2023-5 IF

IF BUT BREXIT HADN’T BEEN BINARY…

PREFACE 

If, oh if only: +          the Dec. 2021 Brexit vote in the House of Commons… or before that +          the 2016 referendum itself… or before that             the 2011 referendum on the electoral system had been preferential…

As this text will demonstrate, binary votes can be manipulated.  With regard to the UK’s relationship with the EU, therefore, maybe the appropriate course of action should be to accept that, as a methodology by which a nation can come to a collective decision, a binary ballot is inadequate.  Accordingly, the question should be revisited, and maybe too that of the UK’s Westminster electoral system, with a more accurate decision-making voting procedure.   

This essay first shows that majority voting, be it in parliament or in a referendum, can sometimes be inaccurate.  Part II compares some of the other decision-making voting processes that have long since been available.  And in conclusion, the text suggests that the question should indeed be reviewed, to be resolved in a multi-option or, better still, preferential ballot.  

Or maybe there should be two polls: a preferential vote on the UK’s electoral system, and a second vote on the UK’s relationship with the EU.

ABBREVIATIONS

                                                              As used in

                                                      decisions           elections

                        approval voting                                   China§

AV*                 alternative vote                                   Australia

BC                   Borda count                                        Slovenia

EEA     European Economic Area

EU       European Union

FPTP                first-past-the-post                                India, UK

            =          plurality voting           Denmark

IRV*                instant run-off voting

MBC                modified Borda count

MMP                multi-member proportional                Germany, NZ

PR                    proportional representation

PR-list                         (single preference)      Netherlands

(preferential, single-party) Belgium

                                    (preferential, multi-party) Switzerland

PR-STV                                                            Ireland

PV*                  preferential voting

RCV*               ranked choice voting                          some US states

                        serial voting‡              Sweden

STV*                single transferable vote

TRS                  two-round system       Norway†, France, NZ^¶

WTO                World Trade Organisation

§          the methodology is used in China’s village council elections

*          all names for the same methodology

‡          a series of majority votes, usually on amendments

†          in parliament, but only once

^¶                 in some referendums, the first in 1894.

CONTENTS 

Preface                                                                                  3

Abbreviations                                                                         4

Introduction                                                                           6

Part I                           The Bane of the Binary Ballot                7

Part II                          Beyond the Binary Bind                       11

Part III                         So What’s Next?                                 20

LIST OF TABLES

Table I                         A Voters’ Profile

Table II                       Another Voters’ Profile

Table III                      Comparing Various Voting Methodologies

Table IV                      A Voters’ Profile with Partial Voting

Table V                       Approval Vote Analyses

Table VI                      A Preferential Ballot

INTRODUCTION

We start in 2011, with the UK referendum on the electoral system.  The choice was binary: either the alternative vote AV or the status quo, first-past-the-post FPTP.  Neither was proportional representation PR.  So for those in favour of PR, that question was similar to that of the waiter who asks a vegetarian, “beef or lamb?” 

There are of course lots of electoral systems, over 300 of them!  When Slovenia had a debate on this topic in 1996, there were three options ‘on the table’.  In 1992, a New Zealand referendum on this topic had five options.  And Britain had a multi-option vote in 1948 in a constitutional plebiscite held in Newfoundland, so legally, the 2011 UK referendum could have been a multi-option or even a preferential poll… if but there had been the political will.  Pluralism is possible.

+          If there had been more choice, the outcome might have been a compromise, as it was in New Zealand: half PR and half FPTP – the German system called multi-member proportional MMP.

+          If, come the 2015 election, the UK had replaced FPTP by a form of PR, it is likely that, instead of the Tory Party gaining an overall majority – 51% of the seats (on only 37% of the vote!) – Westminster would then have had a majority coalition, either Labour/Tory (which was unlikely), a Tory/Lib-Dem as in 2010, an all-party coalition (as during WWII), or a right-wing affair, Tory and UKIP, as in Israel.

+          If there had been a coalition, there might not have been the Brexit referendum, and none of the horrible consequences, not least the division and bitterness it created.

Part 1

The Bane of the Binary Ballot

A History

Binary voting was first devised by the Greeks about 2,500 years ago, and a little later on, quite independently, by the Chinese, during the former Hán dynasty.  In China, it was used only in the Imperial Court, but in Greece, down at the forum, they tried to be more democratic; the people (or rather, the men) voted, and majority votes were either singletons – “Option X, yes-or-no?” – or pairings – “Option X or option Y?”

