[EM] Approval Voting and Classical Mechanics

[Forest Simmons]

Very interesting!

Simulations of these dynamical systems could easily include the candidate
popularity factor.

Would typical initial conditions lead to equilibium solutions? Would they
cycle? Or would they be chaotic?

Simulations can answer many such questions if the analytical approach
below gets too complicated.

Forest

On Thu, 28 Aug 2003, Alex Small wrote:

> Let's talk about a very hypothetical model.  Say that we use Approval
> Voting.  We have N candidates and an M-dimensional issue space.  Say also
> that voters decide which candidates to approve based solely on where the
> candidates stand on the issues, and not based on how the candidates are
> faring in polls.  So, if polls show that two candidates are running
> neck-and-neck, those voters who approved both of them (or neither of them)
> won't make any strategic adjustments.
>
> This may not be very realistic, but bear with me because we'll get to some
> interesting math.  This post is about candidate strategy, not voter
> strategy, and candidate strategy will lead us to interesting math.
>
> Candidate i gets a number of votes
>
> Vi(x11, x12,..x1M, x21, x22,...x2M...xNM)
>
> where xij denotes candidate i's stance on issue j.  Candidate i's goal is
> to maximize Vi.  So he'll adjust his stands on the issues.  His adjustment
> will move him through issue space in the direction
>
> grad(Vi)
>
> where grad is the gradient with respect to the variables {xi1...x1N).  The
> only requirement on candidate i's velocity in issue space is that it be in
> the direction of grad(Vi), so we are free to assert that
>
> (d/dt) (xi1, xi2,...xiN) = grad(Vi)
>
> where (xi1, xi2,...xiN) is the vector denoting i's position in issue
> space, and (d/dt) is a time derivative.
>
> If we take the second derivative of i's position, to get something akin to
> a force in classical mechanics, we get that
>
> (d^2/dt^2) (xi1,...,xiN) = (d/dt) grad(Vi) = grad((d/dt)Vi).
>
> Since Vi has no explicit time dependence, all of the time dependence comes
> in via the time dependence of the variables xij.  So we can rewrite the
> time derivative of Vi as
>
> (d/dt)Vi = (d/dt)(xi1,...,xiN) dot grad(Vi)
>          = grad(Vi) dot grad(Vi)
>          = |grad(Vi)|^2
>
> So the "force" on candidate i can be written
>
> (d^2/dt^2) (xi1,...,xiN) = grad(|grad(Vi)|^2)
>
> The important result is that force on candidate i is the gradient of a
> scalar function, and that scalar function depends only on the positions of
> i and the other candidates.  At this point we can bring in all sorts of
> advanced machinery from classical mechanics to analyze this.
>
> One crucial difference between the dynamics of candidates in issue space
> and the dynamics of massive objects is that although the "force" is the
> gradient of a "potential energy", the "potential energy" has the form of
> the most obvious candidate for the a "kinetic energy".  In classical
> mechanics the kinetic energy is the square of the velocity.  However, the
> velocity is the gradient of Vi, so we can't distinguish between potential
> and kinetic energy.
>
> It's also worth noting that Vi has maxima, hence there are stable points.
> Vi varies between zero and the number of voters (which we assume to be
> fixed).  Vi goes to zero as xij goes to +/- infinity, because as one of
> i's positions becomes more and more extreme we assume that fewer and fewer
> people will vote for him.
>
> Finally, the exact form of Vi is not always that crucial.  Many profound
> insights in classical mechanics depend only on general features of the
> interaction potentials rather than the nitty-gritty details.
>
> Anyway, it may be interesting to use this perspective to gain insight into
> candidate strategy in Approval Voting.  The neglect of voter behavior is
> obviously a non-trivial omission.  Still, it may be interesting.
>
>
>
> Alex
>
>
> ----
> Election-methods mailing list - see http://electorama.com/em for list info
>