[EM] attempt of a grand compromise

[Steve Eppley]

Hi,

[I'm sorry it took so many days before I finished 
this reply to Jobst's message.  I'm also sorry there
are a lot of other messages to which I've wanted 
to reply but haven't found time...]

Jobst H wrote:
> Steve E wrote:
>> Jobst's "immunity" is weaker than the "immunity from 
>> majority complaints" defined in my web pages (linked 
>> from <www.alumni.caltech.edu/~seppley>), which MAM 
>> satisfies but River does not.  
> 
> Of course beatpath or river or any other method designed 
> to elect a single winner don't satisfy that criterion
> simply because they are not designed to construct a
> social ordering but only to find a winner.  However, 
> if you find it important to have a social ordering 
> in addition to the winner, you can easily take a tree 
> of maximal beatpaths (in case of the beatpath method) 
> or the tree-like river diagram (in case of the river
> method) as this social ordering. Then your kind of
> immunity is easily fulfilled. If you require orderings 
> to be complete (which I guess), you can easily complete
> the tree-like ordering in whatever way to get a complete
> ordering which fulfils your criterion.

No, my "immunity from majority complaints" criterion (IMC) 
is more demanding than that.  It can't be satisfied merely
by extending a "single winner only" voting method so it 
also constructs a social ordering consistent with maximal 
strength beatpaths.  The social ordering must be more 
self-consistent than that sometimes is.  My web pages 
provide an example that shows BeatpathWinner is not 
immune even when extended to construct a social ordering. 
(See "Scenario A10" in appendix A of the "immunity 
from majority complaints" webpage linked to the 
www.alumni.caltech.edu/~seppley webpage.)

> Some time ago, I sent you this reply:
>   When I called the criterion you refer to by the name 
>   "immunity" some years ago, I surely didn't mean to
>   interfere with your terminology.  I came to know your
>   excellent web pages only this year... So, it is only 
>   a funny coincidence that we independently termed two
>   very similar criteria by almost the same name :-)
>
>   Anyway, I don't think there is too much confusion. 
>   First of all, I only termed it "immunity" in my
>   postings (and more lengthy "[weak or strong] immunity
>   from binary arguments" in the old paper of mine).
>   Secondly, as you say it can be considered a weaker 
>   form of your "immunity from majority complaints".
>   Thirdly, my immunity is a property of *candidates*
>   given some preference profile, whereas your immunity 
>   is a property of *rankings* which implies that 
>   the top candidate is immune in my sense.

But the reverse isn't true.  When the top candidate
is immune in Jobst's sense, it does not imply that
some social ordering could be constructed (with the 
winner atop the social ordering, of course) that 
is immune from majority complaints in my sense.
Jobst's immunity is significantly weaker, not 
just weaker in the mild technical sense of a
social ordering not being required.
 
>   If we want to make it absolutely clear in the future, 
>   we could stop calling a method itself immune and
>   instead say that a method "elects an immune candidate"
>   or "constructs an immune ranking", respectively.
>   What do you think?
> 
> but I did not receive an answer yet :-)

True, I didn't send an answer.  Sorry about that.  
I haven't found the time to think about the terminology.
It's clear now, though, that that discussion might have 
been premature since it appears from your writing that 
you have had a misconception about my IMC criterion, 
which is more demanding than you appear to have noticed.  
This is probably my fault since the descriptions of 
criteria in my home page are brief and don't always 
match the detailed descriptions linked on my other 
webpages.  My IMC is so demanding that it can be 
satisfied only by MAM (and variations of MAM that 
differ from MAM only in tiebreaking.  This is NOT 
just due to other voting methods being narrowly 
defined so they don't construct a social ordering.
Even if those methods are extended so they construct
a social ordering, they still fail IMC.

BeatpathWinner also fails "immunity from second-place 
complaints" (I2C):

   Immunity from Second-Place Complaints (I2C)
   -------------------------------------------
   Let w denote the alternative that wins.
   Let x denote the alternative that would win 
   if w were deleted from all ballots.  
   The number of voters who ranked x over w must not 
   exceed the number of voters who ranked w over x.

   (I2C and some other related criteria are defined 
   in my webpage that provides the detailed definition 
   of IMC.)
  
Note that I2C does not require the voting method to 
output a social ordering! (Instead, it implicitly 
refers to a natural iterative way to extend any 
single-winner voting method to produce a social 
ordering.)

I2C is in the same spirit as IMC since it could be 
difficult to rebut the complaining majority when I2C
is violated.  Complaints from a majority who prefer 
a "2nd place" alternative over the winner would be 
the most dangerous, I think.  Thus violations of I2C 
can be considered the most egregious way to fail IMC, 
which means satisfaction of I2C is at least as important 
as satisfaction of IMC.

