[EM] Re: majority rule, mutinous pirates, and voter strategy

[Juho Laatu]

Hello James,

Thanks for the excellent mail. I still found some points where 
different definitions lead to different conclusions. See (lengthy) 
comments below.

BR, Juho



On Mar 17, 2005, at 09:51, James Green-Armytage wrote:

> I suggest that most public elections will fall within the region of
> "somewhere". (Please see my 3/14 post.)

Ok. I don't have any firm opinion (due to lack of sufficient 
understanding) although I'm trying to seek evidence of unusable 
strategies and hoping that the best method would be close to the "best 
sincere method". I'd maybe not even try to fight some (non fatal) 
strategies in the hope that many people would vote sincerely just 
because they are supposed to do so, or because they are lazy or not 
aware of the strategies.

> copying your pirate example for reference:
> 101: a>b>x>c
> 101: b>c>x>a
> 101: c>a>x>b
> 100: x

> 	Your logic is as follows: If X wins, and a group of 202 pirates who
> preferred another candidate rather than X wanted to mutiny, there 
> would be
> 201 pirates ready to stand in their way, serving as an effective
> deterrent. However, if A wins, and the 202 C>A pirates (101: B>C>X>A, 
> 101:
> C>A>X>B) mutiny in favor of C, there won't be sufficiently many 
> pirates to
> fight to defend A.

Yes, that's what I thought.

> (I'm assuming that all the
> pirates know each other's expressed ranked preferences, as would be the
> case in any real public election.)

My assumption was that the fact that there are four parties of about 
equal size was known. Since I at some point said that these pirates 
would be from different countries, maybe also the exact number of 
people in each party is known. In most elections that is not known. The 
fact that there is a cycle and the direction and strength of it and the 
fact that one party is not part of the cycle may or may not be known 
(or guessed) in advance. (Pirate example is of course just one example 
to demonstrate the generic idea that in some elections preference 
margins inside the Smith set are stronger than preference margins 
towards the other candidates.)

> 	So I ask you, would the B>C>X>A voters participate in the first mutiny
> against A? I suggest that they would not, because they would realize 
> that
> a victory for C so reached would be unlikely to last.

I'm afraid I don't have a clear answer to this. But I do my best and 
try to explain at least something.

First one obvious counter question:  What if X was elected in the first 
round? Maybe no strong mutiny tendencies at all.

If they would have foresight and they would guess that there is a loop 
of preferences, and the Smith set parties would be able to find each 
others and agree, my proposal to the Smith set parties would be to toss 
a three sided coin before the first election and leave X party out of 
this process based on the fact that it is expected to get one vote less 
than the other parties. (I leave it open if the pirates also believe 
that it is correct to pick the winner from the Smith set.) Then all 
Smith set party voters would vote the lucky winner (A,B or C).

I think cycles of three are really vicious. The next possible strategy 
of B party would be to agree with C party that A will be thrown out 
together, and B party voters will after that satisfy with and defend 
their second best alternative C. This is already a partial answer to 
your question. If someone at B party would invent this strategy and 
convince others about its benefits, they would participate in the 
mutiny against A. On the other hand, if they would invent the logic 
your described first, maybe they wouldn't participate in the mutiny.

There are really many options - and finding an easy solution for a 
cycle of three is not easy. My best guess is that it is all up to luck 
and negotiation skills and quick actions.

Although I took the mutiny case up, answering your question is a bit 
tricky also because in normal Condorcet elections there are no second 
rounds. Well, maybe some countries could have a law saying that new 
election must be arranged when majority of voters is not happy with the 
achieved result. But in that case probably also all old candidates 
would participate in the new election. I think it is characteristic to 
the Condorcet methods that all necessary information will be collected 
already in the first and only election round. Maybe the correct 
interpretation to the term "mutiny" is not to expect people to change 
the result but to ask who would be the best such first choice that 
there would be no interest to change the result. I'm a bit biased here. 
Also any other reasoning (than those favouring X) for picking the 
winner of the election are ok.

