2026-05 Dichotomies and Dualities

THE DE BORDA RULE

Majority voting may be fair, if its dichotomy is a duality.  

On 2nd March in the Centre for Conflict Resolution in Munich:

I was in Tibet last year so, of course, I did some background reading before my visit, and I came across the word ‘non-duality’.  (I’ll give you the full quotation later.)  Now as you all know, 

I ‘speak very well the English,’ but this word was completely new to me.  I goggled.  I googled.  

And the basis of my talk today is a critique of binary decision-making, because so many of our majority votes are not dualities; next a little history, and then some proposals for a non-binary polity.

First things first: definitions.  There are two types of binary voting, singletons and pairings.  A singleton – “Option X, yes-or-no?” – offers just one option; a pairing gives a choice of two – “Option X or option Y?”  

In a multi-option debate with singletons, there’s the possibility of lots of (different) majorities against every option.  Whereas, with a pairing, there’ll always be a definite outcome.  

Nearly 2,000 years ago, the Romans were well aware of many of the problems associated with binary voting.  I quote Pliny the Younger in 105: they learnt “the powers of the proposer, 

the rights of expressing an opinion, the authority of office holders, and the privileges of ordinary members; they learned when to give way and when to stand firm, 

how long to speak and when to keep silence, how to distinguish between conflicting proposals and how to introduce an amendment, in short, the whole of senatorial procedure.”

But they were stuck with binary voting.  So, if we take a simple situation of a motion, option M, a couple of amendments, A and B, and S, the status quo, they devised a procedure:

               +             choose the better amendment:                                A v B

               +             form the substantive:                                             (A v B) v M

               +             make the decision:                                              {(A v B) v M} v S

If, then, we take the very simple scenario: I’m on a committee of a dozen members, and we need to repaint the front door.  The first thing to do, of course, is to choose a chairperson, 

a nice, neutral, non-voting chair, someone who is modest, honest, patient, polite, someone who is old and wise… like me.

               “Ri’ht oh,” says I, “any suggestions?” and:

               5 want red                                                                           R

               4 like white                                                                         W

and          2 prefer blue                                                                       B

so, if we use singletons, there are majorities against everything, of 6, 7 and 9 respectively.  So that’s no good.  And if we use pairings?  OK, so let’s have a look at their 1st-2nd-3rd preferences:

               5 opt for red, white and blue                                              R-W-B

               4 choose white, blue and red                                             W-B-R

and          2 go for blue, red and white                                               B-R-W

which means   red       is more popular than   white                              R > W =          7:4

                        white               ”                      blue                             W > B =          9:2

and                  blue                 ”                      red                               B > R =          6:5

which means, of course, that 

R > W > B > R > W > …….

And it goes round and round in circles, for ever: Le Marquis de Condorcet’s famous paradox of binary voting.

So, in our debate, if the motion is “let the door be red” while the two amendments are “delete ‘red’ and insert ‘white’” or “insert ‘blue’,” the debate for the substantive shall proceed as follows:

{(W v B) v R} 

               or, if the motion was White and the two amendments were Blue and Red

{(B v R) v W} 

or it could be

{(R v W) v B}

and given that

R > W > B > R > W > …….

we have

{(W v B) v R}        =             W v R     =             R

or 

{(B v R) v W}        =             B v W     =             W

or

{(R v W) v B}        =             R v B      =             B

so everything depends on the order of voting.  In other words, it depends on me, the scrupulously fair chair.  But I’m Irish; I don’t like red, white or blue; what I’d really like is Green.  

               So I could suggest this as the possible ideal compromise which everyone is (not) looking for.  Imagine their preferences are now:

               5 opt for red, white, blue and green                                   R-W-B-G

               4 choose white, blue, green and red                                   W-B-G-R

and          2 go for blue, green, red and white                                    B-G-R-W

               Umm, G is definitely not very popular!  Never mind; I’m the charismatic chair; and they all believe in majority voting, so I now propose the following order of debate and voting:

{(W v B) v R} v G

and given that

R > W > B > G > R > W > …….

we now have 

{(W v B) v R} v G                =             {W v R} v G           =             R v G      =             G

and Green it is.  It was close, only 6:5, but it was perfectly democratic!

