After Jean-Charles de Borda, 1733-99 - DeBorda Institute - 2023-18 House of Lords (my presentation)

Thursday

Jun082023

2023-18 House of Lords (my presentation)

Thursday, June 8, 2023

DECISION-MAKING

1 2,500 years ago, when the Greeks of old devised the world’s first democratic structures, they were very aware that, on complex topics, there might well be a majority against every proposal.

We have probably all enjoyed the experience of trying to persuade a bunch of kids to choose the vegetable for lunch. Turnips? No, a majority doesn’t like turnips. And that’s that! Swedes? Another majority against. Another that. Broccoli? And so it goes on: as in Greece, a majority against everything.

2 So, if the questions are singletons – “D’you want ‘this’, yes-or-no?” – there might well be lots of ‘no’ answers. With the other form of majority voting, however – “D’you want ‘this’ or ‘that’?” – a pairing, there will always be an outcome of some sort.

3 So, imagine the scene of a status quo, option S; a motion, option X; and two possible amendments, options Y and Z. Given the above uncertainties, the Greeks devised certain rules using what was at the time the only known voting system, binary or majority voting.

+ Choose the preferred amendment;

+ adopt or reject this preferred amendment; the substantive;

+ decide, this substantive or the status quo; the result.

4 Despite the invention of other more accurate voting procedures, the above rules are still used today. But consider what might happen if 15 people have the preferences shown.

Table I A Voters’ Profile

Preferences	5	4	3	3
1^st	Z	S	Y	X
2^nd	S	Y	X	S
3^rd	Y	X	Z	Z
4^th	X	Z	S	Y

With four options, if asked a singleton, “Option ’this’, yes-or-no?” there will be majorities against everything, of 10, 11, 12 and 12 against Z, S, Y and X respectively. So the answer is nothing.

5 But with pairings – and there are six of them: X/Y, X/Z, X/S, Y/Z, Y/S and Z/S – the answer might be anything: X, Y, Z or S.

So comparing X and Y, with X in the green and Y in yellow, we see that Y > X by 12:3. Next, X > Z = 10:5, and Z > Y by 8:7, and so

X > Z > Y > X > ….

and it goes round and round, for ever; Le Marquis de Condorcet’s famous paradox of binary voting.

Now the Table shows that:

X:Y = 3:12, so Y > X

X:Z = 10:5, so X > Z

X:S = 6:9, so S > X

Y:Z = 7:8, so Z > Y

Y:S = 3:12, so S > Y

Z:S = 8:7, so Z > S

In summary, majority voting might produce nothing, or it might produce anything. Therefore, the Greeks decided, there have to be those rules.

6 So we return to the debate in which, according to those ancient rules, the format shall be like this:

Figure 1 The First Debate

v = …..

Z v = …..

X v = …..

So, three binary votes. After lengthy arguments no doubt, the outcome will be as shown.

Figure 2 The First Outcome

v = Z

Z v = X

X v = S

7 If, however, the debate had been structured such that the motion was Z, and the two amendments were X and Y, the result would have been different.

Figure 3 The Second Outcome

v = Y

Y v = Z

Z v = Z

8 Indeed, whenever there is a paradox, the outcome can be manipulated, simply by adjusting the debate. Furthermore, if the status quo has been changed, democratically of course, so that it was now option Y, another debate could be held:

Figure 4 Another Debate and Outcome

v = S

S v = Z

Z v = Z

So whenever there is no majority in favour of any one option, whenever there is a paradox, the debate can be manipulated to produce any outcome at all. And if there is a majority which the chair doesn’t want, he/she can introduce other options in effect to split the majority, and then, again, adjust the order of voting.

9 That, then, is a mathematical proof to show that, when there are more than two options ‘on the table,’ – which, in a pluralist democracy, should be just about always – binary voting can be inadequate. This was asserted, nearly 2,000 years ago, in a Roman court of law by Pliny the Younger. Little wonder that others have resorted to more sophisticated forms of voting. The first government to do so was Chinese, some 800 years ago, when the Jīn Dynasty used a plurality vote. Europe at that time was just emerging from the Dark Ages, but Ramón Llull then suggested, Nicholas of Cusa invented, and Jean-Charles de Borda developed, in the years 1299, 1433 and 1770 respectively, that which is now called the Modified Borda Count MBC.

10 But let us first return to a contemporary debate, to consider the above voters’ profile, with exactly the same preferences, on the question of the UK’s relationship with the EU.

Table II The Same Voters’ Profile

Preferences	5	4	3	3
1^st	*EEC*	EU	CU	*WTO*
2^nd	EU	CU	*WTO*	EU
3^rd	CU	*WTO*	*EEC*	*EEC*
4^th	*WTO*	*EEC*	EU	CU

11 When David Cameron held his binary vote on the option EU, there was indeed a (small) majority against. But maybe, as with the children squabbling over lunch, there was a majority against everything. Indeed, when Theresa May held her indicative vote on four options, there was exactly that, and partly because she used majority voting, her vote indicated nothing! Lord Desai had already described the taking of umpteen majority votes in a debate on umpteen options as “daft,” – but that was in 2003.

