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

Steve Eppley seppley at alumni.caltech.edu
Sat Dec 24 08:01:44 PST 2005

```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

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