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 > > > _________________________________________________________________ > The new MSN 8: smart spam protection and 2 months FREE* > http://join.msn.com/?page=features/junkmail > > ---- > For more information about this list (subscribe, unsubscribe, FAQ, etc), > please see http://www.eskimo.com/~robla/em > > > ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
