[EM] Later no harm, Condorcet, and randomization

[Kevin Venzke]

Dear Jobst,

 --- Jobst Heitzig <heitzig-j at web.de> a écrit : 
> you wrote:
> > Suppose we're using a method that satisfies Clone-Winner:
> > 
> > 51 A 49 B
> > 
> > A wins. Now replace A with two clones, so
> > 
> > 25 A1>A2 26 A2>A1 49 B
> > 
> > A1 or A2 will win, but only assuming this is how A voters really vote
> >  after the cloning operation. In real life I suspect B has a very
> > good chance of winning.
> 
> I see, very good point! I haven't thought about that before...

This problem worries me a lot, especially if the faction that abandons
the other has a better chance of winning:

25 A1
26 A2>A1
49 B

> >> Yes, Woodall's definition for the non-deterministic case is
> >> different. I was only saying that for deterministic situations both
> >> agree!
> > 
> > I don't think this can be true, since a criterion for
> > non-deterministic cases can still be applied in deterministic
> > circumstances. Revising Woodall's criterion to be deterministic would
> > just say that no candidate above the new preference can turn into a
> > loser.
> 
> Well, what does Woodalls LNH demand when the method always elects some
> candidate deterministically, using randomness only for rare ties? It
> demands that when some later preference is added, the winner cannot
> change to a less-preferred candidate.
>   And what does LNPMDO demand when the method always elects some
> candidate deterministically, using randomness only for rare ties? It
> demands that when some later preference is added, the winner cannot
> change to a less-preferred candidate.
>   That's exaclty the same, they only differ for non-deterministic methods.

I hate to press the point, but Woodall's LNHarm can be read exactly
the same way for a deterministic method as for a random method. When
the vote A>B is changed to A>B>C and this changes the winner from B to
A, it's still a LNHarm failure, even if the voter is happy with it.
(It's also a Later-no-help failure, since A is assisted. But since the
voter probably likes Later-no-help failures, I don't think they are
a huge issue.)

I don't mean to say that your suggested LNHarm interpretation isn't reasonable.

> > I still have some difficulty understanding the sense in which "CW
> > else Random Candidate" satisfies a weakened LNHarm. I'll read it
> > again.
> 
> I'll try to explain again: (i) When A is the CW, then adding a later
> preference to some ballot that already contains A leaves A as the CW, so
> nothing changes. (ii) When there is no CW then all candidates have a
> positive probability of winning, including my last choice. So when I add
> some further preferences to my ballot, then either nothing changes since
> there is still no CW, or I produce a CW which will then get elected with
> certainty. Since that CW cannot be my last choice, I will have the
> positive effect that before the change my last choice was a possible
> winner, but after the change it is not. That's what LNPMDO demands.

Ok, I understand. That is interesting.

> > No, the CDTT can be defined two ways, unless I'm mistaken (and if I
> > am mistaken, I hope some list member will correct me!):
> > 
> > The set of all candidates which have majority-strength beatpaths to
> > any candidate possessing a majority-strength beatpath to them. (In
> > other words, candidate A is in the set unless there is some candidate
> > B with a majority- strength beatpath to A, while A doesn't have such
> > a beatpath back.)
> 
> Ah, OK, I understand the difference now. The Smith set can be defined
> similarly, but using ordinary beatpaths instead of "majority strength
> beatpaths", by which you mean a beatpath in which every beat is
> supported by more than half of all voters, right? Then it will make a
> difference when equal rankings occurr, OK.

Right, equal rankings or truncation. It has the property that if A has
a majority-strength win over B, adding preferences to ballots for B can't 
reverse it. That's a vulnerability for Condorcet/Smith, since all pairwise
wins are considered.

> > The CDTT could be larger or smaller than the Smith set. One reason I
> > suggest Random Ballot is that, except for Woodall's DSC method, I
> > don't know of any other monotonic, clone-independent,
> > LNHarm-satisfying method.
> 
> Do you consider LNH more important than Condorcet-efficiency?

Well, in this context, I think it doesn't make sense to pair the CDTT with
a method that fails LNHarm, since the point of the CDTT (as I see it) is a
good degree of LNHarm. I know of maybe 7 methods that satisfy LNHarm strictly,
which is not so many.

If I had to choose blindly between a Condorcet method and a LNHarm-satisfying
method, I would choose the Condorcet method, since LNHarm alone doesn't
seem to guarantee much (consider FPP).

I'd insist on the property that if a majority vote X>Y and don't vote Y over
anyone (Minimal Defense), then Y mustn't win. The CDTT inherently satisfies
this while not strictly satisfying LNHarm.

> > I have to admit I haven't put any thought into uncovered candidates
> > or their significance.
> 
> I think the significance is only that they possess beatpaths of length
> one or two to all other candidates and so majority complaints can be
> rebutted most easily.

Oh, so that's the goal? Is it mathematically guaranteed that some candidate
will have a beatpath of length <=2 to every other candidate?

> > I have a dumb question, though. You have a ranking (generated
> > randomly, or considering approval, etc.) and an "empty chain." I
> > assume the first candidate in the ranking goes into first spot in the
> > chain. But what if this is the CW? Then you won't be able to add any
> > more candidates.
> 
> There's no such thing as a "dumb" question I guess. What you say is
> completely correct. When the CW happens to be first in the sort order,
> no further candidate would be added to the chain and the CW would thus
> win like s/he should.

Ah, I see. That is interesting, if you managed to get a monotone method
selecting from the uncovered candidates.

> Don't think of the sort order as a ranking, it is only the order in
> which I look at the candidates! When there is a CW, no matter where in
> the order s/he comes, s/he will win. Only when there is no CW, there are
> three or more candidates (=the Banks set) which could be the winner,
> depending on the sort order. The significance of the sort order is then
> that the later candidates have a better chance of winning since they
> need only beat the candidates in the chain, while the first candidate in
> the sort order has to beat all other candidates to win. In other words,
> TACC elects the least approved candidate only when s/he is the CW, which
> is a kind of "intuitive loser" criterion, isn't it?

I guess so. No one has proposed an "intuitive loser" criterion, have they?

Kevin Venzke



	

	
		
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