There were problems, of course, especially when there was no clear majority in favour of any one option.  It is a situation known to us all: when trying to find the collective will of children as to which vegetable is to be served for lunch, a majority may oppose parsnips, an overwhelming number may dislike swedes, while an even greater mass might dismiss broccoli… in a nutshell, there may well be a majority against everything!  It may also be a trifle difficult when it comes to choosing their favourite pudding, with majorities various in favour of ice cream, chocolate cake, blancmange, and so on.  Indeed, in any pluralist and therefore multi-option debate, a single majority vote ‘for’ or ‘against’ any one option can be pretty meaningless, unless (as in Slovenia) that outcome is compared with the levels of support given to other options.

Accordingly, the Greeks devised a set of rules based on what was in those days the only known voting procedure – the binary vote.  Despite the invention of other voting procedures in the intervening years, for some extraordinary reasons (Emerson 2012: 25 et seq), this ancient procedure is still in use today.  It works like this: if some folks do not like the status quo, option S, they may move a motion, let’s call it option X.  Others can propose an amendment or two; options Y and Z.  So the procedure was – and still is! – as follows:

+          choose the preferred amendment, Y or Z;

+          reject or adopt this amendment, X or (Y/Z), to get the substantive as it’s called; and

+          then, in the final ballot, decide on the status quo or the substantive, S or {X or (Y/Z)}.

With one motion and two amendments, the question may thus be resolved in just three binary votes.  Imagine, then, a scenario in which nine (ninety or nine million) persons have the preferences shown in Table I.

Table I                        A Voters’ Profile

Preferences	Number of voters
	4	3	2
1st	X	Y	Z
2nd	Y	Z	S
3rd	Z	S	X
4th	S	X	Y

In binary votes, singletons, on these options, there are majorities of 5, 6, 7 and 9 against X, Y, Z and S respectively; in a nutshell, a majority against everything.  According to the above procedure, the first vote is Y v Z, and while 7 prefer Y to Z, only 2 prefer Z to Y, so Y is more popular than Z, which we write as Y > Z = 7:2; so Y is the preferred amendment.  Next, it’s X v Y, and X > Y by 6:3, so X is the substantive.  And finally, S v X, and S > X, 5:4.  So the answer is S.

In summary, the voters, having decided, initially, that they don’t want option S, then decide, democratically, that they want option S.  In other words, binary voting is, sometimes, hopelessly inaccurate.  Indeed, on occasion, as in the above voters’ profile, the answer is wrong.  After all, everyone prefers Z to S!  Therefore, in politics (and in business), if and when a controversy is contentious, the matter should rarely if ever be resolved in a binary vote or a series of binary votes.

Brexit

In 2016, the debate on the UK’s relationship with Europe was multi-optional, and there were (at least) four options: the UK in…

+          a)         the EU,

+          b)         the EEA,

+          c)         the Customs Union,

or

+          d)         the WTO.

The UK held just one binary referendum on just one option, and there was a (small) majority against.  But there might have been even bigger majorities against the other options.  Indeed, at 48% in favour of option a), ‘remain’ might well have been the winner!  To make matters worse, David Cameron (and others) then assumed that what a (slim) majority did not like, a negative, was actually the positive ‘will of the people’!

Next came Theresa May.  Accepting that the problem was indeed multi-optional, she resorted to her parliamentary indicative votes.  There were four options, so in theory, the House could enjoy a little pluralism.  In practice, however, it was back to dualism: she held four majority votes… and lost the lot!  Yet again, there was a majority against everything.

Enter, stage right, Boris Johnson.  He knew that, if the MPs were asked a singleton – “Is it ‘his deal’, yes or no?” – he might lose.  In a pairing, however, where there is always a winner (unless, as is unlikely in a House of 650 MPs, there’s a draw), he could perhaps get what he wanted.  So he chose both options: ‘his deal’ and something horrible.  Thus it was a parliamentary vote between ‘his deal’ and the most unpopular of all options, ‘no deal’.  December 2021, and ‘his deal’ won.  But of course it won.  ‘Any deal’ versus ‘no deal’ would have won.  Like so many other binary ballots in multi-option debates, that majority vote by itself was also, logically, pretty meaningless.

