[EM] Visualizing Covering

[fsimmons at pcc.edu]

To better appreciate the value of methods that make use of the covering relation it is helpful to have a visual 
interpretation of covering.

Imagine that the voters are distributed in some region of the plane, and that the candidates are scattered 
among them, and that the voters prefer candidates that are close to them over candidates that are farther 
away.

Then the candidates that are beaten pairwise by candidate A  are contained in a delta (like the Nile river 
delta) shaped region on the side of A that is away from the (up river) center of the distribution of voters.  If 
candidate  C  covers candidate A, then  candidate C’s delta shaped region must contain candidate A’s delta 
shaped region.  For this to happen C must be more or less directly between A and the center of the 
distribution of voters.

Also if C is the candidate with the strongest victory over A, and C covers A, then  C is the most natural 
compromise candidate for the A supporters, because it will be both close to A and somewhere between A 
and the center of the distribution of voters.  My original version of UncAAO was designed to take advantage 
of this geometry: if the method generates a covering sequence A, C1, C2, … where each successive 
candidate is the most natural compromise of its predecessor, then there can be little incentive to rank 
insincerely.

In that version, A was the approval winner, C1 the candidate that covered A with the least approval 
opposition from A,  C2 the candidate that covered C1 with the least opposition from C1, etc. until arriving at 
an uncovered candidate.

The simpler version that I call MEA simply initializes a variable X with the approval winner A, and then (while 
X is covered) replaces X with the highest approval candidate that covers X, until finally arriving at an 
uncovered candidate.

In general, instead of approval, one can use some other estimate of candidate strength, and state the 
procedure thusly:

Initialize X as the estimated strongest candidate.  Then (until X is uncovered) replace the current X with the 
candidate with the greatest estimated strength among those candidates that cover the current X.

For example, one can use a version of Bucklin (like MCA) to estimate the strength.  

Or one can estimate the strength as follows:

For each candidate X let S1 be the set of pairwise scores of X against other candidates and let S2 be the 
set of differences of N and the scores against X in pairwise contests (where N is the total number of 
ballots).  [In the case where all of the candidates are completely ranked on all of the ballots, then S1 and S2 
are identical sets.]  Let  m(X) be the minimum number in the union of S1 and S2.

The bigger m(X), the higher our estimate of the strength of X.

Etc.

Forest