[EM] Semantics of voting

Colin Champion colin.champion at routemaster.app
Sun Mar 6 02:58:33 PST 2022

Kevin - you make a lot of points, I will try to reply to a few of them.
    I'm sorry if I didn't make myself clear. My own view is that the 
correctness of electoral decisions needs to be understood evidentially, 
which in the end means Bayesianly. I am pretty sure that Jack Good held 
this view and that Kenneth Arrow did not; but when I try to formulate 
some alternative understanding of the correctness of electoral decisions 
such as may have been in Arrow's head, I honestly struggle. I cannot be 
entirely clear about what a non-evidential semantics would look like.
    I would not say, as a philosophical position, that the semantics 
need to make reference to a model of voting. If it was possible to put 
together a model-free formalistic semantics, it might well be 
acceptable. But in practice I cannot see any viable semantics which does 
not have a model underlying it. Take this as a challenge, not as a dogma.

My notion of a jury model is that each candidate has an objective 
valence (or excellence), which we can take to be gaussianly distributed, 
and that each voter ranks candidates according to his or her own noisy 
estimates of their valences. This is essentially the model Peyton Young 
used, except that he worked with probabilities rather than with 
statistical distributions. (I prefer my own approach because I find it 
hard to keep a clear head when dealing with probabilities. I'm not sure 
Young himself entirely succeeded - I think at one point he says 
"independent" when he means "conditionally independent given... ".)

I'm quite struck by my counterexample to IIA (49% A>C>B, 51% B>A>C, 
which is simply a numerical version of Good's argument). It seems to me 
obvious that A is the rightful winner, and that if C is removed from the 
ballots then B becomes the rightful winner. Certainly anyone who thinks 
that C simply drops out of the analysis, and that my example is 
equivalent to 49% A>B, 51% B>A is making an elementary statistical 
error. I cannot believe that Arrow would have made such a mistake, so I 
conclude that he understood electoral correctness in a different sense 
than Good and I do. If only he had told us what it was!

In fact I haven't verified my numbers. Given a little time I could make 
a plot, similar to the ones on my web page, of isofactors for the pair 
of fractional ballots 0.49 at A>C>B, 0.51 at B>A>C. I assume there would be a 
hill whose summit was in the region in which A is better than B and B is 
better than C. The more ballots you accumulate, the steeper the hill 

Nor have I thought through the application of IIA to spatial models. I 
don't see any reason why it *should* apply, and maybe Arrow's proof 
makes it unnecessary to fill in the details. After all, Arrow never 
limited his theorem to spatial models, and one counterexample is all 
that's normally called for.

I agree that the Borda count is hopeless in the presence of tactical 
voting, even under a jury model. But philosophically I don't feel 
threatened by this. My view is that the right electoral decision is the 
one which is most likely to give the best candidate, or whose expected 
loss is least, or whatever, given - simultaneously - a model of sincere 
voting behaviour and a model of how voters try to beat the system. In 
practice I recognise that the calculation is beyond anything I can 
envisage performing, and that even partial results are likely to 
constitute a significant advance. Under a jury model with tactical 
voting, I have some evidence that the best methods are the ones with the 
worst reputations: FPTP and IRV and their extensions (including 

I think my words about the logical criteria may have come across as more 
dogmatic than I intended. I hadn't realised that participation was 
sometimes recognised as a form of monotonicity. I think I would say that 
some criteria are always and necessarily true; some are always or nearly 
always true, and may may provide useful guidance; and some are totally 
untrustworthy. So long as one doesn't treat a statistical approximation 
as a logical truth I have no real complaint.


More information about the Election-Methods mailing list