On Wed, 23 Mar 2005, Jobst Heitzig wrote:
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!
In Rob's algorithm, once A is in the lead, the ABC voters stop approving B. Then eventually C surpasses B in approval, so the CAB voters stop approving A, so C eventually surpasses A in approval.
When the approval order is CAB, then the A supporters start approving B.
When the approval order reaches CBA, then the B supporters stop approving C.
Etc.
So your kind of equilibrium does not imply equilibrium in Rob's algorithm.
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?
Except those who rank the CW in last place (e.g. by truncating).
It's not easy to show that Rob's algorithm (the batch version) will always converge to the CW.
The ballot by ballot version doesn't always converge to the CW, though for most ballot orders it will.
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