[EM] attempt of a grand compromise

[Jobst Heitzig]

Dear Steve!

you wrote:
> I'm not sure which "strategy criteria" Jobst credited me 
> with, but I suspect they're the ones for which I credit
> Mike Ossipoff: 
> 
>    minimal defense, which is nearly the same as Mike's
>    "strong defensive strategy criterion" (SDSC).
> 
>    non-drastic defense, which is nearly the same as 
>    Mike's "weak defensive strategy criterion" (WDSC).
> 
>    truncation resistance, which is nearly the same as
>    Mike's same-named "truncation resistance criterion".

I admit I did not check again but built my claim on Markus' note that
(my version of) immunity implies the strategy criteria listed on your
site. So I erroneously credited you with them (Sorry Mike!). So, let's see:

The proposal fulfils MINIMAL DEFENSE and NON-DRASTIC DEFENSE: Assume a
majority prefers y to x. When each of them ranks all candidates
sincerely and place their approval cutoff somewhere between x and y,
then y has a defeat of strength >n/2 over x but x has no defeat of
strength >n/2 over any other candidate. Hence x is not immune and cannot
win. QED

As to TRUNCATION RESISTANCE, I fear the proposal might FAIL that because
of the specific definition of defeat strength. With defeat strength =
winning votes, this criterion also follows from immunity, but with
defeat strength defined otherwise, it will probably fail. It seems this
will be a general problem with cardinal weighted pairwise -like methods,
do you agree? If so, we should figure this out more clearly since it may
undermine our hopes that cardinal weighted pairwise will have better
strategy properties...

> 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.

> Unfortunately I haven't 
> yet found time to discuss with him what might be the 
> best set of terms to name and describe these properties.  

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 ment 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.
> 
> 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 :-)

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.

> 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!

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 replacing 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.

> 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...

> 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...

> However, I think it would be simple
> for Jobst to modify his new method to make it as 
> robust as the methods I wrote about that satisfy 
> sincere defense.  That is, the size of the majority 
> that ranks x over y does not need to be ignored.

Hm... I don't understand what you mean by "not ignored". Do you propose
to combine the weighted defeat strength with the unweighted one in some
way, perhaps by counting weak preferences once and strong preferences
twice?

Yours, Jobst