[EM] Re: Approval Questions

[Jobst Heitzig]

Dear Forest!

You answered to me:
>> The point is that when all ways to fill in the ballot are admissible
>> strategies, there is never as group strategy equilibrium unless a
>> sincere CW exists. My question here was: Is there such an equilibrium
>> with Approval Voting when only ballots with sincere preferences are
>> allowed!
>>
> 
> Not in general, but yes when there is a CW.
> 
> The same cycle
> 
> x A>B>C
> y B>C>A
> z C>A>B
> 
> with max(x,y,z) < (x+y+z)/2
> 
> illustrates the lack of such an equilibrium.
> 
> If A is the winner, then the next time around, B and C supporters can
> collude to adjust their approval cutoffs to make C the winner (to their
> mutual advantage), etc. around the cycle.

Well, that depends! B and C voters not always have such an incentive,
since the following situation *is* an equilibrium of the desired kind:
  4 A>B>>C
  2 B>C>>A
  3 C>A>>B
Here A wins and those who prefer C to A can *not* make C the winner
since voting
  2 B>C>>A
  3 C>>A>B
will make B the winner instead!

So, it is *not* true that the existence of an equilibrium of the desired
kind implies that there is a CW, and that was why I asked whether there
might perhaps always be such equilibria. Unfortunately, they're not...

Anyway, I'm still interested in Approval Strategy, because I would like
to find out how much one can strategize in our Condorcet-Approval
hybrids by only moving the approval cutoff around.

> I gave an argument a few years ago for the case of a CW, first in the
> context of a one dimensional issue space, and later more generally.  I
> don't remember the date.  I'm sure that  Abrams, Fishburne, or Merrell
> already published better arguments, long ago.

Hm. I'm not sure I got the point here. To prove that when a CW exists
there is a group strategy equilibrium under Approval Voting seems easy:
The equilibrium is when all approve of the CW and all preferred
candidates, isn't it?

Yours, Jobst