Now back to the question of X vs. Smith set.

What do you say about the following viewpoints:
1) candidate A is good if there is a strong interest to change some 
other candidate to A
2) candidate A is good if there is a weak interest to change A to some 
other candidate

 From a stability oriented point of view 2) looks more tempting. The 
path oriented thinking seems to favour 1). If 1) is used, X has no 
chance. If 2) is used, X has a strong chance.

One type of foresight requiring logic for the pirates would be to elect 
X. Assuming that their thinking was in line with 2).

I have made some studies above and, of course, came back to defending 
X. The best constructive comment I can make at this point is that the 
voting method should elect the best candidate right away so that the 
selection would be done once and for all. I picked mutiny resistance as 
a requirement since it seems to favour X. Also other 
criteria/requirements on which candidate is the best may exist.

My answer to your question is:  Don't know. Maybe there is no answer 
since it is all up to tactics.

I think it is a mathematical fact that if mutiny resistance is accepted 
by a country as the target of the election, one must elect the 
Condorcet loser in some cases. And MinMax (margins) is the correct 
voting method if votes are sincere. In real life votes are not 
necessarily sincere and some other variant may be more useful. But this 
does not contradict the fact that in some cases the Condorcet loser 
should be elected. (Before shooting me because of my strong claims, 
please see also below how I defined the mutiny to mean the first mutiny 
:-).)

> 	Preferences:
> 35: R>S>T>Z
> 33: S>T>R>Z
> 32: T>R>S>Z
> 71: Z>R=S=T
> 	Pairwise comparisons:
> R>S 67-33
> S>T 68-32
> T>R 65-35
> R>Z 100-71
> S>Z 100-71
> T>Z 100-71
>
> 	Candidate Z is the minimax(margins) winner. However, he is in no wise 
> the
> most mutiny-proof candidate.

Here our thinking differs. I'm thinking about the probability of the 
first mutiny, which can be said to correspond to (one measure of) 
dissatisfaction with the election results (in countries where 
revolutions are unlikely and term "mutiny" therefore would mean only 
mutiny in some debates and votings (other votings than what we are 
talking about here), not really throwing out the elected person). For Z 
the probability of first mutiny is still the smallest. Chain of 
mutinies may not lead back to Z, but that does not worry me if there 
will be no mutiny in the first place.

> (Hence they can happily mutiny against Z
> without worrying that it will hurt them in the long run.)

This is a good argument in the pirate world if there are sequential 
mutinies, but as I said, I'm mostly focusing on avoiding mutinies 
altogether (and consider it a problem of the election if there is even 
one).

> So I argue that Captain R would
> suffer less risk of mutiny than Captain Z.

Yes, in the meaning that you described. Z has however slightly smaller 
risk of first mutiny.

> 	I hope that I have disrupted your assumptions concerning the "risk of
> mutiny" concept.

Yes you did to some extent. I however still hide behind the argument 
that the risk of first mutiny is the parameter that some election 
organizers may want to minimize.

Another threatening argument in your example could be the fact that R, 
S and T supporters could be seen as one big party. If that was the 
case, then they would beat Z clearly 100 against 71. What would my 
argument be against this. I think the best explanation in that case is 
that people inside the RST party are not able to agree internally which 
candidate is best and therefore electing Z from the other party might 
be appropriate. It is possible to seek voting methods that would allow 
people to express their opinions in more detail than in the basic 
ranking based methods. In this case RST party members maybe would like 
to have a rule that would somehow eliminate the negative effect of 
loops within each party. But for the time being this is ffs for me and 
I stick to the basic Condorcet rules. => Least first mutiny risk is a 
least mutiny risk, without considering if party internal mutiny risk 
should be considered smaller than mutiny risk between parties.

