[EM] Approval Voting elections don't always have an equilibrium

[Jan Kok]

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