The answer, of course, is horribly wrong.  Everybody, 5 + 4 + 2, they all prefer B to G, by 11:0.  The answer could not be more wrong!  Majority voting is moronic.

PLURALISM

So what would be a better way of doing things?  That question was first asked by Pliny the Younger in the year 105, and hence his plurality voting.  

Europe then went into the Dark Ages, and the first country to actually use multi-option voting was China in 1197, during the Jīn Dynasty; 

the topic was war with Mongolia, there were three options on the ballot, and the 88 members of the government voted:

                                                             war                                          5

                                                             an-alternating-policy                33

                                                             peace                                     46.

So peace it was.  Alas, it didn’t last very long: in 1206, a Mongolian Quriltai or Assembly elected Chinggis Khan, and the rest, as they say, is history.

The debate on pluralism swung back to Europe.  In 1268, Venice started to use approval voting; thirty years later, Ramón Llull spoke of preferential voting, and the first points methodology was Nicholas Cusanus’s 1433 points 

system which, by accident of history, came to be called a Borda Count BC.  The debate moved to France which, in the 18th Century, was in a revolutionary mode.  Folks knew that democracy was ‘on the table,’ and members of 

l’Académie des Sciences looked across The English Channel at Westminster, the only democracy of those days, and they concluded, “Mon Dieu, c’est incroyable!”  The House of Commons was, and still is, binary; its architects 

ignored the example of King Arthur with his round table, and built instead the two-sided, confrontational chamber, sometimes called the mother of parliaments, more accurately known as their grumpy ol’ grandpa.  

But you cannot identify the average age or the average opinion, la volonté général, the common will, in a binary for-or-against vote.  With – “Are you young or old?” – the answer is neither; the question is not a duality; 

the majority is bound to be wrong.  So too, the question – “Left-wing or right-wing?” – will never identify a consensus.  In a nutshell, majority voting is not much good when trying to get an agreement, 

but the only people to recognise this, so far, are the UN environmental jamborees, the Conferences of the Parties, the COPs.

Little wonder then that two members of l’Académie – Le Marquis de Condorcet and Jean-Charles de Borda – proposed multi-option systems.  The first identifies the option which wins the most pairings, 

and the Modified Borda Count MBC is a points system, similar to the BC of 1433, but with one huge difference.  

In a BC of n options, regardless of how many preferences the voter casts, a 1st preference gets n points, a 2nd preference gets (n-1) points,[1] and the option with the most points is the winner.  

In contrast, in an MBC, if the voter casts only m preferences, the 1st preference gets just m points, the 2nd preference gets (m-1).  The difference may be tiny; its effect is huge.  

The n-rule tempts the intransigent voter to cast only a 1st preference; and if every voter does that, the BC is virtually the same as a plurality vote.  The m-rule, in contrast, 

encourages the voters to vote, not only for their favourite, but also for their compromise option(s), and if everyone does that, we can identify the collective compromise… which is what, in theory, democracy is all about!

In an MBC, therefore,            

                               she who casts only 1 preference gets her favourite just 1 point,

                                              (she says nothing about the other options, so they get nothing); 

                               he who casts 2 preferences gets his favourite 2 points,

                                              (and his 2nd choice 1 point);

and so on; accordingly, 

                               she who casts all 5 preferences gets her favourite 5 points,

                                              (her 2nd preference gets 4 points, her 3rd gets 3, etc.).

In binary voting, it’s A v B, only two ways of voting, win-or-lose, so the house divides.

In multi-option voting, well, with three options – A, B and C – there are 6 ways of casting all three preferences; 

with four options, there are 24 ways; and with five or six options, we may relish in human diversity… but still get a definite outcome.

To win, a protagonist will need, not only lots of 1st preferences from supporters, but also a good few 2nd and 3rd preferences 

from other MPs from other parties.  In a nutshell, cooperation can replace confrontation.

AN INCLUSIVE POLITY

So, we now talk of a polity, a democratic structure which is not binary!  And after all, in a pluralist democracy, 

every controversy is bound to be multi-optional.  Consider, then, a multi-party parliament discussing, let’s say, the budget.  

In a 4-/5-party parliament, there could well be 4 or 5 draft budgets ‘on the table,’ whereas, in a 10-/12-party parliament, 

some of the smaller parties may well choose to cooperate with each other, or perhaps join forces with one of the bigger parties, 

and there would probably be about 6 or 8.

