[EM] Some myths about voting methods

[Raph Frank]

On Fri, Jun 5, 2009 at 9:08 PM, Warren Smith<warren.wds at gmail.com> wrote:
>> Nash equilibria...
>
> --"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.

This assumes that

Hitler: 99
Other: 1

is better in some sense than

Hitler: 100
Other: 0

Since for lots of methods, one vote rarely makes any difference, I
think it is reasonable to consider the 2nd result worse than the
first, rather than perfectly equal.

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.

Nash equilibrium assumes that you optimise your strategy assuming that
nobody else changes theirs.  An equilibrium is where nobody has an
incentive to change their strategy.  However, if they could
coordinate, then a large block changing their views could benefit the
whole block.

In your example, the state of system starts with 100% voting for
Hitler, but that isn't an equilibrium, as each voter has an incentive
to switch away from Hitler, no matter what the other voters do (worst
case, Hitler still wins, but there is a chance that it could shift
cause someone else to win).

>
> What we conclude from this example, is game theory and Nash
> equilibria, simply do not work well when applied to voting.
> Please do not use Nash equilibria or game theory in voting theory
> arguments, at least unless you have understood this first and hence
> are working VERY cautiously.
>

Why does plurality lead to the 2 party system but approval voting doesn't?

The condorcet winner should win both, but that doesn't happen in real
life due to coordination problems.