[EM] Strong Sincere Defense criterion (was: Re: attempt of a grand compromise)

[Steve Eppley]

Hi,

In the discussion of Jobst's proposed voting method
that's a compromise between Condorcet and Approval,
I mentioned the Sincere Defense criterion and alluded 
to voting methods that satisfy it...

I wrote:
>> 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.) 

Jobst replied: 
> 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.  Yet I don't think I'm suffering 
from false memory syndrome; my recollection of writing 
about it is fairly strong.  Very mysterious!  Anyway, 
I'm posting this message to ensure that at least some 
of the info about sincere defense is not lost.

I had several families of methods that satisfy Sincere 
Defense.  Some satisfy stronger criteria, such as 
this one:

   Strong Sincere Defense criterion
   --------------------------------
   1. Each voter must be allowed to vote any ordering 
      of the alternatives and optionally include a 
      "dividing line" somewhere in her ordering.  

   For all x & y, let Vxy denote the voters who 
      vote x over y.
   For all x & y, let Vx/y denote the subset of Vxy 
      who place the dividing line between x and y.
   For all x, let Vx/ denote the voters who 
      vote x over the dividing line.

   2. For all x & y, if x is not elected 
      and #Vxy > #Vyx and #Vx/y > #Vy/, 
      then y must not be elected.

I expressed condition 2 in an unusual way, conditioning
y's defeat on x's defeat, so that the wording of the 
criterion also applies to multiwinner methods.

To check that Strong Sincere Defense can be satisfied, 
we must show that the binary preference relation P 
defined by "xPy if and only if [#Vxy > #Vyx and 
#Vx/y > #Vy/]" is always acyclic:

   Proof that P is acyclic: 
   Assume there are alternatives x1,x2,...,xk such 
   that x1Px2Px3...xk-1Pxk.  We must show ~xkPx1.  
   By the definition of P, the following inequalities 
   must hold:
      (1) #Vx1/x2 > #Vx2/
          #Vx2/x3 > #Vx3/
            ...
          #Vxk-1/xk > #Vxk/
   Clearly the following always holds:
      (2) For all y & z, Vy/z is a subset of Vy/.
   By 1 and 2 the following sequence of inequalities 
   must hold:
      (3) #Vx1/ >= #Vx1/x2 > #Vx2/ >= #Vx2/x3 > #Vx3/
           ... >= #Vxk-1/xk > #Vxk/ >= #Vxk/x1
   Thus #Vx1/ > #Vxk/x1, which implies ~xkPx1.
   It follows that P is always acyclic.

To satisfy this criterion, methods like MAM, River
and Beatpathwinner and methods that resolve cycles 
by discarding small majorities can be simply modified 
so they treat majorities that satisfy the inequalities 
in condition 2, if any, as if they're larger than all 
majorities that don't satisfy the inequalities.  MAM, 
for instance, would begin by affirming all the majorities 
that satisfy condition 2, if any, and then would consider
the remaining majorities one at a time as usual.

The clause "#Vxy > #Vyx" in condition 2 may be unnecessary, 
but I like it.  I don't think we should require y not 
finish over x in scenarios where more voters rank y over x 
than vice versa.  A majority might rank y over x but some 
of them might put both below or above the dividing line, 
for instance "z/y>x."  I don't want to attempt to divine
whose preferences are stronger than whose; from social 
choice theory we know how nearly impossible it is to do 
interpersonal comparisons of preference intensities.

There's a similar criterion, in which the expression 
"#Vxy > #Vyx and #Vx/y > #Vy/" in condition 2 is 
replaced by "#Vx/y > #Vyx".  But I don't remember
if I proved this is always acyclic and can therefore
be satisfied.

The dividing line should not be thought of as an 
approval/disapproval threshold, I believe.  It should 
be promoted as a strategic device whose placement 
in one's ballot says nothing about one's approval 
or disapproval or strength of preference.  Then it 
can be placed optimally without raising concern in 
the voter that she is misrepresenting her preferences.

One final comment... I haven't promoted sincere 
defense or methods that satisfy it because of
their added complexity.  I think it'll be tough
enough to persuade voters to adopt a method that 
asks only for orders of preference, and I also 
think orders of preference (tallied well) will 
suffice to defeat greater evils and make politicians 
much more accountable.  That would satisfy me.

--Steve