[EM] Three rounds

[Kristofer Munsterhjelm]

Juho Laatu wrote:
> --- On Wed, 12/11/08, Kristofer Munsterhjelm <km-elmet at broadpark.no> wrote:
>> Schwartz is Condorcet-like because a CW will always be in
>> the Schwartz set, and Smith (and Schwartz) is a reasonable
>> extension of the Condorcet criterion (from "a candidate
>> who is preferred to all others should win" to "of
>> a group where the group is preferred to all outside the
>> group, a group member should win"). Minmax is Condorcet
>> yet not Schwartz, but anything that's Schwartz is also
>> Condorcet.
> 
> 
> Clones of members of the Smith set (I'll ignore ties in this mail)
> are  in the Smith set but Smith set members need not be clones. The Smith set
> is thus not a (unified) group of clones (but just a group of candidates
> that happen to best all the others in a pairwise comparison).

> There are also other potential "reasonable extensions" of the
> Condorcet criterion. One interesting question is if it is more important
> for the elected candidate to have weak opposition or to have a narrow
> opposition.
> 
> (Beatpaths may be considered a "winner evaluation criterion" too -
> although their meaning in real life situations is not clear - maybe they
> are used just to identify clones in some approximate way rather than
> describe the value of the to be winner.)

> One basic example.
> 17: A>B>D>C
> 16: A>D>B>C
> 17: B>C>D>A
> 16: B>D>C>A
> 17: C>A>D>B
> 16: C>D>A>B
> A, B and C are in a strong loop. A, B and C form the Smith set but
> they are not clones. Each of them is beaten badly by another member of
> the loop. D loses to all the Smith set members, but only with a very
> narrow margin.
> 
> One could say that D is the most acceptable choice, and that electing
> the candidate with weakest opposition (against any single one of the
> other candidates) is a natural extension of the Condorcet criterion.
> 
> (D is the Condorcet loser but it is also very close to being the
> Condorcet winner. The visual impression of "being below the top three"
> positions D somewhere deep down at the bottom of the picture and at the
> end of the preference list, but obviously such 2D visualization does not
> describe the cyclic relations in the best way.)

I think that in order to get anywhere on this path, we would have to 
know what it is we actually want from a runoff. There are two reasons 
why you might have a runoff: the honest, that voters can discuss which 
of the two candidates are better without having to consider the others, 
and the strategic, arising from that in a runoff that has only two 
candidates, the optimum strategy is honesty (if we consider the runoff 
one election, not half an election).

I'm going to skip past the strategic reason for now, but I'll note that 
earlier I mentioned some ideas regarding that, with two election methods 
being run in parallel (one resistant to strategy and one vulnerable), 
the winner of each going to the runoff; this could even work if one 
method is vulnerable to a different strategy than the other.

For honesty, then, we have to know which are the two best candidates. 
This sounds like a proportional representation problem with a "council" 
of two; however, such methods cannot be invulnerable to cloning, since 
the Droop proportionality criterion and clone independence contradict 
each other (by http://www.mcdougall.org.uk/VM/ISSUE3/P5.HTM , 
"clone-no-harm").

Also, if we want to retain the properties of the first-round election 
system, and that election system is Condorcet, then one of the 
candidates in the runoff must be the CW (when it exists). I would go 
further and say that there's no need for a runoff if there's a CW, but 
others may disagree. In any event, if the first round method is Smith or 
Schwartz, or more generally picks from a well defined subset of the 
candidates, then one of the candidates of the second round must be from 
that set as well. The former destroys any chance of passing the DPC, 
since Droop proportionality is incompatible with Condorcet (by example 
given in the Voting Matters article linked to above).

It seems difficult to consider a consistent way of picking the other 
candidate, given this. Consider opinions on a line, where the centrist 
at 0.5 is the CW, and assume that runoffs will be held even when there's 
a CW. Then where should you put the other candidate? Not to the right, 
because that would be biased against the left-leaning voters. Not to the 
left, because that would be biased against the right-leaning voters. So 
it must be another centrist, a clone. But what choice is that?

However, that may work only as an argument against "runoffs should be 
held even when there's a Condorcet winner". To my knowledge, on a 
political line and with honest voters that prefer candidates closer to 
them to those farther away, there'll always be a Condorcet winner. That 
means we'll have to consider opinion spaces in greater than one dimension.

Call the candidate that's retained from the first round to pass 
criteria, the retained candidate. Perhaps we could then say that if the 
retained candidate is off-center in n-space, then the right thing would 
be to pick the viable candidate closest to its antipode (reversed 
coordinates) as the other candidate. But what's a viable candidate? Is 
it viable if in Smith (or mutual majority, or whatnot)? Is it unviable 
if it isn't? We need the viable qualifier to keep the runoff method from 
picking an unpopular candidate simply to provide "opposition", e.g 
picking a lone right-wing authoritarian to "oppose" the retained 
left-wing libertarian in a society of (mostly) left-libertarians.

>> If you're going to use party list, I don't see much
>> point in a runoff. Either it'll be multiwinner, in which
>> case a runoff doesn't make much sense, or it'll be
>> single-winner, in which case you can just use the equivalent
>> single-winner method. For ordinary party list, that method
>> would be Plurality; instead of voting for a party, vote for
>> the party's designated "appointee".
> 
> 
> I was thinking of single winner elections only.
> 
> The party lists could be more interesting when breaking Condorcet
> cycles. But in a runoff one could first vote between parties and only
> then between candidates of the winning party. I'm not sure that this is
> very useful, but this way one could e.g. reduce the risk of the best
> compromise candidate of a party being eliminated too early.
>
> For example
> 40: A1>A2>B>C
> 08: A2>A1>B>C
> 07: A2>B>A1>C
> 25: B>A2>C>A1
> 20: C>B>A2>A1
> A2 would be eliminated first in IRV but here A1 and A2 form a party
> (with 55 first preference votes) and therefore C will be eliminated
> first, B next, and then A2 will win.

This seems to be more of a problem with IRV, and so I'd say that a 
better solution would be to switch to another voting system rather than 
try to patch it up with party lists.

If you want to use party lists, you could add dynamics by doing it this 
way: First, voters vote for parties. Then start the compromise rounds: 
in a round, each party provides the name of its appointed candidate (or 
"I support the candidate of party X in the last round", if party X 
provided its own candidate in that round). This may be done sequentially 
in a random fashion or by popularity or some complex device. After n 
rounds, each party contributes [number of voters for that party] points 
to either the candidate it supported in round n-1, or to the candidate 
the party it supported in round n supported in round n-1. Highest score 
wins. That would be a kind of sequential Asset. The question is whether 
the dynamics would be good.