[EM] Voting Power Clarification

[MIKE OSSIPOFF]

When I said that Approval, with 5 candiates, never gives one voter
5 times the possible ballot-expectation of another voter, with any
combination of the extreme and middle utility distributions that I
described, someone could reply "Sure Approval could: What if one voter
rates all the candidates equally? His ballot expectation is zero, and
so every method, including Approval, would give someone else more
ballot expectation by an infinitely large factor."

Yes, but, for one thing, since that's true of all methods, it isn't
helpful for comparing methods. Besides, why should we expect otherwise?
That voter is indifferent between all the candidates, so why should
he care if his ballot can't improve his outcome, since the merit of
his outcome is fixed.

In my examples in the previous posting, for all the utility distributions, 
the high and low utilities were 0 & 10. For comparison
with various voters who value different candidates, for comparing the
voting systems, it seems reasonable that a uniform high & low utility
be used.

So I considered these utility distributions:

10 10 10 10 10 0
0  0  0  0  0  10
10 10 10 0  0  0
0  2  4  6  8  10

All of these have 0 & 10 as their extremes. While varying which
candidates the voters favor, that range should remain uniform, for
fair comparisons.

The first two are the extreme distributions, and the last two are
the two kinds of middle distributions.

When one voter has one of those utility distributions, and another
voter has a different one, when there are 6 candidates, Plurality,
as I showed last time, can give one voter 5 times the voting power
of another voter. Approval's maximum factor by which one voter's
voting power differs from another is only 1.8

With Plurality, the maximum ratio occurs when the voters have the
opposite extreme distributions. With Approval, the maximum ratio occurs when 
one voter has an extreme distribution, and the other has
10 10 10 0 0 0.

Though I didn't say it, the fact that we're assuming all the Pij to be
equal of course means that we can leave them out.

If we ignore the change in the uniform value of Pij, from adding more
candidates...

When more candidates are added, the ballot expectation for the Plurality
voter, when he rates only one 10, increases lineraly with the number
of candidates minus 1. For the Plurality voter who rates only one zero,
the ballot expectation stays the same. So those 2 voters' ballot
expectations, in Plurality, differ by a factor that increases nearly
lineraly with the number of candidates.

The ballot expectation of the Approval voter who has an extreme
utility distribution increases lineraly with the number of candidates minus 
1.

The ballot expectation of the Approval voter who has
10 10 10 0 0 0 increases with the square of the number of candidates.
So, for those 2 Approval voters, the factor by which their ballot
expectations differ varies nearly linearly with the number of candidates.

So, Approval's worst case remains better than Plurality's worst case
by nearly the same factor, as we add more candidates.

Mike Ossipoff





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