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

Elisabeth Varin/Stephane Rouillon stephane.rouillon at sympatico.ca
Mon Dec 2 15:28:47 PST 2002

```So if I understand well, Approval encourages voters to maximize their utility
by voting for half the number of candidates?

Steph.

matt at tidalwave.net a écrit :

> 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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