[EM] Fallacy in proof

[Alex Small]

I argued that for any Approval election where somebody other than the
Condorcet candidate wins a majority of the electorate can better its
position by voting only for the Condorcet candidate.  This is true, but
that doesn't mean the election of a non-Condorcet candidate (assuming the
Condorcet candidate exists) can't ever be a Nash equilibrium.

Problem:  I never defined who the players are.  If the players are
individual voters than any result with a margin greater than 1 vote is a
Nash equilibrium.  That isn't a very useful result, however.

If the players are blocs of voters with common preference orders then my
proof still could fail.  Suppose in a 3-way race we say there are
6 "players", and each has a different number of votes.  Each player has a
distinct preference order.

Saying that a majority can better its position by only voting for the CW
can require the action of more than one player, whereas the Nash
equilibrium occurs when no SINGLE player can unilaterally change the
outcome for his own betterment.

It could still turn out that in a 3-way race all Nash equilibria elect the
CW.  However, I haven't proven it.

Alex