Difference between revisions of "Pairwise Sorted Dyadic Ballots"

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The concept is too complicated for normal people.  I'll let Forest Simmons explain it in his own words.  See [http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2001-April/005741.html this] and [http://listserver.dreamhost.com/pipermail/election-methods-electorama.com/2005-March/015273.html this].
 
The concept is too complicated for normal people.  I'll let Forest Simmons explain it in his own words.  See [http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2001-April/005741.html this] and [http://listserver.dreamhost.com/pipermail/election-methods-electorama.com/2005-March/015273.html this].
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Briefly, dyadic ballots require each voter to sort the alternatives into a binary tree structure, for example
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      A > B >> C > D >>> E > F >> G > H
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The strongest preference relation >>> is the root of the tree structure, and the alternatives A through H are the leaves of the tree.
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After all of the ballots are in, the pairwise sort is seeded by approval, which is determined by the strongest (i.e. root) preference relation only.
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One of several possible ways to finish the process is this:
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1. Make three pairwise matrices, M, M', and M''.  The first of these M makes no distinction in the preference relation strengths, so that (for M) our sample ballot is interpreted as
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        A>B>C>D>E>F>G>H .
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The second matrix M' collapses the weakest relations and makes no distinction among the rest, so that (for it) our sample ballot is treated as though it were
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        A=B > C=D > E=F > G=H .
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The third matrix M'' is the coarsest; it  collapses all of the relations except the strongest (root) relation:
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        A=B=C=D > E=F=G=H
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2. Now take our approval seeded list and sort it according to the pairwise matrix M', and then according to the (finest)pairwise matrix M.
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Remarks:
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In step 2 it makes a difference if bubble sort or sink sort or some other kind of swap sort is used. I suggest a sort that (at each step) always reverses the remaining out-of-order pair with the greatest defeat strength as measured by the next coarser pairwise matrix.
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So while sorting with M', we should measure defeat strength with M'', and while sorting with M, we should gauge defeat strength with M'.
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If symmetry is an important consideration, then the pairwise margin in the next coarser matrix is the appropriate gauge.

Revision as of 19:48, 30 March 2005

The concept is too complicated for normal people. I'll let Forest Simmons explain it in his own words. See this and this.

Briefly, dyadic ballots require each voter to sort the alternatives into a binary tree structure, for example

      A > B >> C > D >>> E > F >> G > H

The strongest preference relation >>> is the root of the tree structure, and the alternatives A through H are the leaves of the tree.

After all of the ballots are in, the pairwise sort is seeded by approval, which is determined by the strongest (i.e. root) preference relation only.

One of several possible ways to finish the process is this:

1. Make three pairwise matrices, M, M', and M. The first of these M makes no distinction in the preference relation strengths, so that (for M) our sample ballot is interpreted as

       A>B>C>D>E>F>G>H .

The second matrix M' collapses the weakest relations and makes no distinction among the rest, so that (for it) our sample ballot is treated as though it were

       A=B > C=D > E=F > G=H .

The third matrix M is the coarsest; it collapses all of the relations except the strongest (root) relation:

       A=B=C=D > E=F=G=H

2. Now take our approval seeded list and sort it according to the pairwise matrix M', and then according to the (finest)pairwise matrix M.

Remarks:

In step 2 it makes a difference if bubble sort or sink sort or some other kind of swap sort is used. I suggest a sort that (at each step) always reverses the remaining out-of-order pair with the greatest defeat strength as measured by the next coarser pairwise matrix.

So while sorting with M', we should measure defeat strength with M, and while sorting with M, we should gauge defeat strength with M'.

If symmetry is an important consideration, then the pairwise margin in the next coarser matrix is the appropriate gauge.