[EM] [RangeVoting] Re: How to fix the flawed "Nash equilibrium" concept for voting purposes

[Raph Frank]

On Wed, Apr 14, 2010 at 12:57 PM, Bruce Gilson <brgster at gmail.com> wrote:
> I tend to think along the following lines. One vote almost never changes a
> result. However, if a sufficiently large number of voters change their
> behavior together to make a change, the result is what one wants to
> consider.

No, that is the point of the Nash equilibrium issue.  It might be in
the interests of a bloc of "players" to change strategy together, even
if one of them is worse off if he changes alone.

(Think driving on the left side of the road, if everyone switches to
the right side
together, then everyone is OK, but not if you switch on your own and everyone
stays on the other side.  They actually had to do this in various
countries, see
http://en.wikipedia.org/wiki/Dagen_H ).

Most methods will elect a candidate if a majority can coordinate and
so, act together to elect that candidate.

We need a process that distinguishes between, e.g. plurality and
approval voting.

With plurality, the top-2 have a massive advantage.  A condorcet
winner who is not one of the expected top-2, would have a very hard
time winning.

It is not possible to change the vote in small steps from a
non-condorcet winner to a condorcet winner.

> I think this may be the same as what Clay means by "moving the result closer."

The point is that you have a sequence of voter changes that cause the
final change, but each step must be an improvement.

With plurality, if you shift your vote from one of the top-2 to the
condorcet winner, this represents a decrease in the expected utility.

Maybe, we could just define the Nash equilibrium probabilistically.

It is a "probabilistic Nash equilibrium", if no change by any voter
would increase their expected utility of the result, subject to each
voter not being certain how the other voters will vote.

Actually, maybe that is the solution in general for Warren's proposal.
 Rather than defining a Gaussian, each voter's vote is just a
probability for each possible legal vote.  Effectively, it allows each
voter's vote to be a mixed strategy.

If A and B are the top-2, then I think the optimal vote under
plurality is (nearly) 100% probability to either A or B.  If everyone
does that, then either A or B will win, with near certainty.

Under approval, if A and B are the top-2, but you prefer C, then the
optimal strategy is (nearly) 100% AC or BC.  If C is the condorcet
winner, then C is guaranteed to have an expected utility score of
greater than 50% and only one of A and B will have that property.
This shifts the top-2 to either A&C or B&C.  C thus becomes one of the
top-2 and cannot be displaced (by the same standard logic as
previously).