[EM] Hybrid/generalized ranked/approval ballots

[fsimmons at pcc.edu]

When equal rankings are allowed, we can distinguish between offensive and defensive strength.  

A candidate can boast of her defensive strength by saying, "In no pairwise contest did any other 
candidate score more than x points against me."  The smaller the minimum value of x for which this 
boast is true, the better the claim of defensive strength. 

Similarly a candidate can boast of her offensive strength by saying, "In every pairwise contest I scored at 
least X points against my opponent."  The bigger the max value of X for which this boast is true, the 
better the claim of offensive strength.

If complete rankings without equality or truncation are required, then these measures of offensive and 
defensive strength are related by  x+X = total number of ballots, so the strongest offensive candidate is 
also the strongest defensive candidate.

Otherwise, it makes sense to pit the one against the other.

Or consider this:

Let o1 and d1 be the respective candidates with the biggest X and smallest x,

If o1=d1, then elect this candidate, else ...

Let o2 be the candidate that scores the most points against d1, and let d2 be the candidate against 
which o1 scores the fewest points.  These candidates have a certain claim as new champions, because 
o2 scored the most against the candidate d1 that was supposedly the hardest to score against, and d2 
held out the best against the candidate o1 that was supposedly the best scorer.  If o2=d2, then elect 
this candidate.

The question is how to continue this process to avoid going in circles.

Perhaps o3 could be the candidate whose minimum score against d1 and d2 was maximal, and d3 
could be the candidate whose max opposition from candidates o1 and o2 was minimal, etc.

Another way to avoid cycling is to make use of the covering relation which is a partial order, so it never 
cycles.

Any other ideas?