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OK, I think this is the solution:

I will demonstrate how this works by a toy example:
x_1 = x_2
x_2 = 0.5 x_2 + 0.5 x_3
x_3 = x_4
x_4 = 0.25 x_1 + 0.25 x_2 + 0.25 x_3 + 0.25 x_4
where these equations describe how each voter x_i delegates their vote to the \
population (including to themselves).

Now set up the following matrix:
Let each row correspond to voter x_i, and each column correspond to their \
delegation weight, with one caveat.  That caveat is that, if the voter has \
delegated a portion of their vote to themselves, put that portion multiplied \
by their current vote (or 0 if they have not voted) in an extra column at the \
end of the matrix.  So the diagonal of the matrix should always be all zeros. \
 Finally, add one final row at the bottom that corresponds to all zeros, with \
a 1 in the last column.
So you get a 5x5 matrix that looks like this.

Now to do your iterative algorithm, we simply take this matrix and multiple \
it by the vector of inputted votes so far.
Let\[CloseCurlyQuote]s say x_1 has not voted, x_2 has voted -1, x_3 voted 1, \
x_4 voted 1. 
Then we make the vector (0, -1, 1, 1, 1) as our current state. (We have \
appended an extra 1 to make 5 total entries).

So what happens when we multiply the matrix by the current state? We get \
this. Notice that this is what you would get if you ran the iterative \
algorithm for one step.  We look at the first four entries: (-1, 0, 1, 1/4), \
which are all correct and take into account people who have already voted.  \
Also note that we replaced that last column in the matrix with the \
appropriate values based on the initial state votes.

So now instead of iteratively running the algorithm, there is a theoretical \
result that tells us that the convergent state will simply be the eigenvector \
of that 5x5 matrix that corresponds to eigenvalue = 1.  So we do this.  And \
our final result is that the votes are: (-0.5, -0.5, 0, 0).

The last row/column in the matrix and the last row in the state vector can be \
ignored when interpreting calculations \[LongDash] they are just for making \
dimensionality purposes.

Sound good?  I am not 100% sure yet if these matrices will have exactly one \
eigenvalue of 1; I think because we are replacing people who have not voted \
yet with a vote of \[OpenCurlyDoubleQuote]0\[CloseCurlyDoubleQuote] here, you \
will always get a unique solution.  If we chose to leave their vote as some \
\[OpenCurlyDoubleQuote]unknown\[CloseCurlyDoubleQuote], then this will not be \
solvable in general I think.

You are right that my previous method allowed voters to delegate to others \
such that even if they vote, it may not count.  This was stupid, but I \
included it to keep things simple.  The better way to implement what I had \
proposed would be to just adjust so that if you see someone has already \
voted, then you ignore all of their delegation stuff and replace their row in \
the matrix with [0 ... 0 1]. (This was for when I said all diagonal elements \
are 0 and you add extra column to matrix corresponding to \
\[OpenCurlyDoubleQuote]self-delegation\[CloseCurlyDoubleQuote].)

Also, for abstentions, 

Are you basically finding where a delegate has abstained, removing their % of \
vote, and then renormalizing?

So if x_1 = .5 x_2 + .5 x_3, and x_3=abstain, then x_1 = x_2?

And what happens if a vote cannot be determined by delegates?  Then they \
count as no vote, right? In my toy example, I replaced no vote with \
\[OpenCurlyQuote]0\[CloseCurlyQuote], but that is not what you want to do b/c \
voting \[OpenCurlyQuote]0\[CloseCurlyQuote] is not the same as not having \
voted yet, so this was the additional thing to tweak with my approach.\
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Cell[TextData[{
 "Need to show simple statistic on how many motions the user has voted on of \
the total proposed over a specified duration.  This will be an important tool \
to eventually ",
 StyleBox["incentivize",
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 " participation, e.g., if you don\[CloseCurlyQuote]t vote in 25% of motions \
(including via delegation) by the end of the semester, uncooperative fine; if \
you vote in 75%+ of motions, +1 hour bonus ",
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Cell["\<\
\[OpenCurlyDoubleQuote]RSS / News\[CloseCurlyDoubleQuote] feed with \
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checked (like Facebook notifications)?  (Issue #163)\
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Allow members to suggest Pros/Cons to be vetted by moderators.  (Issue #164)\
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Personalized Discussion dashboard showing # of unread comments, statistical \
quorum confidence level, vote tally.\
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Sidebar to easily navigate through various discussions while viewing a single \
item (like in Mail app).\
\>", "SubitemNumbered",
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Cell["\<\
Label discussions under categories (e.g. use of space, mural, etc.).  All \
users should be able to label.  Admin should be able to assign specific \
council labels to when that discussion was finalized.\
\>", "SubitemNumbered",
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Allow the ability to distinguish withdrawing a motion prematurely (with a \
note/comment explaining why) from simply closing it and causing it to fail.\
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Cell["\<\
Allow admins to setup an automated emailing service to users, which, at \
specified times, compiles and links to results/open motions (like a smart \
digest).\
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Directly replying to comments; notifying individuals via \
\[OpenCurlyDoubleQuote]@username\[CloseCurlyDoubleQuote].\
\>", "SubitemNumbered",
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Cell["\<\
Active motions/discussions summary page to display on TV @ main entrance \
(integrated with calendar of events + general announcements).\
\>", "SubitemNumbered",
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Cell["\<\
Carefully check numerical precision when computing binomial sums: (n choose \
k) explodes and 0.5^k goes to 0 exponentially fast.  Might need to use fancy \
bounds/approximations for larger binomial sums (e.g. Stirling, or see p. 51 \
of turquoise notebook).\
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*)