[EM] Strong Sincere Defense criterion

[Jobst Heitzig]

Hi again!

>From your definition of the Strong Sincere Defense criterion:
> 
>    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.
>
>    For all x & y, if x is not elected and #Vxy > #Vyx and #Vx/y >
> #Vy/, then y must not be elected.

That criterion follows easily from (my version of) immunity when
strength is defined the way I did in the compromise proposal: when y is
immune and x defeats y, then #Vx/y is considered the strength of x>y,
hence there must be a beatpath y>z>...>x of at least the same strength,
 and it follows that #Vy/ >= #Vy/z >= #Vx/y.

Hence the compromise proposal meets Strong Sincere Defense and it would
still do if the River part of it was replaced by any other immune cycle
breaking technique such as Ranked Pairs or Beatpath.

> 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.  

Hmmm, that would certainly work but seems to be a bit artificial, don't
you think? Using #Vxy as defeat strength seems more direct to me. Also,
I don't quite see whether your proposal would still be monotonic...

> 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.

That seems to be surprisingly difficult -- I tried several hours but did
not succeed to prove it either...

> 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.

I find it dangerous to explicitly tell people to use things on ballots
strategically or to use things on ballots which cannot be explained in
very easy terms such as "do you like this or that better". We would have
to stop talking about sincere preferences or sincere winners then since
we would not be able to tell whether a ballot was sincere or not...

> 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.

I agree with you, but it seems not all here do.

Jobst