# [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
becomes.

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
Condorcet/Hare).

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.

Colin
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