Manipulation

Let us now return to Table I where X > Y, and Y > Z, so we can write

X > Y > Z

Now Z > S and S > X as well, so if we put them all together, we get

X > Y > Z > S > X…

and it goes round and round in circles, for ever: the famous paradox of binary voting.  It basically means that the politician who so wishes can get whatever he/she wants.

Imagine, then. a different manipulator: someone who wants option Z.  With the status quo of S, she could make Z the motion with X and Y the two possible amendments, then sure enough X > Y by 6:3; next, Z > X by 5:4; and finally Z beats S by 9:0 so, yes, the decision would be Z.

Or again, if the desired outcome were Y, then let Y be the motion while X and Z are the two amendments.  X v Z = Z; Z v Y = Y; while Y v S = Y. 

Admittedly, to get the outcome X, a different status quo is required… but to most manipulators, that’s not a problem.   So if Z is the motion, Y and S the two amendments, and X is the status quo… the outcome is indeed X.

The conclusion is stark: majority voting, the decision-making methodology loved by politicians (and majority shareholders) of all shades and none, is primitive, often divisive, always Orwellian, and at worst, as in the Balkans, the Caucasus, and now Ukraine, a false flag, a provocation to violence.  It is time to examine some of the other methodologies.

Part 2

Beyond the Binary Bind

Evolution

Binary voting was first seen to be inadequate in the year 105 when, in a Roman court of law, the jury in a murder trial had three options: A, B and C, Acquittal, Banishment and Capital punishment.  Whereupon Pliny the Younger realised that, with no majority in favour of any one option, there was indeed a majority against every option.  If asked, “Innocent, yes or no?” B and C both vote ‘no’; if it’s “Execute him, yes or no?”, both A and B oppose the death penalty, and so on.

With the Dark Ages in Europe, the first government to put plurality voting into use was in China, in 1197, during the Jurchen Jīn dynasty, (Franke and Twitchett 1994: 266).  A century later, Europe stirred when Ramón Llull spoke of preference voting; this could be based on which option wins the most pairings, later to be called the Condorcet rule, or on which gets the highest average preference, that which is now called a Borda count BC, or rather a modified Borda count MBC (see below). 

In 1433, Nicholas of Cusa advocated just such a preference-points system, and in the 18^th Century in France, Jean-Charles de Borda did the maths.  The other voting system “in use in medieval Europe” (McLean and Urken 1995: 22) was approval voting, which is non-preferential: voters may ‘approve’ of the option(s) which they support; every ‘approval’ has the same value; and the option with the most ‘approvals’ is the winner.

Multi-option Voting

The methodologies to be considered in this paper are these.

+          In plurality voting which, like FPTP, allows the voter just a single preference, the option with the most 1^st preferences is the winner.  So this outcome may have the support, not of a majority, but perhaps of only the largest minority.

+          The two-round system TRS is a plurality vote followed, if no one option has an absolute majority, by a majority vote between the two leading options.  In most instances, therefore, the outcome will definitely have majority support.

+          The alternative vote, AV (IRV, PV, RCV or STV) is a series of plurality votes.  After each count, the ‘least popular’ option is eliminated and its votes are transferred to the other options still in the running, according to its 2^nd and/or subsequent preferences.  In most instances, again, the final outcome will definitely have majority support, but it might not be the same as the TRS winner.

+          In a Borda count BC and modified Borda count MBC, the voter may cast preferences on up to n options.

In a ballot of n options, Jean-Charles suggested, a voter may cast m preferences; and points shall be awarded to (1^st, 2^nd … last) preferences cast, according to the rule

(m, m-1 … 1).

rule (i)

Unfortunately, even during his lifetime, this was changed to

(n, n-1 … 1)

rule (ii)

or

(n-1, n-2 … 0)

rule (iii).

{There is of course little difference between rules (ii) and (iii); the outcome is the same.  Furthermore, if everyone submits a full ballot – that is, if for every voter m = n – there’s no mathematical difference between any of these rules.}  Both of these n rules are called a Borda count, BC, but Jean-Charles invented the m rule, rule (i), which today is called the modified Borda count, MBC.  When one or more voters cast a partial ballot, when m < n, the outcomes might be very different, as we shall see.

So, in a 6-option MBC ballot, she who casts all six preferences gets 6 points for her favourite, 5 for her 2^nd choice, and so on, whereas he who casts only one preference and says nothing about the other options gets just 1 point for his favourite and nothing for those other options: a one-point difference; in a BC, however, he gets 5 or even 6 points for his favourite and nothing for the others: a five- or six-point advantage.  The BC therefore encourages the intransigent voters, whereas the MBC is fair to all voters, consensual, neutral and intransigent.

