Projecting Electoral Vote Counts

By Anthony Chuka Konwea, PhD, PE

This is the dy/dx of votes counting.
For people wondering why the US election results have not been called, it is all about calculus and then finally some statistics.

Put it this way.
Plot the total number of votes counted in a chosen state during the US elections on the x-axis and any candidate’s percentage share of the total votes cast on the y-axis.

Call this graph the Vote-Share curve.
If you are an analyst working for a candidate like Joe Biden, you want the slope of your boss's vote-share curve, i.e. its dy/dx readout, to always be positive and never negative.

A positive dy/dx for Joe Biden means that he is getting a greater share of the votes counted at the expense of his opponent Donald Trump.

Even if your dy/dx is positive, you want it to be strongly positive.

There is a world of difference between dy/dx =+10 and dy/dx = +2.

The former which is a steep gradient is clearly better than the latter which is a milder gradient.

Especially important is the trend of the vote-share curve.

So both Biden and Trump analysts as well as neutrals in AP, CNN and Fox TV will closely be monitoring the moving averages of dy/dx slope readouts of both the Biden and Trump vote-share curves as the total votes counted value changes and the estimated total votes cast i.e. x-axis expands with time.

A 4-point moving average is the average of the last four vote-share slope read-outs. A 5-point moving average is the average of the last five vote-share slope read-outs.

If the moving average of your candidate’s vote-share curve’s slope readouts is decreasing that is a very worrisome sign for you and a hopeful sign for your opponents.

You could have a larger vote-share than your opponent and still have a negative dy/dx slope readout.

This is exactly the situation Joe Biden currently finds himself in Arizona where his vote-share slope dy/dx readout and the moving average trend of his vote-share curve slope readout is negative.

If that slope readout continues to be negative with each new batch of votes counted and the limit of the total number of votes cast, represented on the x-axis continues to grow, eventually Trump will overtake Biden in Arizona.

But realize that the number of votes cast plotted on the x-axis will not grow indefinitely. It has a finite limiting value even if that finite limit is currently unknown because some votes may eventually be disqualified.

Donald Trump’s supporters are praying that with each new vote-share curve point added as the number of votes counted in Arizona or elsewhere continues to expand, the slope readout (dy/dx) of Joe Biden’s vote share curve continues to be negative, in order for Trump to overtake Biden at some point in the near future.

But what happens when a candidate’s vote-share curve slope readout is zero? Essentially, we are asking what happens when the vote-share curve’s slope is as flat as a soccer field with dy/dx = 0?

A lot in fact. A steady flat vote share curve means the candidate is maintaining a steady share of the votes cast.

When the slope of the vote-share curve suddenly becomes flat, at an inflexion point, analysts become very interested in the ‘slope of the slope’ readout, also known as the d2y/dx2 value.

At an inflexion point, where you may recall the slope is zero (dy/dx = 0), if your candidate’s ‘slope of slope’ readout is positive, i.e. d2y/dx2 > 0, then your candidate is at a low spot.

He is about to dramatically surge ahead of his opponent with time, all things being equal.

On the flip side at an inflexion point, if the ‘slope of slope’ readout of your candidate’s percentage vote-share curve is negative, i.e. d2y/dx2 < 0, your candidate is ominously at a peak point of their percentage vote-share curve. They are about to be overtaken by their opponent with time, all things being equal.

So the inflexion peak point is the point where a positive slope (dy/dx) readout of the percentage vote-share curve turns into a negative slope. The inflexion sag-point is the point when a negative slope (dy/dx) readout of the vote share curve turns to a positive slope.

Joe Biden was at an inflexion peak point in Arizona a couple of days ago, when a dump of results from Maricopa County suddenly flipped the slope of his percentage vote-share curve from a positive value to a negative value. It has remained weakly negative in Arizona ever since.

Similarly, Biden was at an inflexion sag point in Georgia a couple of days ago when a dump of results from the heavily black populated Clayton County flipped the slope of his percentage vote-share curve from negative to positive. It has remained weakly positive ever since.

Arizona, Pennsylvania or indeed any other state cannot be called for Joe Biden if the number of outstanding votes yet to be counted is greater than the current gap-margin between the two candidates respective vote-share curves.

Once the total number of legitimate, admissible votes cast is known, or reaches its maximum limiting value and cannot be stretched out any further, the state elections in any of the four contested states Nevada, Georgia, Arizona or Pennsylvania will be called for Joe Biden if all of the following conditions are met.

One, if the Joe Biden percentage vote-share curve is higher than the Donald Trump’s vote-share curve in that state.

Two, if the statistically determined maximum possible slope read-out, dy/dx, of Trump's vote share curve multiplied by the number of votes yet to be counted cannot make Trump’s vote-share curve greater than Biden's vote-share curve.

Once that unique moment arrives for Joe Biden in Pennsylvania or in Nevada and Arizona combined, Joe Biden will be officially declared as the President-elect of the United States. Apparently, the margins are so tight in Georgia that a mandatory recount has been triggered.

From the balance of probabilities, I can proclaim with a very high degree of certainty that except there is a huge statistical event of gigantic proportions, Biden is poised to meet the two key conditionalities listed above in either Pennsylvania alone or Nevada and Arizona combined.

That being the case with certainty, I say congratulations to Joseph Biden for what seems almost inevitable – A Joe Biden Presidency.

To Donald Trump, I concede you put up an extraordinarily strong challenge with your legion of not so well concealed Trump voters.

Better luck next time, if any.
Anthony Chuka Konwea, Ph.D., P.E., M.ASCE, MNSE, FNIStructE, MNICE.

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