# [EM] UncDMC

Forest W Simmons fsimmons at pcc.edu
Sat Mar 10 12:16:14 PST 2007

```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

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

```