[EM] Steph--One more thing re: voting power

[MIKE OSSIPOFF]

Steph--

Suppose that, in the CR point system that we call 'Approval', you
give 1 to just one candidate, and give 0 to all the rest. And say that
i give 0 to one candidate, and 1 to all the rest.

Does that give me more voting power than you have? We've both voted
on equal numbers of pairwise comparisons. Sure, we haven't voted the
same way, but we've both voted equal numbers of pairwise preferences.

now, say someone else votes for exactly half of the candidates. Admittedly, 
s/he has voted among more pairwise comparisons than either
of us has. But if you think that gives him/her more power, then you
can vote for what you perceive as the best half of the candidates.

if you don't do that, it's because you feel that you can do better for
yourself by voting otherwise. And isn't doing better for yourself the
meaningful interpretation of 'voting power'?

Sure, maybe that other person, by his ballot, improves his expectation
more than you do. most likely in no voting system does everyone have
equal ability to improve his/her expectation. But, as i said twice in
the previous message: Approval reduces by a large factor the ratio
of the amounts by which different voters are able to improve their
expectation by their ballot, when compared to plurality.

let me give one brief example:

Say there are 6 candidates: A,B,C,D,E,F

here are your utility ratings for them:

A10, B10, C10, D10, E10, F0

here are my utility ratings of them:

A0, B0, C0, D0, E0, F10

let's define ballot expectation as your expectation for what you
can do for yourself by your ballot.

in Approval, if you vote for i and not for j, your ballot expectation
is Pij(Ui-Uj)/2  , with respect to i and j,

where Pij is the probability that my vote for i and not for j will
turn a j victory into an ij tie, or change an ij tie into an i victory.

We can ignore the factor of 1/2, since it's present in all those terms.
i'll begin leaving it out.

To find your total
ballot expectation, sum that over all pairs of candidates for which
you're voting for one but not for the other.

obviously different sets of Pij estimates could give wildly different
ballot expectations, given a certain set of utilities. So let's just
say that the Pij are all equal, for a best guess for the purpose of
this comparison of ballot expectations. After all, some Pij could be
greater than others, or it could be the other way around, so why not
just assume they're equal, to get the most likely, typical neutral
guess,for our comparisons.

Say the method is Approval. you'd vote for the candidates you rate
10, and not for the one you rate 0.

you are voting between 5 pairs of candidates, and for each of those,
the utility difference is 10. Calculating your ballot expectation as
described above, it's 50.

likewise, i'm voting among 5 pairs of candidates, and the utility
differences are all 10. my ballot expectation is also 50. Approval
gives the same opportunity to get ballot expectation.

Say the method is plurality. Since we're ignoring the Pij, assuming
they're equal, i have no reason to do other than vote for my favorite
in plurality. you don't want to vote for F, but it makes no difference
which of the others you vote for.

What's my ballot expectation in plurality? As in Approval, i'm voting
among 5 pairs of candidates, each with a utility difference of 10.
So again my ballot expectation is 50. What about your ballot expectation?

you're still voting among 5 pairs of candidates, but now only one
of those utility differences is more than zero. one utility difference
is 10, but the rest are all zero. your ballot expectation, in plurality,
is 10. in plurality, my ballot expectation is 5 times yours.

We've been looking at extreme utility distributions, and we could
look at more inbetween ones, such as if you rate half the candidates
10 and the rest 0, or if you assign gradually increasing utilities
to the candidates from A to F, etc. But whichever of those you look
at, you aren't going to find any example in which our ballot expectations in 
Approval could differ by anywhere near as much as by
a factor of 5. only plurality does that. No matter which of those
utility distributions we assign to you and to me, you won't find a
combination of utility distributions in which Approval can give us
ballot expectations that differ as much as they can in plurality.

So, far from making voters have different voting power, Approval
reduces the factor by which voters' voting power can differ.


mike ossipoff


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