2023-21/24 La méthode Borda, in English

La Méthode Borda

France, in the 18^th Century, was soon to change.  Binary voting had been used – and abused – in the 140 member Assemblée des notabless, where in 1788, Louis XVI’s Finance Minister, M Calonne had the support of a minority of just 44 (we’ll call them les blues; the majority, the 96, we’ll call les rouges).  He divided the 140 into seven committees of 20.  In three of them, all the members were red; the other four committees were mixed, with 9 red and 11 blues.  {In total, then, the reds were 3x20 + 4x9 = 96, and the blues numbered 4x11 = 44.}  All seven committees took their decisions by majority vote, and the final decision was taken by a majority of the seven chairpersons; sure enough, the blues won by 4 to 3.

Meanwhile, l’Académie des Sciences was looking across La Manche at Westminster, the only known democracy at the time, and Mon Dieu, c’est incroyable, they concluded, because you cannot find la volonté générale in a binary vote: the result of asking “Are you left-wing or right?” for example, will fall to one side or the other of might otherwise have been the consensus.

So Jean-Charles de Borda and Le Marquis de Condorcet both invented preferential forms of voting.  On a ballot of say five options – A, B, C, D and E – the voters list them in their order of preference.  The Borda rule is a points system – a 1^st preference gets 5 points, a 2^nd gets 4, etc., and the winner is the option with the most points.  The Condorcet rule analyses the options in pairs: is A more popular than B? which we write as A > B? is A > C?… is D > E? ten pairings in all, so to find the option which wins the most pairings.

Both systems are pretty accurate.  They can be compared to the rules of a rugby competition: the champion, the Condorcet winner, is the team which wins the most matches (pairings), while the Borda social choice is the team with the most points (tries etc.).  In many contests, the Condorcet winner is also the Borda winner, but not always.  Binary voting in contrast would be more like a knockout tournament, with much dependant on the draw… but no-one surely would want a knock-out world cup with France playing Ireland in the first round!

L’Académie adopted the Borda Count BC, and it worked quite well.  In a ballot of n options, le chevalier suggested, a voter may cast m preferences, where obviously:

n > m > 1,

and points shall be awarded, he continued, to (1^st, 2^nd … last) preferences cast, according to the rule,

(m, m-1 … 1).

Indeed, it is brilliant when everyone casts a full list of all five preferences; for we all have preferences; and the BC was designed, in M de Borda’s words, for voters who are “honest.”  Unfortunately, however, his rule was soon changed to

(n, n-1 … 1).

or

(n-1, n-2 … 0).

Worse than that, these n rules came to be called the BC, whereas Jean-Charles had advocated the m rule, which is now called the Modified Borda Count MBC.  The n rules prompt the voters to truncate their ballot.  The m rule, in contrast, encourages (but does not force) them to cast all their preferences; to state not just their favourite option but also their 2^nd and perhaps too their subsequent preference(s)… their compromise option(s)… and if everyone does that, we can identify the collective compromise… which is what politics is all about!  Or should be.

Now nothing’s perfect, and what happens in rugby if France beats Scotland, Scotland beats Ireland, and Ireland beats France?  Or, in a vote, what happens if A > B, if B > C and if C > A?  Le Marquis himself recognised this in 1793.  So the Condorcet rule is vulnerable to what is called the paradox of binary voting, when as here

A > B > C > A > B …..

and it goes round and round, for ever!

Preferences	Ms i	Mr j	Ms k
1st	A	B	C
2nd	B	C	A
3rd	C	A	B

In a BC analysis of this table, all three options get a score of 6.  But what happens if another team, option D the Kiwis, joins the fray?  If New Zealand always gets beaten by the Irish, if C > D, option D could be seen as an ‘irrelevant alternative,’ to quote the jargon.  If nevertheless D does play, if it is inserted into the table just below C, the outcome is different, the final Borda analysis is C 9, B 8, A 7, D 6; so what was a three-way draw is now a win! 

Preferences	Ms i	Mr j	Ms k
1st	A	B	C
2nd	B	C	D
3rd	C	D	A
4th	D	A	B

For this reason, some scientists have said parliaments should use both voting systems: the Condorcet rule never suffers from an irrelevant alternative, and the Borda rule is not vulnerable to the paradox.  If then the Borda and Condorcet winners coincide, we can be 99% sure, we’ve got the accurate outcome.  Sadly, however, many politicians say that because nothing is perfect, they will continue to use that which is most imperfect – the 2,500-year-old binary vote.

It was the same in France.  There was the revolution, so would decision-making now pass to les citoyens, as L’Académie might have wished.  Alas, one individual didn’t like this preferential stuff, and he reverted to majority voting.  This meant that he could control the agenda; indeed, he could control everything, because he and he alone would chose the option.  He chose himself.  And thus, in 1803, with a mere 99.7% of the vote in his third binary referendum, he became l’empereur. Napoléon.  A ‘democratic’ dictator, one might say.  So, as suggested in Coutances after a lecture I gave there in 2006, maybe majority voting should be renamed le système Napoléonique.