 candidates A and B, they ask voters in one half of the ward to
vote ‘AB’, and in the other half to vote ‘BA’.

It would avoid or much reduce these latter two problems if voters were allowed
to express equal preference between candidates. STV allowing equal preferences
is perfectly possible, and has been in use for elections of trustees in some
member organisations in the UK for 20 years.

##  8 Calculation and presentation of results

###  8.1 Software

Traditional hand-counting of votes is well known to suffer from errors: for
example, the count in the UK’s North East Fife constituency in 2017 required 3
recounts, in each of which the winner changed. Physical ballots plus
electronic counting is arguably best: it can be audited by checking physical
batches of ballots against corresponding computer data; and allowing access to
the electronic vote files so that anyone interested can do their own check of
the officially calculated result.

All STV methods are straightforward to program. Meek STV, which as already
mentioned needs electronic counting, is at the same time more elegant and
robust; it is conceptually simpler, avoiding arbitrary choices and messy
programming. New software is discussed in the Supplementary material, and can
be found on Github (Mollison, 2022).

###  8.2 Presentation of results

Clear presentation of results is important for public engagement and
acceptance. The software cited above (Mollison, 2022) produces graphs for each
stage of an STV count (e.g. Figure 12), showing where any transferred votes
originated, for both Meek STV (with or without equal preferences) and the
Weighted Inclusive Gregory (WIG) system used for Scottish Council elections.

Figure 12: Example of graphic output for one stage of a Scottish Council
election

One of List-PR’s advantages is that its results are very straightforward to
present. MMP is much more of a problem, especially if one wishes to show how
near or far the vote distribution is from having disproportional overhangs.
The Supplementary material includes an attempt to visualise the results,
identifying crucial contests, for the Scottish Parliament election of 2021.

##  9 Discussion

###  9.1 System choice

This paper has evaluated the main proportional representation systems against
the ‘good systems’ criteria discussed in §2, principally voter choice,
proportionality and local representation. And for the key criterion of
proportionality, it has gone beyond using one simple statistic to considering
the three aspects that matter most for fair representation - linearity, and
thresholds for representation and for a majority.

It remains to summarise the relative advantages of the three main systems
considered. Because their fit to criteria depends significantly on how they
are implemented, particularly on districting and certain system-specific
rules, it is important for a fair comparison that we consider best-practice
versions of each system.

### STV

For STV, we have found that there are clear advantages in using variable-sized
districts based on natural community boundaries, with sizes in the range
either 3-5 or 4-7 seats, but allowing some flexibility, especially for
isolated or sparse areas. As to detailed rules, see the discussions in §s 1.2,
3 and 7.

Because STV is defined in terms of voter choice, it is no surprise that it
comes out best on that criterion. Nor is it surprising that it gives good
local representation, as judged by the proportion of voters getting a
representative they voted for, because the concept of STV is based on making a
best choice of representatives (§1.1). What is perhaps surprising is that STV
- whether with fixed or variable district size - comes out best for all three
aspects of proportionality, even though only first preferences are considered
in this comparison. The explanation lies in the transfer system of STV, which
avoids wasted votes and gives a smoother threshold for representation (e.g.
Figure 6). The latter, meaning greater uncertainty in how many first
preference votes are required for election, is perhaps a downside for the
individual candidate, but seems to be intrinsic to getting better
proportionality for parties overall.

Perhaps the most novel, and certainly the most satisfying, piece of analysis
of this paper has been the fresh look at the problem of districting for STV.
The approach of fitting electoral districts to communities without worrying
about parity has probably been avoided in the past because it is assumed that
it must come with significant downsides, particularly as regards
proportionality. But it turns out to have almost exclusively upsides,
including being better for proportionality in several ways. Perhaps its most
important advantage is easy updating (§4), which avoids the biases in
representation that arise when, as is the norm with traditional boundary
reassessments, elections take place with allocations that are many years out
of date ((Baston, 2010)).

The variation in size that is necessary for the natural communities approach
also turns out to be an advantage in most respects. It is likely to provide a
better fit to the national vote distribution because different sizes have
different cutoff levels (§5.5, Figure 9). Further, the use of natural rather
than arbitrary boundaries can be expected to give greater variation in vote
distribution between districts, which in turn means that the discretisation
approximation to that distribution provided by STV is likely to be more
accurate, while at the same time giving better opportunities to minor parties
and locally grounded candidates. The variation in threshold for representation
(§5.2) that goes with variation in size - and is larger for STV in any case
(Figure 6) - means that for an individual the probability of election as a
function of first preference votes increases from 0 to 1 over a wide range, as
opposed to the rather sharp - almost deterministic - cutoff of list systems.