+          The Condorcet rule is another preferential procedure in which a voter may again cast preferences for up to n options.  The count consists of pairings: is A more popular than B; is it more popular than C; than D, etc.; next, is B more popular than C, than D, etc.; and so on.  Sure enough, the Condorcet winner is the option which wins all the pairings, while the Copeland winner is the option which wins most of them.

Comparing the Borda and Condorcet rules is a bit like analysing a football league.  In a Condorcet count, every team (option) plays (is compared with) every other team (option), and the team (option) which wins the most matches (pairings) is the winner.  Whereas in an MBC, the winner is the team (option) which scores (gains) the most goals (points).  They are both pretty good rules, and in many seasons (voters’ profiles), the Condorcet winner is the same as the MBC champion (social choice).

Let us now, therefore, compare all of the above methodologies and, to do so, let us consider the rather more diverse voters’ profile shown in Table II, involving 20 voters expressing their views on six different electoral systems:

F          first-past-the-post

M         multi-member proportional (i.e., ½ FPTP and a ½ PR)

P          PR-list[1]

T          TRS

A          AV

S          PR-STV, PR with the single transferable vote (and STV

without PR is the same as AV).

Each group of voters casts its preferences in what (I hope) is a reasonably logical manner: so the four voters whose 1^st preference is for P (a PR-list system), have a 2^nd preference of S (PR-STV), a 3^rd for A (AV), a 4^th for M (MMP), and so on.[2]

We will analyse the votes according to each of the voting systems.

Table II                      Another Voters’ Profile

Preferences	Number of voters
	6	5	4	2	2	1
1st	F	M	P	T	A	S
2nd	T	S	S	F	S	A
3rd	S	P	A	A	P	P
4th	M	A	M	S	M	M
5th	P	T	T	M	F	T
6th	A	F	F	P	T	F

+          Plurality voting         The winner is F on a score of 6; this is far short of the minimum majority score of 11; it is indeed only the largest minority.

+          TRS                             The two leading options, F and M, go into a majority vote play-off, and if everyone’s preferences stay the same, the winner is now M on a score of 12, which is, of course, a majority.

+          AV                               AV is, yes, a series of plurality votes.  With a score of only 1, S the ‘smallest’ is eliminated, its vote goes to its 2^nd preference A for a stage (ii) score of F-6, M-5, P-4, T-2, A-3.  Still nothing with a majority so that’s the end of T, and its 2 votes go to F; so stage (iii) is F-8, M-5, P-4, A-3.  So now out goes A, and stage (iv) is F-8, M-5, P-7.  Whereupon the demise of M leads to a win for P on 12 to F’s 8.  So P has a majority, just as M had a majority of 12 under TRS.

+          MBC                            In this 6-option ballot, in full ballots, a 1^st preference gets 6 points, a 2^nd gets 5, and so on.  So option F, for example, gets (6 x 6 + 2 x 5 + 0 x 4 + 0 x 3 + 2 x 2 + 10 x 1) = (36 + 10 + 0 + 0 + 4 + 10) = 60 points.  And the winner is S on 91. 

+          Condorcet                  In the pairings, as we already know from the TRS count, M > F = 12:8.  Likewise, as it happens, T > F = 12:8 as well; but T loses all of its other pairings, so T’s Condorcet score is 1.  And the Condorcet social choice is again S on a score of 5.

A comparison of all the results is shown in Table III.

Table III                     Comparing Various Voting Methodologies

Methodology	Social choice	Social ranking 1st          2nd          3rd          4th         5th          6th
Plurality  voting	F	F-6	M-5	P-4	T/A-2		S-1
Two-round system, TRS	T	T-12	F-8	-
Alternative vote, AV	P	P-12	F-8	-
Modified BC MBC	S	S-91	M-73	P-70	T-64	A-62	F-60
Condorcet rule	S	S-5	M-4	P-3	A-2	T-1	F-0

So when using a plurality vote, the most popular option is option F, and the most unpopular is option S.  Whereas with an MBC and/or Condorcet, the result is the exact opposite!  So something is wrong.  The BC/MBC and Condorcet are the only methodologies which always take all preferences cast by all voters into account so, needless to say, they are the most accurate.

