[EM] Approval Voting and Classical Mechanics

[Alex Small]

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