[EM] Re: Equilibrium in Approval Voting

[Alex Small]

I can prove by contradiction that all Nash equilibria result in the
election of a Condorcet candidate when using Approval Voting:

Suppose that, for a finite set of n candidates {C1, C2,..Cn} C1 is the
Condorcet winner.  The allowable outcomes are the election of any of those
candidates, or a tie resulting in a messy situation that no voter wants
(i.e. assume that all voters prefer the election of ANYBODY to the election
of nobody, or a messy Supreme Court case).

If a candidate other than C1 wins, a majority of the electorate can improve
its lot by changing strategies so that they vote only for C1.  C1 will then
win.  If no candidate wins any voter could remove the tie and improve their
outcome by not voting for one of those tied.

Since any set of strategies yielding an outcome other than the election of
C1 is not a Nash equilibrium, and since a Nash equilibrium must exist, any
Nash equilibrium must involve the election of the Condorcet winner.

I have not proven that the Nash equilibrium involves sincere strategies, or
that all outcomes electing the Condorcet candidate are Nash equilibria.
For instance, consider the following:

36 A>B>C
14 B>C>A
34 C>A>B
16 B>A>C

A is the Condorcet winner, and also the plurality winner.  If all voters
only pick their first choice, A wins.  However, since some people in the B
camp consider C their second choice, if 3 people from the B>C>A category
vote for B and C, C now wins.  So not every election of the Condorcet
candidate is a Nash equilibrium, but every Nash equilibrium elects the
Condorcet candidate.

Is every Nash equilibrium sincere?  Have to think about that.  Seems like
every Nash equilibrium should be sincere under approval voting, but I'll
have to get to that later.

Alex