[EM] Some myths about voting methods

[Warren Smith]

>> --"Nash equilibria" are an attempt to salvage game theory in N-player
>> games with N>2.
>> But it works badly for voting purposes.
>> My usual example is, suppose everybody realizes Adolf Hitler is the
>> worst candidate but still (idiotically) everybody votes for Hitler,
>> who wins.
>> OK, this election is a Nash equilibrium representing, in the sense of
>> Nash equilibria,
>> "best voting strategy" for all.
>
> Huh?
>
> If everyone else follows the "vote for Hitler" strategy, your optimal
> strategy isn't vote for Hitler.  (Well assuming that the secret ballot
> hasn't been compromised).
>
> You might as well vote for someone else.

--A game player situation is a "Nash equilibrium" if no player can get
more reward by changing their strategy.  In the all-vote-for-Hitler
situation, no player can alter the
election result by changing their vote.  Therefore this is a Nash equilibrium.

Is this "best strategy"?  Well, only in the sense that Nash-advocates
proclaim that
in a Nash equilibrium situation, each player's strategy is "optimal"!

The point is, as you yourself have just observed, this whole
conception often is silly and useless when applied to voting.

> In most real elections, you don't know the exact way the others are
> going to vote.  If all the other voters were using "vote for Hitler
> with a 99% probability and a random candidate otherwise", then your
> optimal vote is to vote for your favourite.

--true.   This actually sounds like a good way to perturb the Nash equilibrium
notion/definition to make it become more sensible than the official
definition.

So the new improved Raphfrk+Nash notion would be, assume each player will
play whatever strategy they select, or with probability epsilon they play
a random strategy.   Now we only have equilibrium if no player can
improve their expected reward, and this includes improvements by very
tiny amounts
proportional to epsilon^4 or whatever.

This would get rid of stupid equilibria like "all vote for worst."

Here's another nasty Nash equilibrium which still applies for the
Raphfrk-Nash version:
In a plurality election consider say, "all vote for Gore or Bush about 50-50"
but a higher reward would come if Nader won.   In this scenario  I guess
you cannot improve expected reward, and in fact will worsen it, by
switching your vote to Nader.


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