What’s more, as we might have foreseen, Table III shows that the MBC social choice is the same as the Condorcet winner; and their social rankings are almost identical as well.  Both of these methodologies are indeed pretty robust; (as noted earlier, the winner of the football league invariably has a very good goal difference).

Partial Voting

There will of course be some instances when, for reasons of conscience, party vested interest, or something altogether more dubious, some voters submit only a partial ballot.   Consider the case when the four P supporters decide to vote for only their 1^st preference, as shown in Table IV.

Table IV                     A Voters’ Profile with Partial Voting

Preferences	Number of voters
	6	5	4	2	2	1
1st	F	M	P	T	A	S
2nd	T	S	-	F	S	A
3rd	S	P	-	A	P	P
4th	M	A	-	S	M	M
5th	P	T	-	M	F	T
6th	A	F	-	P	T	F

According to the rules laid down for an MBC, P’s four 1^st preferences get only 1 point each; under rule (ii), they get 6!  The difference, of course, and its effect, can be huge.  When expressing full ballots, P had come out in third place overall; when not casting any other preferences, it does itself harm, and finishes in fifth place!  If, however, the methodology is just a BC, then, intransigence pays, and under rule (ii), option P is now the winner!

MBC    Table II           S-91     M-73    P-70     T-64     A-62    F-60

MBC    Table IV          S-66     M-61        F/T-56         P-50     A-41   

BC       Table IV          P-70     S-66         F/T-56         M-61    A-41

So, while the rules remain those of an MBC, such a tactic by the P bloc

spells disaster!  The rules of the BC, however, (I repeat), incentivise the non-consensual.

In this regard, the BC is similar to approval voting.  When taking all preferences cast into account, an approval voting analysis of Table IV would give option P an approval vote score of 20, while every other option would get only 16; so under approval voting and under a BC, option P could win. 

Approval voting, however, varies: it sometimes takes just the 1^st and 2^nd preferences into account, and sometimes rather more.  So, as shown in Table V, the social rankings may vary, making the methodology somewhat unstable.

Table V                      Approval Vote Analyses

The Count	Social Rankings
1st and 2nd preferences	F/T/S-8			M-5	P-4	A-3
1st/2nd/3rd preferences	S-14	P-12	F/T-8		M/A-5
1st – 4th preferences	S-16	M-14	P-12	A-10	F/T-8
1st – 5th preferences	P-18	M/S-16		T-14	F/A-10
All preferences	P-20	F/M/T/A/S-16

In a nutshell, both approval voting and the BC are very nice, if every voter is also very nice.  But neither methodology is suitable for the rough and tumble of politics (and business).

Conclusion

The fairest systems of all are the MBC and the Condorcet rule.  Of the two, the former measure is more nuanced, and its methodology is non-majoritarian.  At best – that is, if everyone submits a full ballot – it identifies the option with the highest average preference, and an average, of course, involves every member of congress/parliament, not just a majority of them. 

The MBC, therefore, could be the catalyst of a system of governance based on all-party power-sharing, broad coalitions, governments of national unity.  The important factor is that, in any decision-making process, the choice of options must be done, not by the prime minister or anyone(s) else with a vested interest in the outcome, but independently.

Part 3

So What’s Next?

Electoral Reform

A future referendum on the electoral system could easily be multi-optional, just as it was in Slovenia.  As noted, they used three majority votes and, yet again, there was a majority against everything; so they decided their social choice was the system which got the largest minority, which was TRS.  As noted above, New Zealand also enjoyed a measure of pluralism, a rather unusual form of TRS; it involved a run-off between the winner of the first round MMP and not the runner-up, PR-STV, but the status quo, FPTP.  The overall winner was then MMP.

But back to the UK.  As in New Zealand, the choice of options should be made by an independent commission.  The public can make submissions; everything may be ‘on the table’; but the commission itself should make the final selection of a short list of about five options.

Pluralism

As we saw in Part 1, asking just a single binary choice in a singleton majority vote, “remain or leave?” – in effect, “in the EU, yes or no?” – was an obvious nonsense.  So, in suggesting a second but preferential referendum, the independent commission (or whatever) should decide how many and which options should be on the ballot.

But first, the lessons from 2016.  In effect, there was the vote on Brexit, and there then followed a huge and often bitter debate.  Logically, it would have been wiser to have the debate before the vote.

That debate was, and still is, multi-optional.  Therefore the vote should also be multi-optional and ideally, as we argued in Part 2, preferential.

