[EM] Approval Voting and Nash equilibria

[Alex Small]

First, there are a lot of newcomers on the list.  This discussion might
not make sense, so you might examine the following posts from the
archives:

http://groups.yahoo.com/group/election-methods-list/message/9907
http://groups.yahoo.com/group/election-methods-list/message/9335
http://groups.yahoo.com/group/election-methods-list/message/9653

Here's a question for those who have in the past discussed Nash
equilibrium and Approval Voting:

If, for a given electorate, there's a Nash equilibrium where all voters
with the same ordinal preferences vote the same way, we could say that the
Nash equilibrium involves pure strategies.  If, on the other hand, an
equilibrium involves some of the voters in a faction (group of people with
identical preferences, but not always identical absolute utilities)
breaking ranks and voting differently, then we could say that the
equilibrium involves a mixed strategy.  (Note that this is not exactly the
same as the definition of mixed strategies in conventional game theory,
but I'm going to recklessly borrow the term anyway since the conventional
usage bears similarities to the situation I'm discussing.)

Sure, the individuals within each faction followed pure strategies (each
person either did or did not approve a candidate).  However, it's
conventional on this list to aggregate into a single "player" all voters
with identical preferences.  Otherwise, any election with a margin greater
than 1 vote would be a Nash equilibrium, since no single "player" or voter
could change the outcome while all other voters keep their strategies the
same.



OK, with those preliminaries out of the way, here's my question:

Are there any electorates for which ALL Nash equilibria involve at least
one faction using a mixed strategy?  It has been proved on this list that
if there's a Condorcet winner there will be at least one Nash equilibrium
that elects the Condorcet winner, and all voters use pure strategies in
that equilibrium.  (see
http://groups.yahoo.com/group/election-methods-list/message/9907).  Even
if there's no Condorcet Winner there can still be situations where at
least some Nash equilibria involve all factions using pure strategies. 
e.g.

40 A>B>C----All 40 voters approve A and B
35 C>A>B----All 35 voters approve C and A
25 B>C>A----All 25 voters approve B only

(It's easy to verify that this is a Nash equilibrium.)

Can anybody come up with a situation where ALL Nash equilibria involve at
least one faction using a mixed strategy?  Or, can anybody prove that
every electorate using Approval Voting will have at least one Nash
equilibrium in which all factions use pure strategies?



Alex