[EM] A Ballot Space Method Based on CR Ballots

[Forest Simmons]

1. As usual in ballot space methods, convert each ballot into a vector of
CR ratings for the respective candidates.

2. Get candidate vectors as weighted averages of all of the ballots that
rate that candidate at 100 percent.

A ballot's weight is the reciprocal of the number of candidates rated at
100 percent on the ballot in question (i.e. the symmetric completion
weight).

3. Decide on a metric for the vector space.  For best results a metric
which is relatively insensitive to clones should be chosen. [The
L_infinity norm metric is completely insensitive to clones, but may be too
strong.]

4. Construct a complete weighted graph where the vertices or nodes of the
graph are the candidate vectors (with their respective weight totals) and
the edge weights are the distances between nodes (distance according to
the metric chosen in step 3).

5. Start the process of building up a minimum edge_weight spanning tree
for the graph, but stop as soon as any connected component of the graph
reaches fifty percent of the total node weight (i.e. 50 percent of the
total top support of the voters).

[Or equivalently, start deleting edges from the complete graph in order of
largest weight to least weight, until every remaining component of the
graph has less than fifty percent of the total node weight. Then add back
the last deleted edge.]

6. The winner must be chosen from the candidates making up this component
with fifty percent or more of the total node weight.

7. (option a) The other candidates are eliminated from consideration, but
their (node) weights are transferred to the candidate nodes closest to
them.

7. (option b) Similar to option (a) but transfer on a ballot-by-ballot
basis instead of wholesale.

8. (option a) The candidate that minimizes the node weighted average
distance to the other candidates (i.e. other nodes in the remaining part
of the graph) is the method winner.

8. (option b) Reduce the remaining part of the graph to a minimum edge
weight spanning tree.  Then calculate the winner as in option 8(a).

Note that if option 8(b) is used, then one can proceed by collapsing the
tree (while accumulating node weights), and as soon as nodes with fifty
percent of the node weight have been collapsed down to one node, that node
will be the node of the winning candidate.

Enjoy!

Forest