[EM] (not totally successful) attempt to summarize MDDA, DMC, ICA

[Kevin Venzke]

Warren,

Some comments and corrections... Your subject heading says that this is
"not totally successful," but I can't really tell what you don't consider
successful, so I'll just ignore that you put that there.

--- Warren Smith <wds at math.temple.edu> a écrit :
> I consider MDDA often
> unacceptable in many situations because it will generically lead to ties
> (with substantial probability).  Ossipoff replies that that will not
> happen because it will be vanishingly 
> rare for two or more undisqualified candidatess to be ranked on exactly the 
> same number of ballots.
> I counter that I believe there likely are (or will be) a substantial subclass of 
> public elections in which zero ballots are truncated (in fact in most of Australia 
> truncation is forbidden by law).

That's an absurd argument. You might as well say that Approval will lead to
huge ties since it depends on voters truncating some candidates, but e.g. in
Australia voters are not allowed to truncate.

I.e., hopefully no one would be so stupid as to ban truncation when the used
method requires it.

> I shall, in all the MDDA- and DMC- discussion below, forbid ballot truncation.

This seems to make your analysis less relevant.

> Here is the definition of a related (to DMDDA) voting method devided by Warren D Smith,
> (actually originally due to a misunderstanding of DMMDA, but I'll take it:)
> call it WMDDA:
> 1. A candidate is "disqualified" if not in the "Smithmaj set"
> The "Smithmaj set" is the candidates S such that each candidate in S is 
> ranked over each candidate not in S, by a majority.  [Different Smith - not me.]
> Warning: again we need to be careful: 50% of voters say A>B for each A in S and each B not in S.

This is called Smith(gross) or CGTT by Woodall. I guess you mean that it is the
*smallest* such set.

> Virtues of DMDDA and WMDDA:
> 
> FBC compliance ("favorite betrayal")
> means that no one ever has any strategic incentive to vote someone else 
> over his/her favorite, i.e. there always exists a strategically-optimal vote
> rating that voter's favorite top or coequal-top.

Although I don't think Mike Ossipoff noticed that you were only acknowledging
majority-strength wins, he is still right that top cycles aren't compatible
with FBC.

Why not? Because it's no longer obvious and clear-cut whether a voter will
be helped or harmed by creating or removing a pairwise win. The use of beatpaths
makes it impossible to make FBC guarantees to a given voter.

> SDSC compliance means that if a majority of the voters prefer X to Y, then 
> they have a way of ensuring that Y won't win, without reversing a preference 
> or failing to vote their genuine preferences among all the candidates whom 
> they vote over other candidates.
> 
> Schulze's method and MDDA and Range Voting (and Approval) all meet SDSC.

The way SDSC is usually interpreted, Approval doesn't actually satisfy it
due to the fact that you can't express all your preferences. But I agree that
Approval seems to satisfy the spirit of SDSC.

> PROOFS of above 3 DMDDA claims (by Venzke) with WMDDA add-on proofs by WD Smith:
> * satisfies SDSC because if a majority prefer X to Y, and don't approve Y,
> then Y will have a majority-strength defeat. If not all candidates have such
> defeats, then Y can't win. If all candidates have such defeats, then Y still
> can't win, because X has greater approval than Y.
> WDSmith adds: WMMDA also satisfies SDSC because of the same proof with this addition:
> Y is either not in the Smithmaj set (in which case Y cannot win) or is, in which case
> X and Y will both be in the Smithmaj set.
> In that latter case Y cannot win because X has greater approval than Y.

I think you're right about this.

> * satisfies SFC because if no majority prefers anyone to X, then X will
> not be disqualified. If X has a majority-strength win over Y, then Y will
> be disqualified, so that Y can't win.
> WDSmith adds: WMMDA also satisfies SFC because of the same proof.

No, the proof doesn't work for WMMDA. If X has a majority-strength win over
Y, this doesn't mean that Y will be disqualified. Suppose there are three
candidates XYZ and X>Y is the only majority-strength win. Then Smith(gross)
or Smithmaj is {x,y,z}. Y can still win.

> * satisfies FBC because if X wins, and a faction has ranked Y insincerely
> low, then if this faction raises X and Y to the first position, the winner
> will then be either X or Y. This is because X and Y can only lose defeats in
> this way, and other candidates can only gain them. Also, X and Y's
> approval can only increase.
>   NOTE: this FBC proof only works if the "ranking" is allowed to include EQUALITIES not
>   just ">" relations, e.g. A>B=C=D>E=F>G=H.
> WDSmith adds: WMMDA also satisfies FBC because of the same proof.

The proof doesn't work, because of the unpredictable way that raising candidates
can affect who is not disqualified in WMMDA.

> More properties of DMDDA and WMDDA:
> 
> IT IS NOT TRUE THAT it elects a Condorcet Winner if one exists, although
> it would be true if "=" were disallowed in rankings.  Note that the subcase
> where  "=" is disallowed is actually of considerable importance, since with honest 
> non-ignorant voters (assuming generic real utility values) "=" will never occur
> in any vote.

I don't think this case is very important. MMPO also always elects the CW
in this case.

> IT IS NOT TRUE THAT it is Clone-immune, although
> it would be true if "=" were disallowed in rankings.
> If candidate A wins, it is possible that when A is replaced by a set of candidates in 
> a cycle, the winner won't be in this set.

This might be true for WMDDA (i.e. CGTT//Approval), but I don't think it is
true for MDDA.

> It does NOT satisfy incentive to participate, in the sense that you can be better
> off by "not showing up" to vote, because your honest vote can actually hurt you.