I do not know if River satisfies I2C.  

> You continued:
>> But being a weaker property, perhaps it would be better 
>> to call his criterion "resistance to majority complaints" 
>> rather than "immunity from majority complaints."
>> That would be in the same spirit as the use of 
>> that word in the "truncation resistance" criterion.
> 
> I don't think it is in the same spirit since "truncation 
> resistance" refers to a strategy being applied at voting
> time where "immunity from binary arguments" refers to
> arguments being given after the election.

Yes, in that sense truncation resistance differs from 
these immunity properties.  But what I had in mind when 
I wrote "same spirit" was an analogy to a "truncation 
immunity" property stronger than truncation resistance.  
For instance, the truncation immunity satisfied by IRV:

   Truncation Immunity
   -------------------
   Suppose x does not win given votes V.
   Let V' denote a collection of votes that is 
   the same as V except some votes in V' are 
   truncated somewhere below where they rank x.
   Then x must also not win given V'.

All I meant was that, to me, the word resistance
connotes something weaker than does the word immunity,
so it might be a good name for a criterion weaker 
than my immunity criterion.

>> Since the strength of pairwise "defeats" in Jobst's 
>> proposed compromise method is determined by "approval" 
>> rather than by preference, it's not obvious to me 
>> that that method satisfies all the criteria Jobst
>> listed.  Does it really satisfy either of the two 
>> immunity criteria, for instance?
> 
> Immunity from binary arguments means the method elects 
> an immune candidate. A candidate x is immune when for
> each defeat y>x, there is a sequence of defeats x>...>y
> all having the same or larger *strength*.  So, beatpath,
> ranked pairs, and river all are immune no matter how
> strength is defined, but the meaning of "immune" changes 
> with the meaning of "strength" of course!

That's what I meant when I said I wasn't sure your 
(weaker) immunity criterion would be satisfied.  One 
must redefine it in order to claim satisfaction, and 
that's not quite the same as satisfying the original 
criterion.  The justification for one criterion won't 
necessarily hold for an amended criterion.

> So, to be accurate, "immunity from majority complaints" 
> only implies "immunity from binary arguments" when
> strength is defined as winning votes. I guess it will 
> be almost impossible to define a cardinal weighted
> pairwise derivative which fulfils "immunity from majority
> complaints" in the specific sense of your site. But when
> we follow the idea of using cardinal information to
> define defeat strength, then any appropriate definition
> of immunity will also take this into account. My 
> motivation for imposing immunity is this: When there 
> is a number of people prefering some candidate y so 
> much to the winner x that they support an argument 
> to replace x by y, then there should be arguments
> of the same kind leading back to x in order to be able 
> to show that the argument would be of no use to its
> supporters. Now, when we distinguish between weak and
> strong preferences, it seems natural to me to assume
> that only those with a strong preference y>x will support 
> the argument to replace x by y, hence the definition of
> immunity should then also refer to strong preferences
> only.

You may be right.  But I'm thinking that since people are 
so familiar with "majority rule" that the main danger is
from complaints based on a "majority rule" argument.  
Also, it may be that people with a mild preference for 
y>x will also tend to support the argument to replace 
x by y.  Not just people with a strong preference.
 
>> There's another "compromise" method that may be worth 
>> comparing to Jobst's, which I wrote about long ago
>> when I defined the "sincere defense" criterion
>> (which is stronger than minimal defense and 
>> Mike's SDSC.) 
> 
> How wonderful! Whenever I have some idea, I assume many 
> people must have had the same idea before :-) So what 
> is that method? I tried to find it in the archives 
> but didn't succeed...

I searched too and couldn't find any messages about the 
sincere defense criterion.  So a moment ago I posted a 
separate message with more info about sincere defense.

>> My first impression of Jobst's proposed compromise method 
>> is that it satisfies sincere defense but that it would
>> not be as robust as the methods I wrote about (that 
>> also satisfy minimal defense) in the case where a
>> significant number of voters know their sincere order 
>> of preference but do not know where to strategically
>> place the dividing line.  
> 
> When they don't know where to strategically place the line 
> but place it sincerely at the position of their strongest
> pairwise preference, then that is just fine since we are
> trying to keep voters from voting strategically, aren't
> we? But perhaps I just don't understand what you mean
> here...

I suggest you read the message I just posted about sincere 
defense before continuing to read this...

