[EM] UncDMC

[Forest W Simmons]

Here's a Monotone method (UncDMC) that chooses from the uncovered set, 
and always picks the DMC winner in the three candidate case:

1. List the candidates in approval order, highest to lowest, top to 
bottom.

2. Modify the list according to the following rule: as long as some 
candidate in the list is pairwise defeated by its immediate inferior in 
the list, swap the members of the highest such pair in the list.

3. Initialize a set S with the highest member of the modified list.  As 
long as no current member of S is uncovered, add to S the highest 
member of the modified list that covers each of the current members of 
S.

4.  The last candidate added to S is the winner.

This UncDMC method is monotone, clone proof, independent from Pareto 
dominated alternatives, and independent from Smith dominated 
alternatives, and always picks from the uncovered set.

If I am not mistaken, previously, the only known deterministic method 
to satisfy all of these criteria was Jobst's TACC.

A careful comparison of UncDMC and TACC in the three candidate case 
would be helpful.

The UncDMC winner is either the DMC winner or a candidate that covers 
the DMC winner.  In the three candidate case the DMC winner is always 
uncovered, so it is also the UncDMC winner.

We ought to examine three candidate cases where DMC and TACC produce 
different winners.

Now a proof of UncDMC's monotonicity:  

Suppose that the UncDMC winner X improves in approval or in pairwise 
defeats relative to the other candidates (which retain their same 
relative approvals and pairwise defeats relative to each other).  

Then X is still uncovered, X still covers all of the candidates that it 
covered before, and the part of the modified list above X is a subset 
(in the same order) of the part of the modified list that was above X 
before X's improvement.

So X will still be the last member added to the set S, retaining the 
win.

Thanks,

Forest