[EM] Nash Equilibrium

[Blake Cretney]

Here's the definition of Nash Equilibrium I got from
http://william-king.www.drexel.edu/top/eco/game/nash.html

*DEFINITION: Nash Equilibrium* If there is a set of strategies with the
property that no player can benefit by changing her strategy while the
other players keep their strategies unchanged, then that set of
strategies and the corresponding payoffs constitute the Nash Equilibrium.

There is an obvious problem when applying this to voting.  It is very
rare for an election result to be changed by a single vote.  But any
result that cannot be altered by a single "player" is a Nash Equilibrium.

So, we need a similar idea that is more useful in games like voting.
  Here's one suggestion.

*DEFINITION: Electoral Equilibrium* If there is a set of strategies with
the property that no set of players can benefit by changing their
strategies while the other players keep their strategies unchanged, then
that set of strategies and the corresponding payoffs constitute an
Electoral Equilibrium.

By the "set of players" benefiting, I mean each must benefit
individually, not that the set benefits in some aggregate sense.

Now, electoral equilibria have a clear connection to the Condorcet
winner.  Let's say you have what I will call an M class method.  An M
class method is a method where a majority of voters has some way to vote
in order to elect any given candidate.  In plurality/IRV/RP this means
ranking that candidate first.  In approval, this is done by voting only
for that candidate.  Borda and most other positional methods are not M
class.  Most other methods are, even the bad ones.

Anyway, let's say a majority of voters prefer X to Y, but Y still wins
(either because of strategy or because there is no sincere CW or because
this isn't a Condorcet criterion method).  If this is an M class method,
that majority can cause X to win, by definition.  And this is a better
result for every member of that majority.  So, this isn't an electoral
equilibrium, and Y doesn't win for any electoral equilibrium.

So, if we have full preference rankings, then the only equilibria will
be at sincere CWs.  But does every CW have an electoral equilibrium?
  Let's say that X is the CW and we are using an M class method.
  Everyone is voting in the way that a majority can use to make X win (in
other words, putting X first on their ballots).  Now, can Y be made to
win?  Only those voters who prefer Y to X can be part of the set
mentioned in the electoral equilibrium definition.  By the definition of
CW, this is less than half the voters.  So, since a majority of voters
are still voting for X first, X still wins, since we are using an M
class method.  Therefore, this (and every) sincere CW is the winner for
some electoral equilibrium.

Note that I don't claim that every strategy set resulting in a win for a
sincere CW is an electoral equilibrium.  This is false.

BTW, "electoral equilibrium" might be a bad name.  This definition isn't
really restricted to voting; it can be used for normal game theory
problems.  To me it seems like such a general concept, that I suspect it
already has a name.

---
Blake Cretney (http://condorcet.org)