### List_PR

Most of the advantages for local representation in using natural community
boundaries (including more meaningful representation and better
proportionality because easier to update) apply equally to List-PR and MMP.
Indeed, such use of natural boundaries is constitutionally enshrined in many
electoral systems, for example those of the Nordic countries (List-PR) and
Germany (MMP). Where such mandated boundaries do not exist, the size of
districts to choose will need to trade off the desiderata of localism and
proportionality. For both criteria, detailed consideration suggests a slightly
higher number than for STV, so perhaps a maximum of around 10 for List-PR
districts. Using St-Lague with constituencies electing 6-10 representatives
should ensure good proportionality between major parties, with a threshold for
representation of at most about 5% of the national vote, and considerably
lower for parties with local concentrations of support. All constituencies
will be competitive, so that votes matter equally everywhere.

‘Good overall proportionality’ here is of course ignoring the wasted vote
problem of all single-preference systems, whereby votes for below-threshold
parties contribute nothing to results, a problem compounded by the deterrent
this implies against expressing real preferences - which in turn threatens the
concept of proportionality.

The key criterion on which List-PR and MMP fall down is voter choice: voters
should be able to express support for a large or small party or none without
fear that their vote will be wasted, and to express preference for individual
candidates within parties.. These problems can be mitigated to some extent by
(respectively) allowing parties to present combined lists, and by the use of
open lists; the electoral system of Finland includes both these features.

In sum, if the great majority of voters have strong party allegiances, and are
happy to vote for a party rather than for individuals, then List-PR is a
reasonably satisfactory system.

### MMP

For districting in MMP, similar considerations apply, except that somewhat
larger districts are appropriate. For example, a district with 9
constituencies and 9 list seats will give voters the same number of
representatives as a List-PR district of size 10. Having such larger districts
should also somewhat reduce the probability of occurrence of overhangs.
However, the use of more politically coherent districts, which tends to
improve the proportionality of STV and List-PR, works against MMP because
coherent districts are more likely to have overhangs.

MMP retains the single local representative that some find attractive, but at
the expense of major problems: these include safe seats, having two different
kinds of representatives, and difficulty for voters in understanding the
effect of their votes (§3.2). However, the biggest problems with MMP are the
potential for overhangs, with their nonlinear effect on proportionality
(§5.3), and for gaming the system (§3.2). Neither STV nor List-PR has such
problems.

### In conclusion

To summarise with a broad brush: FPTP, alias plurality, fails on on all three
of the basic criteria. Its strongest suit is in providing very local
representation, but the costs of this include non-natural boundaries needing
frequent adjustment; around half of voters not having a representative they
voted for; and probably worst, some voters - swing voters in marginal
constituencies - have much more influence than others, thus significantly
distorting politics.

List-PR addresses all three of these problems, with votes mattering equally
everywhere, and introduces the major improvement in fairness of broad
proportionality between votes and seats.

MMP is a halfway step from FPTP towards List-PR, a compromise that brings with
it the problems summarised above.

STV on the other hand can be seen as a refinement of List-PR (see §1.3), in
which allowing voters to more fully express their preferences and allowing
transfers so as to avoid wasted votes significantly improves the
proportionality of the system as well as voter choice; and can be implemented
at the smallest local scale that gives most voters a representative that they
voted for.

###  9.2 Technical challenges

The statistical analysis of this paper has concentrated on the socially most
important questions of systems analysis in relation to basic democratic
criteria. But as a coda, here are a few problem areas of interest to the
theoretician - and like much theory, progress in these areas may lead in time
to socially useful conclusions.

The body of nearly 1100 sets of full preference data for Scotland’s recent
council elections is a huge resource, whose surface I have only scratched
here. The dependence of second on first preferences has been touched on
briefly in §6; a rather different approach is taken in a parallel paper on
electing a single choice (Mollison, 2023), where the occurrence or (spoiler
alert) mainly non-occurrence of cyclic preferences is of key interest.

In a multi-party context, such as the Scottish Council elections where there
are five principal parties and the relations between them have changed over
successive elections (Figure 9), it would be interesting to look deeper into
the structure of preferences.

Finally, in the analysis of the vital threshold for a majority (§5.4) it was
pointed out that the support for a potential majority is not just a matter of
first preferences; under STV it depends on the balance of final preferences.
The question of how first and final preferences are related, and what it tells
us about the relations between parties and the voters’ perceptions of these,
is of significant social interest.

## Supplementary material

  1. §4

An STV scheme for Westminster

  2. ”

The Islands Review proposals 2021 and their implications

  3. §5

Data, R code and additional figures for proportionality analysis (to come)

  4. §6

Data, R code and additional figures for preference analyses (to come)

  5. §7

Community-centred democracy: fine-tuning the STV Council election system

  6. §8

pref R package manual

  7. ”

Results of the Scottish Parliament election of 2021

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