[EM] Composite methods (Re: Eric Maskin promotes the Black method)

[Kristofer Munsterhjelm]

robert bristow-johnson wrote:
> 
> On Jun 21, 2011, at 7:56 AM, Markus Schulze wrote:
> 
>> Hallo,
>>
>> Eric Maskin, a Nobel laureate, is currently very
>> active in promoting the Black method.

> and we've all been groping for a name for this primary voting criteria 
> that is not this non-American, Frenchie, probably sorta pinko-socialist 
> secular humanist "intellectual" (did i mention *not* American?) whose 
> heresy is leading us away from the One True Faith of the Single 
> Affirmative Vote.  we have sects in the One True Faith, some of us 
> believe in the sanctity of the Two Party System: "if yer ain't fer us, 
> you agin' us.  and pass da ammunition, Ma."

I've mentioned it before, but I think Condorcet enjoys an additional 
advantage here. Say there's a CW and he is not elected. Then that means 
a majority prefers the CW to the candidate who was elected, and if that 
majority is annoyed enough, it could try to repeal the voting method in 
question. However, if the method always elects the CW, any attempt to do 
so must face a majority who did prefer that CW to all the other 
candidates, and if that majority feels the candidate is good enough, 
they can block the repeal by virtue of being a majority.

> i don't have a better idea than "true majority rule".  but there must be 
> a better one than that.  Warren, i remember you like "beats-all winner" 
> for the CW.  i wonder if the "beats-all method" is a good label.

Alas, as Jameson has pointed out, the IRVistas have muddied the waters 
by saying that the candidate that makes it to the last IRV round *is* a 
majority winner. (By extrapolation, every candidate that is not the 
Condorcet loser is a "majority winner", because given an arbitrary 
loser-elimination method, you could make any non-CL win, but never the 
Condorcet loser.)

>> The Black
>> method says: If there is a Condorcet winner, then
>> the Condorcet winner should win; if there is no
>> Condorcet winner, then the Borda winner should win.
>>
> 
> i hadn't heard of the Black method before, but just reading this shows 
> pretty superficially a problem.  above is one way to say something...

[snip]

> at the core, let's assume that we are already disciples of Condorcet, we 
> all agree that method X is best for domain X, he doesn't say squat about 
> why method Y is preferred in domain Y.  if we're nowhere near to a 
> conclusion that Borda is good for anything (he might have been a good 
> general, i dunno), then how do we conclude that it is preferable to 
> everything else when there is no CW?  sorry, i haven't even got past 
> this block.

I guess Maskin thinks Borda is the best on domain Y. Why, I don't know.

>> Maskin's argumentation doesn't work because
>> of the following reason: Whether an election
>> method is good or bad depends on which criteria
>> it satisfies. Most criteria say how the result
>> should change when the profile changes. Now it
>> can happen that the original profile and the
>> new profile are in different domains. This
>> means that, to satisfy some criterion, election
>> method X for domain X and election method Y for
>> domain Y must not be chosen independent from
>> each other.
>>
> 
> but, this is the fundamental argument of those who claim that it is 
> natural for an election to be spoiled, to be dependent upon irrelevant 
> alternatives.  isn't that what the fundamental issue is about for why 
> Condorcet (assuming a CW exists) is consistent with any simple-majority, 
> two-candidate election where every vote carries equal weight?  that's 
> what's fundamental about it, it is consistent to the concept that if 
> Candidate A is preferred to Candidate B, Candidate B is not a winner, 
> and being consistent with the result when the profile changes in that 
> manner is both tangible and operational (we can get a handle on it and 
> doing it differently, like using IRV instead, makes a difference).

The point is that the transition between the X- and Y-domain also 
matters, and just sticking methods together doesn't take the transition 
into account.

>> Example:
>>
>> The participation criterion says that adding
>> some ballots, that rank candidate A above
>> candidate B, must not change the winner from
>> candidate A to candidate B.
>>
> 
> does Black do this?

Nope. Condorcet is incompatible with Participation, even though 
Condorcet is compatible when there is a CW, and Borda is compatible on 
its own.

Consider it analogous to having a function that's made out of two 
horizontal lines, but the rules (impossibility theorems) forbid the two 
lines from having the same height. Then, although both the first 
(Condorcet) and second (Borda) line is flat (passes Participation), 
there's no way to combine lines (base methods) so that the function 
(composite method) is flat, as a whole, along its entire domain. There 
will always be a jump between the first and second domain.

> okay, since adding a positive number to the margin increases the size of 
> the margin,  and since, if there is no cycle (domain X), the Condorcet 
> winner is decided *solely* by the margins (even the signum function of 
> the margins), that sufficiently satisfies the participation criterion, no?

Indeed. A similar argument holds for Borda. If you show up and vote 
A>B>C.., that can only make B  win if B or a lower preference would 
otherwise have won (since you give more points to A than to B, more 
points to B than to C, and so on).

>> Election method Y satisfies the participation
>> criterion in domain Y, because the Borda method
>> satisfies the participation criterion in general.
>>
>> However, election method Z doesn't satisfy the
>> participation criterion since the Condorcet
>> criterion and the participation criterion are
>> incompatible.
> 
> whoa.  the Condorcet criterion is not assumed for Z unless domain Y is 
> empty.  and isn't participation and Condorcet with no cycle 
> compatible?   of course it's not guaranteed to satisfy participation for 
> all possible situations in domain Y.  that's not even the point is it?  

By sticking "Elect the Condorcet winner if there is one" to the front of 
Borda, one makes the resulting method pass the Condorcet criterion. In 
essence, one overrides part of the domain that Borda would otherwise 
cover so that the result (Z) always passes the Condorcet criterion. In 
the remaining Y-domain, the Condorcet criterion doesn't apply, because 
there's never a CW there.

> Maskin isn't saying that the participation criterion makes Borda the 
> *best* alternative method from the POV of disciples of Condorcet, is he?

I don't think Maskin has mentioned the participation criterion at all 
(though I haven't checked the links Markus gave). Markus picked the 
Participation criterion because it's something where X (electing the 
Condorcet winner when there is one) passes, Y (Borda when there is no 
CW) does as well, but the two domains stitched together (Z) does not.

There are other such criteria. Unless I'm mistaken, Condorcet passes LNH 
(Harm and Help) when there is a Condorcet winner (since margins and wv 
is the same), and Plurality also passes LNH, but "Condorcet winner if 
there is one, otherwise Plurality" does not pass LNH because both LNHarm 
and LNHelp are incompatible with Condorcet. There's no way to "line up 
the facets" between the two domains so that it always works.