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 ---- Election-methods mailing list - see http://electorama.com/em for list info