[EM] Semantics of voting
Colin Champion
colin.champion at routemaster.app
Thu Mar 10 01:11:18 PST 2022
Kevin – I suspect we don't disagree about anything. It would have been
clearer if I'd said that my jury model was a special case of the
Condorcet/Young model with added distributional assumptions for my own
conceptual convenience.
A simpler example of IIA has occurred to me which I mention for what
it's worth. It doesn't refer to voting but brings in all the important
features.
Let x, y and z be 3 values. We have no idea which is greater than
which. Our initial beliefs can be expressed by giving them independent
standard Gaussian priors. E[x-y]=0.
Now we're given a single piece of information that x>z. Our
posterior beliefs about y are unchanged from our prior, but we now have
a joint posterior distribution on (x,z) which is a standard bivariate
Gaussian restricted to the half-plane in which x>z. E[x]>0, E[z]<0;
E[x-y]>0.
Here x is the merits of automobiles, y is the merits of buses, and z
is the merits of teleportation. If we are interested in choosing between
automobiles and buses then teleportation is an irrelevant alternative;
according to IIA comparisons with an irrelevant alternative can be of no
help in choosing between automobiles and buses; Bayes' theorem disagrees.
Colin
On 09/03/2022 00:14, Kevin Venzke wrote:
> Hi Colin,
>
>> 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... ".)
> Ok. A sort of hidden rating, and the voters use them to populate whichever type
> of ballot is provided.
>
>> 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!
> Doesn't IIA hold *within* a jury model? I mean the estimates of valences.
> Perhaps Arrow intends such a model, and means that it would be reasonable to
> hope that a property of the model could also be reflected in the procedure.
>
>> 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.
> My view is pretty similar.
>
> Kevin
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