Note that if one of the voting methods I described in 
that separate message is used, for instance MAM suitably 
modified, the minimal defense strategy would still be 
effective to defeat "greater evil" candidates in the case
when a majority fails to coordinate their use of the 
dividing line to defeat such candidates.  For example, 
suppose Nader voters, who were part of the majority who 
preferred Gore over Bush, had voted "Nader / Gore > Bush," 
stubbornly refuse to rank Gore over the dividing line.  
It doesn't follow that they have only a weak preference 
for Gore over Bush, or that their preference for Gore 
over Bush is weaker than the preference for Bush over 
Gore of voters who vote Bush / Gore > Nader.  In other 
words, what if two voters have these preference 
intensities:

   Z >>>> Y >>> X

   X >> Y > Z

   If the top voter places the dividing line at the point 
   of strongest preference, between Z & Y, then she does 
   not provide as much help to Y over X as she could.
   Yet her preference for Y over X is stronger than
   the preference for X over Y of the bottom voter, 
   who would place the dividing line where it is 
   strategically optimal.

Attempting interpersonal comparisons of preference 
intensities given only expressions in ballots is too 
difficult for me!  And I'm not certain it's the right 
goal anyway.  Suppose selfish evil people tend to
have preferences that are more intense than those 
of socially responsible people?  I don't know that 
that's the case.  But I'm putting my faith in the 
greater number of people who have some preference,
including in that count the people who don't claim
to have a strong preference.  Here's my heuristic:

   The greater the number of people who say 
   x is better than y, the more likely it is 
   that x is better for society than y.

Won't implementing that heuristic with a method such as
MAM (or MAM with the dividing line, if the voters can
tolerate its extra complexity) suffice to align the 
incentives of the politicians with the interests of 
the people, weeding out corruption, as well as any 
voting method could?

(You can infer that I hope there is no need to 
"compromise" with advocates of Approval or with 
people who want to let voters express ratings.)

By satisfying both (strong) sincere defense and my 
second "nearly equivalent" wording of minimal defense, 
that majority is given two ways to defeat the "greater 
evil" candidates, and the two ways are not mutually 
incompatible.  Both strategies can be attempted.

The compromise method that Jobst proposed does not 
satisfy my second wording of minimal defense, because 
his method measures the strength of a majority for 
x over y solely by the voters' placements of the 
dividing line: #Vx/y.  Thus, in his method, a majority 
who rank x over y may be treated as smaller than #Vx>y, 
their actual size.  Since I put my faith in the size
of the larger majority rather than in the claimed 
intensities of a smaller majority (or minority), 
that seems unnecessarily risky.  But I think I'm 
open to persuasion, if I see a good argument for 
heeding the smaller majority's "cheap talk" claim 
of stronger preferences.

Perhaps I should summarize here an argument I've posted 
several times...  Society has had a lot of experience
with a voting method extremely similar to Approval: 
we vote "yes" or "no" on ballot propositions. (It's
not identical to Approval since the option to continue
the status quo is not explicitly on the ballot and
wins if no proposition explicitly on the ballot 
earns more yeses than nos.)  Conventional wisdom
is that when two or more conflicting propositions
are placed on the ballot, they can spoil each other's 
chances because some voters tend to vote no on 
compromise alternatives that they'd vote yes on 
if a more preferred alternative were not also 
on the ballot. (I cited an example from several 
years ago: The LA Times recommended voters vote yes 
to expand the LA City Council from 15 to 25 but no 
to expand the Council to 21, while simultaneously
opining that either 25 or 21 would be better than 15.
They explained their no on 21 by saying they didn't
want the compromise to defeat their favorite, 25.  
But on election day, both lost.  Would 21 have won 
if the Times had recommended yes on 21?  If so, 
25 was a spoiler.)  Whether or not the conventional 
wisdom will be true of Approval, if the elite actors 
(politicians, rich donors, etc.) believe it's true 
then their incentive would be, as it is now, to 
avoid placing too many candidates on the ballot.  
In other words, we'd still have two big parties 
each nominating one candidate per office, and thus 
voters would not be given the chance to sort candidates 
having similar & winnable platforms from least corrupt 
to most corrupt. (Similar logic holds for simple 
cardinal rating schemes.)

But Jobst's compromise method looks much better than 
Approval (neglecting its extra complexity) since it 
pays attention to the existence of majorities, differing 
from other good methods only in how it measures the 
strengths of the majorities (and the extra complexity 
of its dividing line).  So it satisfies a lot of 
important majoritarian criteria, such as top cycle, 
and maybe it would suffice to bust the incentive that
maintains the "only two viable candidates" system.

--Steve