Below is an extract of a piece written by Dr Peter Emerson, de Borda Institute, author of The Punters’ Guide to Democracy (Springer, Heidelberg).
I live in North Belfast. Imagine a hypothetical community group on the Ballysillan, a committee of a dozen, and we’ve just installed a new door. So, what colour shall it be? The decision shall be democratic, and chaired, impartially of course, by a wise old soul… like me.
So, any suggestions? I ask. 5 want it to be red; 4 prefer white, and 2 say blue. Oh dear; with singleton majority votes – “Option X, yes or no?” – there are majorities of 6, 7 and 9 against red, white and blue, respectively, a majority against everything. (As in Theresa May’s first ‘indicative vote’ on Brexit, when 8 options were ‘on the table’.) Ri’ht oh, I suggest, let’s try pairings – “Option X or option Y?” – in which case, what are your preferences?
The 5 have 1st-2nd-3rd preferences of red, white and blue R-W-B
4 ” white, blue and red W-B-R
and 2 ” blue, red and white B-R-W.
So, with majority voting, it’s:
“R or W?” and R is more popular than W, which we write as R > W by 7:4
or “W or B?” ” W > B ” 9:2
or “B or R?” ” B > R ” 6:5.
And this means that
R > W > B > R > …
which goes round and round for ever! The paradox of binary voting.
Needless to say, every committee member believes in majority voting. And in any debate on three options, with four possible outcomes – R, W, B or the status quo – a decision will require three majority votes. This procedure was devised over 2,000 years ago by our Greek and Roman ancestors, and it’s still in use today.
In any multi-option debate:
+ first, choose the favoured amendment;
+ secondly, adopt or reject this to get the substantive, as it’s called;
+ and finally, make the decision: the substantive or the status quo.
So, if R is the motion and W and B are the two possible amendments, we first debate W v B, and to get the substantive, the winner of that pairing will face R:
(W v B) v R.
There again, if W is the motion, it could be
(B v R) v W
or again
(R v W) v B.
But this means the substantive depends upon the order of voting. After all,
(W v B) v R = W v R = R
while
(B v R) v W = B v W = W
and
(R v W) v B = R v B = B.
In other words, I, the totally unscrupulous chair, can get whatever I want. Now in fact, I don’t want R, W or B, I want Green, G. So I could have said, OK everybody, what about a compromise, what about green? Umm – on the Ballysillan? – green is not very popular. And when I ask for their preferences,
5 have 1st-2nd-3rd-4th preferences of red, white, blue and green R-W-B-G
4 ” white, blue, green and red W-B-G-R
and 2 ” blue, green, red and white B-G-R-W.
Oh dear: all 11 prefer B to G.
But I’m the chair. And I choose the order of voting. So I suggest:
{(W v B) v R} v G
because that gives
{W v R} v G
which leads to
R v G
so the answer, the democratic will of all 11, is
G.
Ha! I’ve won. But, as we saw earlier, all 11 prefer B to G, the answer is wrong.
In summary, if I the chair can dictate the order of voting, and if no one option has majority support, I can get whatever I want. Furthermore, if something which I don’t like does have majority support, I can suggest other alternative colours, split the majority, adjust the voting order, and still get whatever I want.
Further info
Read the full piece here: https://www.tasc.ie/blog/2026/07/24/democracy-is-for-everybody-not-just-a-majority-and-definitely-not-just-the-majority/