[EM] How to fix the flawed "Nash equilibrium" concept for voting-theory purposes

[Warren Smith]

John Nash's idea for trying to salvage multiplayer game theory,
was the so-called "Nash equilibrium."  A situation is an Nash eq. if
each player cannot improve her expected utility (payoff at end of
game) by altering her strategy (with all other player strategies
assumed to stay fixed).  Nash's theorem is a Nash eq. always exists.
They gave him the Nobel Prize in Economics for that.

Problem: The Nash equilibrium is a nearly worthless idea when applied
to voting & elections (viewed as an N-player game where there are N
voters).
For example, consider a 2-way election Gandhi vs Hitler in which everybody votes
for the (unanimously agreed to be) worst choice: Hitler.

Well, that is a "Nash equilibrium" because no single voter can change
the election result!

Indeed, essentially every possible vote pattern in every possible
large election, is a Nash equilibrium.  So Nash says almost nothing
about voting.  It is worthless.

But now here is  a very simple and highly effective fix, apparently
suggested here for the first time (and thus proving the stupidity of
all voting theorists including me).

Have each voter cast, not "one vote" but rather each voter casts "a
standard gaussian random variable" number of votes of each possible
type.  The voter does not get to control her vote, she only gets to
control the mean of the Gaussians.   So for example, in the
Gandhi-Hitler example, she can use the mean +1 or -1 (and fixed
variance) and that is all.

In a rank-order 3-way ballot (6=3! choices), we could make her employ
mean=1 for the vote she likes, and mean=0 for the 5 votes she does
not.

This "Smith fix of Nash" seems to work for any election method based
on a finite number of kinds of "vote totals."

In my Gandhi-Hitler scenario, 100% Hitler votes now is NOT a Nash eq,
and the only Nash eq is 100% Gandhi votes.    This idea is really a
pretty excellent fix.  It gets rid of
all the huge number of "stupid" Nash equilibria and seems to leave
only the "sensible" ones.

In the DH3 scenario
http://rangevoting.org/DH3.html

I am not sure what the Nash equilibrium (or equilibria?) are, but I am sure that
honest voting is not it, because each individual voter finds burial to
be an "improvement."   Presumably the Nash strategy in that scenario
will be a probability-mixture of honest and strategic votes.

-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html