[EM] expectation with sincere voting, table.

[MIKE OSSIPOFF]

Bart wrote:

>I don't know about the IRV figures, but the Borda figures don't seem
>right to me.  For example, in the u(B) = 0.9 case, the Borda figure
>should probably be 2, if calculated in the same way as the others.
>
>I am assuming the probability of a tie between AB, AC, and BC are equal,
>and that the probability of a three-way tie is negligible.  For u(A) =
>1.0, u(B) = 0.9, and u(C) = 0.0, the utilities in the case of a tie
>(broken randomly) if the voter doesn't participate are shown in the
>first column.
>
>---------------------------------------------------------
>Outcome utility for voter with u(A)=1, u(B)=.9, u(C)=.1
>where participation breaks a two-way tie:
>
>Non-participation   Plurality   Approval   Borda
>AB       .95           1.00       .95       1.00
>AC       .50           1.00      1.00       1.00
>BC       .45            .45       .90        .90
>
>mean(x3) 1.9           2.45      2.85       2.90
>---------------------------------------------------------
>Utility gains for ideal strategies:
>
>Non-participation   Plurality   Approval   Borda
>AB        0           .05         0         .05
>AC        0           .50         .50       .50
>BC        0            0          .45       .45
>
>mean(x3)  0           .55         .95      1.00
>---------------------------------------------------------

But you've got to consider not only the utility gains,
but also the probability of getting it, which varies with
how many votes of vote-difference you're voting between
that pair of candidates, and how many votes of vote difference
the other voters are averaging between each pair, on the average.
That last one depends on how many pairs each voter votes between
and how many votes of vote difference he votes between each
pair that the votes for. Only in Borda does he ever vote a
vote difference of more than 1. Those considerations differ
between methods, so they affect the comparison of the methods
in that regard.

But now that I've found out the sincere voter's utility
expectation improvement with 0-info favors IRV over Approval,
I've decided that that standard of comparison doesn't mean
anything :-)  How convenient. But it doesn't mean anything,
because, for comparing methods according to your expectation
in the election, you're interested in genuinely big expectation
differences, the kind that we find when we compare different
methods' social utility and use that to judge our expectation,
right now, in a future election some time after the new method
is enacted.

The tiny expectation difference due to the fact that your tiny
effect of voting is a little larger with IRV is negligible in
comparison. What the expectation effect of your voting _is_
relevant to is your voting strategy of course. That's all.
It isn't a way of comparing voting systems as I was trying to
do.

Of course, with 0-info, everyone's expectation, with any method,
is the mean utility of the candidates, plus the tiny amount that
you raise your expectation when you vote a certain way.

That's from the point of view of someone who doesn't know anything
about the voters. If you don't, then your strategy has to
be based on that lack of information. But just because you don't
know where the voters are in policy-space, that doesn't mean that
they aren't somewhere. They have unknown positions, and
in any particular election there'll be a social utility maximizing
candidate. And if, for instance, Condorcet has a good automatic
ability to pick that candidate, then it still does so even if
we know nothing about the voters.

Of course, if the candidates, like the voters, have 0-info about
the voters' positions in policy space, then we can't expect them
to arrange themselves with their mean position near the voter
median. Not by design, at least. But of course that still can be
said to be the most likely place for their mean position to be,
even if we & they don't know where it is--just by chance, wouldn't
their mean position tend to vary around the point that's the'
voters' median & mean? So to that extent Approval still has some
reason to be expected to do reasonably by social utility when no
one knows anything about voters' positions.

Anyway, the candidates in reality have some idea where the voters
are, and we can observe them every day scrambling to be where the
voters are.

Merrill's simulations showed Approval doing better than IRV
in social utility.

>
>***

About considering factions instead of individuals or pairs of
individuals, maybe that could make a difference by making some
of our simplifying assumptions less valid, like the assumption
that, in Borda, casting a vote-difference of 2 between 2 candidates
doubles our chance of changing which of them wins, as compared
to casting a vote difference of 1. And, similarly, the assumption
that if 2/3 times as many people vote between 2 candidates,
then the chance of our vote difference between them changing which
of them wins is 3/2 times as great.

But aren't those assumptions only important in Borda, and for
comparing different methods as I was mistakenly trying to do
today? For calculating strategy in a particular method, aren't
those assumptions less needed, and wouldn't consideration of
factions have less effect on our results?

Mike Ossipoff



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