[EM] 15 reasons to support DMC

[Jobst Heitzig]

15 reasons to support DMC
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1. Allows to distinguish important from minor preferences.
Unlike rankings-only methods like Beatpath or MinMax, DMC allows voters to give a full ranking and still make clear where their most important preferences are by specifying an approval cutoff. This will make it easier for voters to give full rankings instead of ballot-voting.

2. Immunity from second place complaints.
Unlike in MinMax and Beatpath, the DMC winner always defeats the candidate which would win if the winner were not present. This can be paraphrased to “accepting the mandate has always majority support”.

3. Smith-efficiency.
Unlike in Condorcet//Approval or MinMax, the DMC winner always has a beatpath to each other candidate. More specifically, the DMC winner has a beatpath to candidate X leading thru more-approved candidates than X only.

4. Good defendability of the winner against contenders.
In addition to Immunity from second place complaints and Smith-efficiency, the DMC winner also always defeats her most approved contender (unlike Condorcet//Approval, Smith//Approval, Beatpath).
More generally, the DMC winner always defeats or is more approved than any other candidate (unlike in Beatpath).

5. Strange winners are seldom.
Unlike in Beatpath, the least approved candidate cannot win in DMC unless she defeats all other candidates.
Unlike in Condorcet//Approval, a candidate defeated by all others cannot win in DMC (“Condorcet Loser Criterion”).

6. Robustness against “noise” candidates.
In addition to being clone-proof (see below), DMC cannot be manipulated by strategically nominating weak candidates in the hope of influencing the result. More precisely, adding some candidate X does not change the DMC winner whenever there is some other candidate Y which is (i) more approved than X, (ii) defeats X, and (iii) defeats every candidate which is defeated by X and less approved than X.
In particular, unlike Beatpath, DMC fulfils the IPDA criterion: Adding a Pareto-dominated candidate does not change the winner.

7. Easy and transparent algorithm.
For DMC, the candidates need only to be sorted by approval score and then each pair of candidates needs to be inspected at most once to decide which candidates are doubly defeated. Methods like Smith//Approval or Beatpath involve an iterative and more complicated procedure.
To speak mathematically: With n candidates, DMC finds the winner in O(n^2) time, whereas Smith//Approval and Beatpath need at least O(n^3) time.

8. Robustness against counting errors.
Since DMC uses only the ordering by approval score and not the precise approval scores, and uses only the direction of the pairwise defeats and not any kind of “defeat strength”, it is more robust to small changes in the individual preferences than Beatpath.

9. Avoids “margins/winning votes”-debate.
(For the same reason)

10. Avoids a discussion of “cycles”.
In DMC, the winner is found by an intuitively plausible algorithm which does not need an understanding of the concept of majority cycles, whereas Smith//Approval and Beatpath do.

11. Allows to construct a complete ordering.
If necessary, one can also assign final ranks to all candidates such that the k-th ranked candidate is the DMC winner when all k-1 candidates above her are removed from the race. 
This ordering arises naturally from the following “resorting” procedure: List the candidates from top to bottom by their approval score. As long as there is a pair of neighboured candidates in the list such that the lower candidate defeats the upper candidate, swap the topmost such pair. When no such swappings are indicated further, each candidate defeats the next one, the DMC winner is listed on top, and the candidate listed at k-th position would become the DMC winner when all k-1 candidates above her were removed.

12. Combines instead of separates the available types of information.
DMC combines the approval and ranking information from the beginning: When both kinds of information indicate that candidate X is “better” than candidate Y, then Y is considered “doubly defeated” and cannot win. Condorcet//Approval and Smith//Approval use the two kinds of information separately instead, first only looking at the pairwise defeats from the rankings to test for a Condorcet Winner or find the Smith set, and only afterwards using the approval information to resolve the remaining ambiguity.

13. Monotonicity.
Unlike IRV, DMC is monotonic, that is, reinforcing the DMC winner on some ballots cannot turn her into a loser.

14. Clone-proofness.
Unlike Condorcet//Approval and MinMax, DMC is cloneproof: Assume a candidate X is added which is a “clone” of some other candidate Y who is already in the race. That is, both are approved by the same voters and no voter ranks a third candidate between them. Then this cannot change the DMC winner except that when Y won before then now X may win instead.

15. Defeats other method’s winners.
In every situation, the DMC winner is either identical to or defeats the winner of each of the following methods: Approval Voting, Condorcet//Approval, Smith//Approval, DFC, TAWS.


Jobst


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Remarks: 
I post this mail to both the standard EM list and the new Condorcet list at Yahoo.
Strictly speaking, some of the above arguments are only exactly valid when there are no ties (equal approval scores or equal number of votes for both candidates in a pairwise comparison).
The method referred to as “Schulze” or CSSD by others is called Beatpath here.



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