[EM] Group strategy (was Re: Approval Voting elections don't always have an equilibrium)

[Steve Eppley]

Much of the work on strategy-proofness and equilibria is only 
about *individuals* not having an incentive to change their own 
vote, given an assumption that no one else' vote will change.  
That neglects the incentive for a (coordinated) group to change 
their votes, as in Jan Kok's example below.  
--Steve
--------------
Jan Kok wrote:
> In Rob Brown's "Movie Night" introduction to election methods, Rob
> suggests that allowing people to watch the current vote results and
> change their votes as often as they like would lead to a stable
> situation where no one would feel a need to change their vote.  (I
> believe that situation is called a Nash equilibrium, is that right?)
> 
> Here is a situation where there apparently is no such equilibrium.
> 
> 6 A>B>C These voters initially approve A
> 3 B>A>C approve B and A
> 8 B>C>A approve B
> 10 C>A>B approve C
> 
> So the Approval votes initially are
> A=9 B=11 C=10
> Now the C>A>B voters approve A
> A=19 B=11 C=10
> The B>C>A voters approve C
> A=19 B=11 C=18
> The C>A>B voters un-approve A
> A=9 B=11 C=18
> The B>C>A voters un-approve C
> A=9 B=11 C=10
> ...and we are back where we started.
> 
> (B is winning at this point.  What if the B>A>C voters attempt to
> freeze the situation by un-approving A?
> 
> A=6 B=11 C=10
> The C>A>B voters approve A
> A=16 B=11 C=10
> The B>C>A voters approve C
> A=16 B=11 C=18
> Now the B>A>C voters re-approve A!
> A=19 B=11 C=18
> ...and we are back in the previous sequence.)
> 
> So, it seems an Approval election can have NO equilibrium, and
> obviously there will often be ONE equilibrium.  Question: can there be
> more than one equilibrium?
> 
> Cheers,
> - Jan