On Mon, 29 Apr 2002, Richard Moore wrote: > Forest Simmons wrote: > > What if the polls could tell us (for each i and j) what percentage of the > > voters approve both candidates i and j. If that percentage is not close > > to the product of the percentages of approval for i and approval for j, it > > would tell us that that approval for i and j are statistically related; > > perhaps the nature of this relationship might be useful information for > > approval strategy. > > > > This information wouldn't require additional questionaires, only summing > > n*(n-1)/2 combinations from each existing questionaire (where n is the > > number of candidates). > > See my April 12 post. I defined Bij(X) as the probability that i will beat > j if i has exactly X votes. If we know nothing about the relationship > between i and j votes, then for this value we can substitute the cumulative > probability that I called Cj(X). > > A correlation (or an anti-correlation) of i and j votes would skew Bij(X). > So if you had the n*(n-1)/2 sums, then perhaps you could determine how > to skew the Cj(X) values to get accurate Bij(X) values. > > I don't know what that calculation would look like, though. How good are > you at statistical formulas?
I'm pretty rusty, but Joe Weinstein might know where to start. BTW I've been thinking a lot about your [Richard's] Democracy Potential ideas lately. In fact I didn't sleep much last night because of it. I was thinking about it in connection with ranked ballots. Suppose that we make the simplifying assumption that all of the voters and candidates place themselves at corners of the issue space, i.e. they generally think of themselves as either for or against an issue, and they think of the candidates in the same way, either for or against. Then why does it sometimes appear that candidates and voters are strewn along a one dimensional spectrum? Well, suppose that the issue space is an ordinary 3D cube, and that the greatest concentrations of voters are located at two diagonally opposite corners. Then the diagonal through the center of the cube connecting those opposite corners determines the spectrum onto which the non-extreme candidates and voters are projected. Where a voter projects onto that line relative to the candidates roughly determines the voter's preference order among those candidates. The other effect in this model which can affect the preference order is that different voters consider different issues to more important, so they scale the three axes differently. Geometrically, for different voters the cube turns into boxes with various lengths, widths, and heights. To see that this can affect the preference order, drop down a dimension. If A, B, C, and D are the four corners of a rectangle, and A and C are diagonally opposite, then there are two possible projection orders along the AC diagonal, namely ABDC, ADBC (as well as their reversals CDBA, and CBDA), depending on whether the AD or the AB dimension is considered greater, i.e. more important to the voter. It seems to me that if you were situated at A, i.e. you agree with candidate A on both issues, then your preference order would either be ABDC or ADBC depending on whether you considered the AD issue or the AB issue to be more important. You would definitely agree with candidate A on first and last place, but you might differ on the order of the two other candidates if you placed different importance on the two issues. So each of the four corners gives rise to two realistic preference orders, yielding a total of eight realistic preference order possibilities out of a total of 4*3*2 permutations, i.e. only one third of the twenty-four possibilities are rational according to this model. In other words, if the issue space is truly two dimensional, then (in this model) the candidates are divided into four clone classes, and there are no more than eight preference orderings of these clone classes. Here's where Democracy Potential comes in. Suppose that in some election we are able to discern that this model is apt, and that the issue space is essentially two dimensional. [That doesn't mean that there are only two issues, it just means that the issues are correlated in such a way that there are only two effective dimensions; if you tell me your stand on the the two key issues, then I can reliably predict your stand on the remaining issues.] Continuing ... suppose that there are six candidates, and that ABC are clones at one corner of the issue square, D and E are clones at another corner, and F is at a third corner, and no candidate occupies the remaining corner. To do a Democracy Potential calculation, just beef up all the corners with virtual candidates until they all have three clones apiece before applying Copeland. This is still in the very rough stage. One result that interested me was this. Condorcet Cycles are possible in this two dimensional model, but only by taking into account that different voters will differ on which issue is more important. Suppose that candidates A, B, and D are not clones, so that they occupy different corners of the issue square in such a way that the path DAB forms a right angle. Then place a virtual candidate C diagonally opposite A. Suppose that the side AB is longer than the side AD. Then the possible preference orders along the AC diagonal would only be ADB(C) and its opposite (C)BDA, and the only orders along the other diagonal would be the opposites B(C)AD and DA(C)B. So leaving out the virtual candidate, we would be limited to ADB, BDA, BAD, and DAB. Two of these ADB and BAD are in reverse cyclic alphabetical order, and the other two are in the other cyclic order, so no cycle of three is possible. But if the group of voters aggreeing with candidate A sees AD as more important than AB, then the preference order ADB is replaced with ABD, and a cycle ABD, BDA, DAB is formed. Now a confession: I started thinking about this in connection with proxy methods. It seemed to me that if there are more than a dozen or so candidates, and the effective dimension of the issue space is only two or three, then there should be plenty of options for the typical voter among the preference orders of the candidates themselves, without having to consider all of the n factorial possible orders. This would be true (in my model) if each voter residing at a corner of the issue space shared the same relative sense of importance of the respective issues with some candidate residing at the same corner. It seems to me that this condition might be approximated pretty well in reality, and that the exceptional voters might be willing to either go with the flow or be satisfied with Approval ballots to express their non-conformity. In other words, I'm thinking hybrid Approval / ProxyApproval would work handily in elections with large numbers of candidates. In a tangential thread, I would like to see Alex Small's symmetry cancelling idea tested by Majority Potential simulation. It seems to me like it might do well in that setting. Forest ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