One hint for the RST party, in case we end up using MinMax (margins), 
is that they should consider setting only two candidates, and maybe 
arrange a pre-election within the party before the actual election. 
Note that I keep looking alternative strategy reduction methods that 
would allow me to still use the sincere method (or something close to 
that) and sincere votes in the elections.

I might mention here also how I want to disrupt your thinking. There 
are at least three points.
1) One should consider separately what is the best ideal / strategy 
free / sincere method, and after that what is the method that is best 
in real life. The latter differs from the former because it tries to 
defend against various strategies.
2) "Least first mutiny risk" is a possible and rational target (= as 
the ideal case of the previous point) for some elections (if some 
country wants to emphasize this aspect)
3) If one accepts point 2) then in those elections Smith set is not a 
target. It could be used for defensive purposes as described in point 
1) above but better if we could do without it.

Maybe you can comment which ones of these you find ok.

> 	If there is a majority rule cycle, then one cannot avoid ignoring at
> least one majority preference. However, one can always avoid ignoring a
> majority preference that is not contradicted by another majority
> preference (via a cycle).

Why is it more important to avoid the loopless majority opinions than 
the looped ones? One can always avoid the loopless ones but then one 
has to violate one of the Smith set internal majorities (that might be 
stronger). (I'm back to square one. Don't want to end up in a debate 
loop though :-).)

> In
> Condorcet systems, we should not assume that this polarization will
> remain; rather, it seems logical that compromise candidates will 
> emerge,
> which they haven't done in your example.

I agree. And my example is an extreme case. I would expect that in most 
elections there are no loops. If there is a top loop, then it is 
probable that it is weaker than the preferences between the top loop 
and the other candidates. As a result the pirate case is highly 
unlikely to occur in real elections.

Furthermore I don't want ever to face the situation where a Condorcet 
loser or anyone outside the Smith set is elected. (But is extreme cases 
that may be the least bad option.) Actually my target is to prove that 
these strange cases are so rare that we probably need not care about 
them to much. The most probable reason why we might have loops is 
intentional strategies. As you know, also here I hope that most of them 
would be so unusable that we could simply forget most of them. If you 
want to shake my world a bit more, I could be most vulnerable to 
attacks that would show practical use cases where strategies could be 
useful in normal elections (typically without detailed information of 
the preferences of the voters). Your recent mail on one vulnerability 
of margins was a good one. I hope to come back to that.

> 	Please read and consider my recent post about strategic vulnerability 
> in
> "margins" methods before you state so unequivocally that it is "the
> correct voting method". Actually, even then you might want to be 
> careful
> about calling anything "the correct voting method" without some sort of
> qualification.

I note that I already restated that claim earlier in this mail. Sorry 
for using strong terms, but I did that because I believe that is a 
mathematical fact and I wanted to point that out. In my sentence I 
however missed the assumption that this requirement points out the 
correct _sincere_ voting method, i.e. I was talking about the ideal 
case and strategies were excluded in this claim. Fight against 
strategies may lead to deviation from the ideal sincere method. Another 
point that was certainly confusing here is the definition of risk of 
mutiny. I meant the first mutiny after election.

> X is a
> slightly worse choice, because choosing X unnecessarily violates 
> majority
> rule.

I'm still a bit confused about the use of term "unnecessarily". It 
seems to mean that in the loops violation is necessary and in linear 
preference chains unnecessary. This makes sense as long as we have a 
Condorcet winner and we can pick the candidate that at the better end 
of the linear chain. Then it would be an unnecessary violation to pick 
someone else than the Condorcet winner. But if there is a loop at the 
end of the chain (Smith set), then avoiding violation of majority 
preference in the linear structure leads (now necessarily!) to 
violation of a majority preference in the loop. And in this situation 
where we must violate some majority preferences we could as well 
violate the linear ones.

(In the pirate example we would need to violate only one but strong 
majority (margin) preference or alternatively three weak linear 
preferences.)