[EM] In defence of IRV

[Colin Champion]

Forrest wrote: "If people interested in voting methods reform would take 
the time to digest these basic facts .... they would never accept any 
'ranked choice voting' method unless it satisfied the Condorcet 
Criterion." I think that a defence can be offered for people who reject 
the Condorcet principle.

Firstly, some published evaluations show no appreciable difference in 
performance between Condorcet methods and IRV. In Tideman and 
Plassmann's 2012 paper, Table 1 appears to show that when the number of 
voters is large, Condorcet methods are 94.46% accurate and AV/IRV is 
94.41% accurate. A twentieth of a percent difference! In Table 2 of 
Green-Armytage et al (2015) you get similar numbers: Minimax is 95.19% 
accurate and "Hare" 94.45%. If the difference really is that small, then 
IRV has the advantage of known resistance to tactical voting and has the 
benefit of having been widely used in practice. There should be no 
argument that IRV is the way to go.

In my view there are two weaknesses with this argument. I think (i) that 
having 3 candidates in a multidimensional space makes the problem 
unrealistically easy, and (ii) - notwithstanding (i) - that these 
results could only have been produced by buggy software. I'll say more 
about this later. But Tideman is probably the most respected worker in 
the field. You can't argue against IRV if you don't admit that his 
evaluations have been seriously misleading.

Secondly, it seems to me that supporters of IRV generally acknowledge 
the defects of their method (while probably not being aware of their 
magnitude). They consider that at least IRV has a realistic chance of 
adoption. They point to the fact that their critics are divided into 
1000 factions, that they attach too much importance to theological 
arguments about logical criteria, and that they often end up advocating 
unrealistic methods. This allows Condorcet supporters to be portrayed as 
lacking social responsibilty: see 
https://rangevoting.org/SchulzeComplic.html.

Finally, voting methods are generally part of larger electoral systems, 
and decisiveness is often as important as fairness. In the UK, electoral 
reformers start off thinking they have irresistable logical arguments, 
and end up getting bogged down in unwinnable debates about the need for 
'firm government'. (I can say nothing about the USA, whose institutions 
I don't understand.) Unless you can argue that a change of voting method 
will not do more harm through indecisiveness than it does good through 
fairness, you haven't really got an argument in favour of Condorcet methods.
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On evaluations. The Median Voter Theorem (which I think Forrest was 
alluding to) predicts that if a large number of voters come from a 
symmetric distribution (eg Gaussian), then all Condorcet methods will 
elect the candidate closest to the point in space which minimises the 
sum of distances to voters (ie. the centre of the distribution). This is 
not quite the same as electing the candidate who minimises the sum of 
distances to voters, who will be identified as the rightful winner under 
standard evaluation procedures. But the difference is tiny, and cannot 
account for the 5% error rate attributed to Condorcet methods by Tideman 
and his coworkers.

I described an evaluation of my own in a previous post. I used Gaussian 
mixture models instead of pure Gaussians precisely in order to escape 
from the Median Voter Theorem; otherwise the Condorcet systems would 
have been indistinguishable from each other. But it's a trivial change 
to revert to a single Gaussian. When I do so, the accuracy of Condorcet 
methods is 99.73% and that of IRV is 95.81% (3 candidates, 30001 voters, 
a million trials). Similar to Tideman et al for IRV; totally different 
for Condorcet.

Now see what happens when we go up to 9 candidates. The accuracy of 
Condorcet systems drops to 99.43%... and the accuracy of IRV to 56.92% 
(these figures from 100k trials).

To be quite clear: I am not saying that Tideman's evaluations are wrong 
because they contradict my own; I am saying that they are wrong because 
they contradict the Median Voter Theorem; that my own evaluation 
attaches numerical figures to an otherwise qualitative argument; and 
that the Tideman results are also difficult to reconcile with those in 
other published evaluations (Chamberlin and Cohen (1978) and Darlington).

Incidentally there was also a much earlier evaluation by Samuel Merrill 
III. He defined utility as 'decreasing linearly with (Euclidean) 
distance' (p26) which brings him under the purview of the Median Voter 
Theorem, but he nonetheless found the Borda count to outperform 
Condorcet methods. In his own words (p24) "we will see that this 
criterion [ie. of maximising utility] and the Condorcet criterion need 
not agree". I don't trust Merrill's evaluation either, and in this I 
have some powerful support from Warren D. Smith, who wrote ("Range 
voting", p24) "That suggests that Merrill's computer program had bugs".

CJC.