[EM] Proposal

[wclark at xoom.org]

Forest Simmons wrote:

> To convert this to a fully deterministic but pseudo random method,
> eliminate steps (4) and (5) and simply take the approval winner from the
> last pass through the For loop.

...but why 100 iterations?  Why not 50, or 400?  Using 100 makes
everything work out for nice round percentages, but other than that it
seems arbitrary to me.

What type of behavior does this system actually produce?  I'd guess that
we'd see convergence to a small (singleton?) set very quickly, so that
additional iterations past a certain point only serve to bias the election
in favor of the members of this set.

In other words, each iteration results in a winner, and this list of
winners will eventually converge to a single winner repeated over and over
(or perhaps cycle between multiple winners).

If that convergence is what's desired, why even bother keeping track of
the initial rounds at all?  Why not wait until the system has settled down
before we start adding marbles to the bag (breaking the relationship
between marbles and weights in the process).

> In fact, the method was designed to estimate the approval equilibrium
> candidate winning probabilities: the candidate weights after the last pass
> estimate those probabilities as percentages, i.e. the number of each
> candidate's marbles divided by one hundred, estimates that candidate's
> equilibrium approval winning probability.

I assume it's related to your work described in this post:

http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-December/011372.html

This really sounds VERY much like the types of algorithms used at search
engines, particularly the type of ranking systems used by Google and
Teoma.  They use various algorithmic tricks to vasty simplify their
calculations, so perhaps you might be able to adapt something to your
purposes.  A search for "Google PageRank" will turn up plenty of details,
much of it as mathematical as you'd like to get, noise-floors and
eigenvectors and all.  The Wikipedia, as always, is a good place to start:

http://en.wikipedia.org/wiki/PageRank

> Observe that in both versions a candidate has to win at least one
> approval round before having any chance of being the ultimate winner.

I like your earlier goal of making sure any candidate with any approval at
all would have a positive chance of winning.  Why did you reject it?

> Note that the deterministic version fails participation in one sense:
> adding ballots favorable to the winner could change the value of J for
> which this winner wins to J=99, and then some other candidate wins
> on the 100th pass.

Under what circumstances do you see this happening?  I'm still not clear
on the behavior of this system.  Do you have any simulation results you
could share?  Or even just some intuitions you might have?

I'm confused as to why you're talking about "the value of J for which this
winner wins" rather than who wins in round J.  Your way makes it sound as
if the number of wins for each candidate is fixed, and I don't see why
that would be the case.

> However, the total weight of the winner would not be decreased by the
> added favorable ballots, so the his winning probability (in the
> non-deterministic version) would not decrease, and hence the prior
> probability of winning in the deterministic case would not decrease
> either.

I don't follow your reasoning here.  If the winner originally won in both
the 99th pass as well as the 100th pass, but because of some ballot
changes in her favor she no longer wins in the 100th pass... doesn't she
get one less marble?

Or are you implying that additional favorable ballots are only capable of
shifting around the order in which candidates win each iteration round,
and not the actual number of rounds won by particular candidates?

> So the spirit of Participation is met: you don't decrease the prior
> probability of your candidate's winning by participating.

Everything else aside, we're still left with the issue of how to interpret
probabilities for unique events.  A strict Frequentist might accuse you of
using too Bayesian an interpretation, in the deterministic (or perhaps
even non-deterministic) case.

If my candidate loses after I vote -- but provably would have won had I
not voted -- I may or may not be willing to buy the Bayesian line about
the relevence of prediction and prior knowledge and such.

More on the Bayesian-Frequentist debate:

http://en.wikipedia.org/wiki/Bayesian

On the interpretation of probability in general:

http://plato.stanford.edu/entries/probability-interpret/

-wclark

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