[EM] A distance based method

[Juho Laatu]

On 11.7.2011, at 2.34, fsimmons at pcc.edu wrote:

> First find a clone consistent way of defining distance between candidates.
> 
> Then while two or more candidates remain
>  of the two with the greatest distance from each other
>  eliminate the one with the greatest pairwise defeat
> EndWhile.
> 
> Various variants are possble.  For example, you could count defeats only from the remaining 
> candidates.  Also there are various possible measures of defeat strength.  In that regard, if you say that 
> any defeat by covering is stronger than every non-covering defeat, then the method will always elect a 
> covered candidate.
> 
> To get a distance estimate in a large election you could just ask each voter to list the pair of candidates 
> that seem the most different on the issue or combination of issues of most concern (to that voter).  The 
> pair submitted by the greatest number of voters would be the first pair considered, etc.

The approach of finding the distant ones and thereby eliminating the extremists first makes sense. But I wonder still if also other approaches would make sense, like measuring which candidates are near each others. It may be easier for a voter to tell which candidates are close to his opinions rather than tell how distant two random candidates (that are nowhere near that voter) are to each others.

> 
> What potential for manipulation does this direct approach introduce?
> 
> Perhaps voters would try to pit their favorites' rivals against each other.  Would that be insincere?  Not if 
> they consider their favorite to have a reasonable middle of the road position, while viewing the rivals as 
> being at opposite unreasonable extremes.

If we assume three candidates that for a loop (CW cases are not interesting), then the winner will be the candidate that was not included in the first comparison. So it would make sense to make a comparison of one's worst rivals first. The best strategy might be to forget the really different candidates and point out those candidates that are most likely to beat your own favourite.
> 

> 
> What indirect measure of distance could be used?
> 
> If we count the number of ballots on which candidates X and Y are ranked at opposite extremes (top 
> rank for one versus unranked for the other), the monotonicity of the method would probably be 
> destroyed.  Is there a more subtle way of inferring the distance that wouldn't destroy the monotonicity?

My sincere ranking is A>B>C>D>E. Candidates D and E may well be most distant from each others from my point of view. I might consider myself to be in the centre of a one or two dimensional space. D could be far left from my point of view and E far right. Distance from A to both of them is shorter than distance between D and E.

In the proposed method cloning of the candidates helps them. For this reason it is not in my interest to emphasize the clone relationships of candidates that I do not like. And as seen from the D/E example above, candidates D and E may well not be clones (similar) to me, but candidates A and B are likely to be near clones (nearly similar) to me. One could thus measure the clone relationship starting from the beginning of my ranking order. Some "clone points" could be added to A-B, A-C, B-C and so on with decreasing weight. Maybe already here A and C are more likely to be similar than B and C. One possible simple rule to order the similarity pairs could be to say that pairs whose more liked candidate in more liked than the more liked candidate of the other pair represents higher similarity (otherwise the popularity of the second candidate is used). This rule would be also in line with the loop of three strategy example above, i.e. it is good to make a comparison between the worst competitors first.

Instead of eliminating one of the most distant candidates one at a time, one could also use a tree approach here. Measure the similarity of the candidates and form a corresponding binary tree. One would proceed in the comparisons starting from the most similar candidates that form the smallest branches and then proceed towards the root of the tree.

One question is how to measure the strength of the pairwise comparisons in the whole electorate. Should one use some points and weights or should one make majority decisions and pairwise comparisons between the similarity pairs in Condorcet style. Points might be a quite natural approach here. But let's assume that candidates A, B and C form a loop. I want B and C to be compared first (see my preference order above). If positions close to the beginning of my ranking order make candidates more similar, then I would have some interest to bury both B and C under numerous less risky candidates (this is risky of course). If the ranking distance between B and C is important then I would have some interest to bury C (risky again, and that increases also the distance between A and C). If the weight of my vote in determining the clone relations is always the same if the relative order of A, B and C stay the same, then I will not have similar strategic incentives to bury. Strategic nomination of clones of A (and maybe B too) could be also interesting. There are thus various strategic opportunities. But these incentives may also be small enough and the strategies difficult enough so that I will not bother to use them.

It could be also possible to allow the voters to indicate their clone estimates as part of their rankings. One could use a cutoff to indicate which candidates the voter considers good enough to be clones from his point of view. Multiple cutoffs and different preference strengths like A>B>>C>>>D>E are not very handy since D and E may not be clones as discussed above. But instead of this king of linear ordering / linear space one could use at least in principle also two-dimensional (or even n-dimensional if the voters are really competent :-) ) ratings to indicate preferences and similarity. The voter himself would be described as a point in the middle of the paper, or maybe a screen. Different candidates would be put on different places on the screen so that similar candidates are next to each others. This would allow the voter to indicate whether D and E are similar or different. One could also alternatively (in theory) allow the voters to indicate the similarity level of each pair of candidates. But maybe the geometric approach of n-dimensional ratings (and derived rankings) is easier, more natural, sufficient, maybe also offers less opportunities for strategies, and already complex enough.

Juho