In my opinion, this isn't a big deal, since if you have enough information to
determine that you'd be better off staying home than voting sincerely, you
probably also have enough information to find an insincere way of voting that
is at least as effective as not voting.

> It fails "add top", i.e. adding votes all ranking A top, can harm A.
> Venzke clarifies: MMDA fails Mono-add-top because adding A-top votes could cause a lower-ranked
> candidate to lose a majority-strength loss, so that this candidate becomes the only
> one not disqualified.

I don't think Mono-add-top is especially interesting. It's implied by Participation
and isn't much easier to satisfy.

> Now this does not technically count as an FBC failure, because I daresay there
> exists some way to rank A top and dishonestly order the remaining candidates,
> which still leaves A the winner.   However, in practice, it perhaps might have some 
> vaguely similar effects to FBC failure.  

Why do you say this? I already pointed out that A being ranked top on these
ballots isn't the *cause* of A losing; it's just something that happens to be
true about the ballots which make A lose.

> Finally, consider what I call the "DH3 scenario" 
>    http://math.temple.edu/~wds/crv/DH3.html .
> This is a horribly-common and horribly-severe problem that Borda and all Condorcet methods
> based on full-ranking-ballots suffer from in the presence of strategic voters.  

This again, verbatim?

> But Deluxe MDDA might be able to avoid it because it is not just a 
> ranked-ballot method -

On the contrary, DMDDA is friendlier to burying strategy (ranking opponent
frontrunners strictly last, below even worse candidates) *because* of the placeable
cutoff.

> there are also those approval threshholds.  Let's see.
> The situation is there are 3 main rival candidates A,B,C with comparable 
> support, and a "dark horse" D whom all feel is inferior and has "no chance."
> The pathology is the A-supporters strategically rank A>D>B&C
> and so on (although they honestly feel  A>B&C>D), resulting in D winning.
> With Deluxe MDDA, they presumably still will act that way, BUT only approving
> the top one and disapproving the bottom three.  In that case D is the Condorcet
> winner and wins despite zero approval.  So it appears Deluxe MDDA still suffers
> from the horribly-common and horribly-severe DH3 pathology.

So here's the concrete scenario you wanted: Try this scenario instead with
plain MDDA or ICA.

> This DH3 failure is a very bad
> thing and would severely increase its Bayesian Regret.
>  
> CLARIFICATION:  I claimed WMDDA and DMDDA "fail" DH3
>    http://math.temple.edu/~wds/crv/DH3.html .
> However, at the time I failed to appreciate that these methods permit equality rankings.
> Then the A-supporters could strategically rank A>B=C=D instead of honestly A>B=C>D
> (and approve A only).  Result with that voter behavior would be good: C would win,
> not D.  If voters rank A>D>B,C however, then DH3 test is failed.  This latter is
> more strategically sensible so I still conclude DH3 is failed.

Again, this is precisely why Mike and I do not suggest to allow disapproving
B and C while ranking them over D.

> Venzke:
> I'm not a fan of an explicit approval cutoff for MDDA, since it makes burial
> strategy risk-free unless your opponents also use burial strategy.
> 
> WDSmith: I do not understand this, too much jargon is used for my poor mind.
> ("burial strategy"? "risk free"?)
> I'm not saying it is wrong, I'm just saying let's have some explicit examples so I know
> what we are talking about.

Please see above.

> ---------------------------------------------
> 
> THE DMC VOTING METHOD:

By the way, Forest gets credit for this method.

> THE ICA VOTING METHOD  (Venzke May 2005)
> "improved Condorcet Approval" method does not suffer from favorite
> betrayal incentive. 
> 
> WDSmith: I regard ICA as often unacceptable due to its lack of an explicit approval cutoff
> or due to its need for ballot truncation, precisely because I believe there will
> be a substantial subclass of public elections without any truncated ballots - in which
> case ICA would often yield a tie.

This has to be a joke. That's like opposing range voting because many districts
do not currently use rating ballots, and can't be expected to even if they
switch to range voting.

> IT IS NOT TRUE THAT it is Clone-immune, although
> it would be true if "=" were disallowed in rankings.
> If candidate A wins, it is possible that when A is replaced by a set of candidates in 
> a cycle, the winner won't be in this set.

ICA isn't clone-independent even if you disallow equal rankings.

Example:

60 A>B|
40 B|A

A wins.

20 A>C>D>B|
20 C>D>A>B|
20 D>A>C>B|
13 B|A>C>D
13 B|C>D>A
13 B|D>A>C

Now B wins.

> Satisfies SDSC.
> 
> Satisfies SFC.

ICA doesn't satisfy SFC. Example:

49 A
25 B>A
26 C>B

Assuming no vote falsification (reversal or expression of a strict preference that
one doesn't have), it is possible that B is the sincere CW, and B has a majority-
strength win over A. Yet A is the ICA winner.

> It does NOT satisfy subdistrict consistency, that is, if district 1 elects winner W
> and so does disjoint district 2, that does not imply that the combined winner is W.

Is this really interesting? I believe it has been shown that consistent rules
are all point-scoring rules.

> WDSmith: I have not checked these two:
> incentive to participate?
>  "add top"?

You can count the methods that satisfy Participation almost on one hand. They
all score points: FPP, Approval, range, Borda, and Woodall's DAC and DSC.

> ICA fails DH3.

Does it really?

It seems to me that DH3 says that voters using a certain strategy must not be
able to wreck the outcome (from everyone's perspective), no matter how useless
this strategy is under the given method.

Kevin Venzke



	

	
		
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