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

Tue Nov 26 18:48:50 PST 2002

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.  
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:

> 
> 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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