[EM] Matt: voting power with more candidates

[MIKE OSSIPOFF]

Matt--

You wrote:


Following Mike's example of setting the utility range from 0 to 10, then
simplifying by
giving all candidates utility ratings of either 0 and 10, it can be shown 
that
the
maximum utility gap for Approval increases as the number of candidates
increase.

I reply:

That's what I'd said. I said that the maximum ratio of ballot expectations 
both in Approval and in Plurality increases nearly
linearly with the number of candidates. Because it increases in the
same way with both methods, Approval remains better than Plurality
in terms of maximum ratio of ballot expectations.

By the way, someone could say that another meaning for voting power
would be, for any pair of candidates Smith & Jones, the power to
influence whether Smith outpolls Jones or vice-versa.

In that regard, Approval could be said to be equal to Plurality, since
in either method a voter has equal ability to influence that.

But really, Approval is ahead in that regard because Smith or Jones
might be one voter's favorite, while another voter might have someone
else as favorite, and want to vote for the favorite. Approval allows
better freedom to vote in a particular pairwise contest, because it
allows voters to vote in more pairwise contests.

Ballot-expectation takes that into account, along with the relative
importance to the voter of those pairwise contests.

Mike Ossipoff






For 6 candidates the maximum gap, excluding meaningless ballots that are all 
0
or
all 1, is 90 (1 1 1 0 0 0) versus 50 (1 0 0 0 0) which is less than double.
For 8
candidates it is 160 versus 70 which is more than double.

For a given number of the candidtes the maximum gap utility can be reduced 
by
limiting the maximum number of 1 or 0 votes to a number less than half the
number
of candidates. For example, only Approval ballots with two or less 1 or 0
votes can
be permitted when there are ten candidates. A single 1 vote represents
utility=90.
Two 1 votes could, at most, have utility 160 (those two 10 and the the other
eight 0).
Contrast that with the maximum utility of 250 for five 1 vote ballots when
there are
no restrictions. So in addition to having a smaller maximum utility gap than
Plurality, limited Approval balloting also enables limiting the size of the
utility gap by
not allowing ballots with equal or near equal numbers of 1 and 0 votes where
the
largest utility values reside. Of course, to achieve this result it is
necessary to
permit both 1100000000 and its inverse 1111111100.

On 24 Nov 2002 at 3:19, MIKE OSSIPOFF wrote:






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