Then came Boris. He probably would have lost on a singleton, “D’you want ‘my deal’, yes-or-no?” He therefore manipulated a pairing: “D’you want ‘my deal’ or ‘no deal’?” So he won. But of course he b….y won. ‘No deal’ was the most unpopular of all options. ‘Any deal’ versus ‘no deal’ would have been a victory for ‘any deal’.

In a nutshell, majority voting is hopelessly inadequate.

* * * * *

A BETTER POLITY

12 If the above voters’ profile is analysed by other voting procedures – if, in other words, if we count things in a different way – the outcome might also vary.

But first, it is surely pretty obvious that with 5 people thinking it is the best while the other 10 think it’s most definitely not the best, option EEC is very divisive. Opinions on options CU and WTO are varied. But a full dozen gives the tinted option EU a 1^st or 2^nd preference, a total of 12 think it’s pretty good, and only 3 regard it as the worst possible option, so maybe the EU best represents the voters’ best compromise.

And sure enough, if you change the way you analyse voters’ preferences, you might get different outcomes. As Prof. Don (Saari) will show, for this particular profile, the ‘democratic’ outcome depends not on the voters’ preferences, but on the voting procedure!

Table II The First Voters’ Profile

Preferences	5	4	3	3
1^st	*EEC*	EU	CU	*WTO*
2^nd	EU	CU	*WTO*	EU
3^rd	CU	*WTO*	*EEC*	*EEC*
4^th	*WTO*	*EEC*	EU	CU

Table III The Analyses

Procedure		Social Choice	Social Rankings
Plurality voting		*EEC*	*EEC*-5	EU-4	CU-3	*WTO-*3
Two-round system		*EEC*	*EEC*-8	EU-7
AV (or STV or RCV)		*WTO*	*WTO*-10	*EEC*-5
AV (or STV or RCV)		EU	EU-10	*EEC*-5
Approval voting	1^st and 2^nd	EU	EU-12	CU-7	*WTO*-6	*EEC*-5
Approval voting	1^st, 2^nd, 3^rd	*EU/CU*	*EU/CU*-12		*EEC*-11	*WTO*-10
MBC		EU	EU-43	CU-37	*EEC*-36	*WTO*-34
Condorcet		*EU/EEC*	*EU/EEC*-2		*CU/WTO*-1

In summary, the MBC and Condorcet are the two systems which always take all preferences cast by all voters into account. Of the two, the MBC is the more nuanced. Furthermore, it can identify the option with the highest average preference; it is non-majoritarian. It could be, therefore, the basis of a more consensual polity.

A CONSENSUS POLITY

13 In a five-party parliament, any one party may move a motion. Other parties may have other ideas but, rather than amending ‘this’ and ‘that’, ‘these’ and ‘those’ paragraphs, each may propose a complete package. The methodology may be compared to what happens in a German constructive vote of confidence: if you don’t like (option or) candidate A, suggest an alternative, B, C or whatever. So, in such a pluralist democracy’s 5-party parliament, any one debate might see up to five options on the table (and computer screen, if not too a dedicated website).

In the debate itself, the Lords or MPs may suggest amendments, or composites, or even a deletion, but a change shall be adopted only if the original proposer(s) agree thereto. As the debate proceeds, the number of options under consideration may vary. If it comes down to a singleton, this may be regarded as the verbal consensus. If not, the Speaker shall draw up a balanced list of options to represent everything still on the table – either verbatim, amended or in composite – and, if all concerned agree to this list, the members may proceed to the vote. In the analysis, if the most popular option has surpassed a predetermined threshold, it may be adopted.

CONCLUSION

14 Pluralism is possible. Politics does not have to be an adversarial contest. And in these days of conflict, Covid and Climate Change, the need for a more consensual polity cannot be over-emphasised. Suffice here to say that, in identifying the option with the highest average preference, the non-majoritarian MBC is indeed inclusive, literally. It is also robust, extremely accurate, and very difficult to manipulate. Furthermore, given the possibilities catered for by the advent of electronic voting, and the dangers we all face under AI, this more sophisticated voting procedure of preferential voting could and should be adopted.

If it were to be the international norm, if decisions in every (multi-party or multi-faction) parliament or congress, in London, Washington, Moscow and Beijing, were to be based on the options which received the highest average preference levels, majority rule could be replaced by all-inclusive coalitions, and party-political politicians could cooperate, just as the world’s nations are learning to cooperate in the COP conferences… without majority voting! Granted, some politicians prefer politics to be win-or-lose. But democracy should be for everybody, not just the winners. Politics should be a game of no trumps.

Thank you.

15 An answer to a possible question.