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Constant-rate Voting

/u/momoro123

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 .

So

x(t) = x0 + rx · t

and

y(t) = y0 + ry · t

Let px (t) be the percentage of x’s votes as a function of time.

Then

x0 + r x · t

x

=

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) =

t→∞

rx

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.

rx

Define Fx to be the voting force of x with respect to y i.e. Fx = .

ry

Then

Fx

rx

=

ex =

rx + ry Fx + 1

1

or, in reverse:

Fx =

ex

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.

2

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