Constant rate Voting (PDF)

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Constant-rate Voting
Let x(t), y(t) be the number of votes for the two candidates as a function
of time. Since voting at the beginning is a bit random, we’ll pick the point
at which it stabilizes to be t = 0. We’ll label x(0) = x0 and y(0) = y0 .
Assume that from t = 0 and onwards x generates votes at a constant rate
rx and y at a constant rate ry .
x(t) = x0 + rx · t
y(t) = y0 + ry · t
Let px (t) be the percentage of x’s votes as a function of time.
x0 + r x · t
px (t) =
x + y x0 + y0 + (rx + ry ) · t
If we take the limit as t → ∞, then we get a horizontal asymptote of
lim px (t) =


rx + ry

This means that given a constant voting rate for each side, the vote will
tend to an equilibrium as time goes on. We’ll label this equilibrium ex .
As an example: If Dr. Haran gets 100k votes an hour, and O’Brien gets
40k votes an hour than the resulting equilibrium would be 71.43% in Dr.
Haran’s favor.
Define Fx to be the voting force of x with respect to y i.e. Fx = .
ex =
rx + ry Fx + 1

or, in reverse:
Fx =

1 − ex

So, given a winning percentage of 81.83% for Dr. Haran, this means that
in a simplified model of the vote, Dr. Haran’s followers were voting at about
4.5 times the rate of O’Brien’s followers.


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