The debate could be conducted in parliament, and/or in a special independent commission, and/or in the country at large as well, with times allocated to discuss each of the options in some detail.  It must be emphasised, however, that the final choice of options must be done independently.

The Final Ballot 

The number of options should definitely be more than 2.  If the final number were 3, it might imply that the one ‘in the middle’ was the obvious compromise, a fix.  So the number should be at least 4, and subject perhaps to a maximum of 6 options.  Maybe 5 would be a good number, as it was in NZ.

The final list should be ‘balanced’, with a range of options covering the entire spectrum of the debate.  Digressing for a moment, if Northern Ireland were to have a multi-option referendum on its constitutional status, then obviously, it would be wise to have a couple of ‘British’ options, a couple of ‘Irish’ ones, along with some of both and/or neither.  Likewise, when it comes to the relationship of the UK with the EU, a similar degree of balance would be important.  The author’s suggestion is shown in Table VI, a set of five options, all arranged in a spectrum, ready as it were for a single-peaked analysis.

Given the 48% given to ‘remain’ in 2016, it could be argued that any multi-option referendum should have an equal number of ‘remain-type’ options.  To do so, however, would be to pursue a binary logic, the very malaise which this paper is trying to overcome.

Table VI                     A Preferential Ballot

The UK’s relationship with the EU. You may vote, in your order of preference, for one or more options, as you wish.  You should cast a 1 for your 1st preference; you may also cast a 2 for your 2nd preference, a 3 for your 3rd preference, and so on, again, as you wish. If you cast just one preference, your 1st preference gets 1 point; if you cast two preferences,      your 1st preference gets 2 points, and                                                   your 2nd choice gets       1 point; and so on.  Therefore, if you cast all five preferences, your 1st preference gets 5 points,                                                   your 2nd choice gets       4 points,                                                   your 3rd choice gets       3 points,                                                   your 4th choice gets       2 points, and                                                   your 5th choice gets       1 point.
OPTION		Preference(s)
A	The UK to be under the rules of the WTO
B	The UK to be in the Customs Union
C	The UK to be in the EEA
D	The UK to be in the EU, with Sterling
E	The UK to be in the EU, with the Euro
In the count, the preferences cast will be translated into points, and the option with the most points will be the winner.

Conclusion

In majoritarian democracies, it is sometimes said that a majority of only 1 is enough.  In a consensual polity, the requirement is rather greater: the option not with more, but with the most points!

Pluralism is indeed possible.  In February 2016, four months before that binary referendum, the de Borda Institute issued a press release to suggest that any binary ballot would be rejected  http://www.deborda.org/press-releases/  (press release No 7) – a majority against everything – and to propose in its stead a multi-option referendum. 

Furthermore, it is probably fair to say that if the 2016 had involved more than two options, the late Jo Cox MP would probably still be alive. Oh if only the ballot hadn’t been binary.  

The need for the world to adopt preferential decision-making cannot be over-emphasised.  It was obvious from the Balkans, where “all the wars in the former Yugoslavia started with a [binary] referendum,” (Oslobodjenje, Sarajevo’s famous newspaper, 7.2.1999), and the same mistake is still being repeated in the current conflict in Ukraine.

Peter Emerson

Belfast, 13.2.2023

References:

Emerson P,                              (2012), Defining Democracy, Springer, 

Heidelberg.

Franke H and Twitchett D,     (1994), Cambridge History of China, Vol

6, Cambridge, CUP.

McLean I and Urken A,          1995, Classics of Social Choice, Ann

Arbor, University of Michigan Press.

If but Brexit hadn’t been binary…

---

[1]           PR-list systems may be closed, and the voter votes for a party only, as in Israel.  In an open list system, the voter may vote for a particular candidate of one particular party (as in Netherlands), for more than one candidate of one party (in Belgium), or for more than one candidate of more than one party (in Switzerland).

[2]           The technical term is single-peaked preferences.  In some debates, the options can be placed on a spectrum, cheap to expensive, low to high, etc., as with prices or tax rates.  It is a little difficult to do so with our six electoral systems, but perhaps we can list them from simple to sophisticated:

F-T-M-A-P-S

In which case he whose 1^st preference is A will probably have a 2^nd preference of either M or P; and if he votes 1^st/2^nd A-M, he may well have a 3^rd preference of T or P… and so on.

            If every voter casts a single-peaked set of preferences, the collective will is bound to be single-peaked as well.