[EM] 15 reasons to support DMC

[Jobst Heitzig]

Dear Warren!

You wrote on approval cutoffs:
> This extra info may have nothing to do with how satisfied the voter 
> is with any candidate.
Just as an expressed preference X>Y may have nothing to do with which of
X and Y is preferred by the voter. Of course that is so since voters are
free to vote anything they consider sensible for whatever reason. As we
know, the only truly strategy-free voting method is Random Ballot, but
even with Random Ballot we can not be sure whether the voter voted
egoistically the candidate which is best for her or socially the
candidate which she considers best for society.

So, we can never know what the information on the ballot "means".

*However*, I think it absolutely necessary that there is a meaning to
what is *asked* of the voter which must be clearly understandable by the
ordinary voter so that sincere voters will know what they are asked to do.
In my experience, people are usually not game-theoretically thinking
egoistic utility-maximizers, so to me it would be absolutely
inacceptable if the terms used on the ballot were explained only in
reference to strategy, as you seem to suggest.
Those terms ("prefer", "be indifferent", "consider equivalent", "be
undecided", "approve", "rate", etc.) must be defined so that the voter
knows what a sincere answer to the questions on the ballot is.
I know whether I prefer a candidate to another or not; if not, I know
whether I consider them equivalent or not. Even here is some ambiguity:
(When asked whether I "prefer" X or Y, does that mean for myself or for
society? Perhaps even this should be explained on ballots...)
I also usually know whether I approve of a candidate or not, although I
readily admit that "approval" is a more vague term than "preference".
Nonetheless it has a clear meaning.
I have much larger problems when asked to "rate" a candidate on
a scale larger than "approved"/"not approved". Perhaps I could try to
understand the scale "-1"/"0"/"1" as "not preferred/equivalent/preferred
to the prototype" if some prototype is given or suggests himself. But
surely I can never say what it means to give a candidate a rating of "4"
as compared to "5" on a scale "0"/"1"/.../"10". Could you please explain
the meaning of this!

> I think in this case "knowing what it means" would mean "having some
>  understanding of voting stragey in DMC".

That seems to be the crucial point: I strongly oppose this view of
yours! Voters are asked not for strategic things but for meaningful
statements upon their true preferences.

> For example suppose I asked a voter "in addition to your vote, please
>  tell me a color from the set {red, blue, green}."  Well, I don't see
>  that this would make it a better voting method.

Of course not. Colours have nothing to do with preferences. Approval
does. Rating may have if you can explain their meaning to me without
using the word "strategy".

> I am sorry to be so stupid.  I don't think I saw the 15 reasons, by 
> the way, in spite of the subject line.

Here they are again, including three more by Forest -- feel free to
discuss them with us on EM:

> 15 reasons to support DMC ------------------------------------
> 
> 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 bullet-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.

> 16.  Like any method that makes germane use of both ordinal and
> approval information it is well adapted to three-slot ballots, i.e.
> voters that don't want to submit complete rankings can opt to have
> their approval order extended by the order of their favorite.
> 
> 17.  It is resistant to the burying strategy that plagues some
> Condorcet methods.  This is related to reason number 9.
> 
> 18. "DMC" is a better name.  It doesn't matter if the D stands for
> Definitive, Definite, or Democratic, "D Majority Choice" is a good
> sell.


You continued:
> In the Hitler/Stalin/Harding example, the voter is satisfied with 
> nobody.  It is clearly stupid for the voter to say that honestly.

No, it's not. First of all, whereever such an example as the above is
possible, there are much more serious problems than the choice between
those three candidates... However, if you consider this realistic, you
should simply add a requirement to the method saying that only
candidates with more than 50% approval can win and that the election
must be repeated if there is no such candidate. With such a method, it
is not only sensible but absolutely *necessary* to disapprove of all
those candidates instead of voting for the lesser evil.

> --- Re range voting, an "honest" range vote would consist of 
> utilities for all the candidates. (Of course, there would have to be
>  some kind of agreed-on units for measuring utility, etc, so this is
> a fantasy.) 

You got it. Utilities are just fantasy (or, more precisely, a model used
by econometrists because of the nice mathematical conveniences they come
with). Show me a voter who can sincerely assign numeric values to Bush
and Kerry!

> If all range votes were honest, and if votes going
> outside the allowed range were not an issue, then the winner would be
> the uniquely best candidate for society in terms of maximizing human
>  happiness. 

I must admit I can't stand this ever and ever repeated seeming
triviality. How on earth can you suggest "human happiness" should be
defined as a sum of individual utilities! The sum is such a non-robust
statistics that a single over-pleased individual can make it arbitrarily
large while all others have zero utility. If you want to define social
utility (my term for your "human happiness"), then at least use a robust
measure such as the median.

> This I think answers your question of why "sum" is 
> preferred to "Windsorized mean", "median" etc etc. 

To the contrary (see above).

> You may say that 
> in reality votes are just moves in a game and there need be no 
> connection to utilities... 

No. That's *your* line of reasoning which is opposed by me.

Yours sincerely,
Jobst