Consider, then, a debate of five options.  Each option shall be debated in turn.  In discussing any part of any proposal, 

MPs may suggest amendments or even composites, but nothing shall be adopted unless the original proposer(s) agree 

to such a change.  Then, when all is said but nothing yet done, the Speaker shall ask all concerned to cast their preferences, 

which will then be analysed according to the rules of an MBC.  If every MP does cast a full ballot, the option with the most 

points is also the one with the highest average preference.  And an average involves every voting MP, not just a majority of them.  

The MBC is egalitarian.  And if the MBC were adopted, there would be no further justification for majority rule, especially in the Knesset!

In real politic, some MPs may well cast only partial ballots, so ‘average preference scores’ would be inadequate, and a different 

measure is used: the consensus coefficient CC.  For any option, like option T, it is defined as follows:

               CCT         =             .Option T ’s  MBC score     

                                              maximum possible score

So, in a five-option ballot

               CCMAXIMUM            =             1.00

               CCMEAN                    =             0.60

and          CCMINIMUM             =             0.20

A five-option dead heat is highly unlikely.  If the winning score is only, let’s say, 0.63, then obviously most if 

not all the other options have scores of a similar magnitude, so there is no consensus.  Accordingly, as a general guideline:

If the:                     CCWINNER                <             0.65        there’s no consensus; if

0.65        <             CCWINNER                <             0.75        it may be the best possible compromise; if

0.75        <             CCWINNER                <             0.85        it can be called the consensus, and if

0.85        <             CCWINNER                                               it may be the collective wisdom.

CONCLUSION

Majority voting is the most primitive, divisive and often inaccurate measure of collective opinion ever invented.  At worst, ‘majoritarianism’ was a cause of violence:

+             in days now gone, in the USSR, the very word translates as bolshevism, большевизм;

+             sadly, China repeated many of the mistakes of the Soviet Union… and it was all very binary;

+             in Northern Ireland during The Troubles;

+             in the Balkans where “all the wars in the former Yugoslavia started with a referendum,” (Oslobodjenje, 7.2.1999) as too did the conflict in Ukraine;

+             in Rwanda, where the 1994 genocide was initiated by majoritarianism;  and 

+             in Gaza, where the recent war was (and still is) pursued by Israel’s majority coalition.

How sad it is that many of the world’s decision-makers know so little about voting theory.  But you wouldn’t have a majority vote on Yīn or Yáng?   

The two are not a duality; they are more like a unity.  In like manner, there should never have been the referendums of 2014: 

“Are you Russian or Ukrainian?”  Those two groups are, or were not a duality; they were cousins; in Europe anyway, they were both mainly Slav; 

their languages are so similar, and their scripts are almost the same; and their religions are often almost the same, if that matters (which it shouldn’t).  

And you certainly wouldn’t have a singleton binary vote in the multi-option debate which was Brexit; at the very least, it should have been a series of majority votes or, better still, just one multi-option, preferential vote.

               Accordingly, this paper would like to suggest:

THE DE BORDA RULE Majority Voting may be fair  if, and only if, its dichotomy is a duality.

We humans, and especially we Europeans, often tend to divide things into supposed opposites like land and sea, night and day; 

but in many of these pairings, each is in part defined by the other.  In other instances, supposedly of an either/or nature – like 

the nature of light, wave or particular – we now know that it’s both.  Or consider another pair, body and soul.  Or yet another, 

male and female; they, of course, are not opposites – rather, they are complements, what’s more, the two cannot be pro-creative 

if they do not cooperate with each other, ideally complimenting and at best loving each other.  In like manner, democracy is for 

everybody, not only a majority.  Our decision-making should be non-majoritarian; our voting procedures should be egalitarian.  

But maybe everything is connected, to quote the Ukrainian-Russian, Vladimir Vernadsky.  Всё связанно (Vsyo svyazanno).  

Maybe we shouldn’t be trying to separate things into for-or-against.  Maybe the future of humankind rests in our collective 

ability to cooperate.  And “…the nonduality of right and wrong [is] the state of a buddha.” 

---

[1]           Some use a different formula – (n-1), (n-2)… – but the social ranking is the same.