What Do You Believe?

Given our last election and some of the rather unusual events I thought I would create a mathematical model to explore this strange space. So the first thing to do is to express the thing in some form and see if it makes sense. An exponential model first comes to mind such as:

I like this one because it represents a lot of natural functions and has some good attributes that work for what I'm doing. First, it will be asymptotic to some final value so close to A any further extensions are meaningless. Second, I can express x as the number of precincts counted for a vote tally. I created a 3 term form of this since most states have 3 different forms of voting. In person, by mail, and provisional. It doesn't affect the final tally but the shape of the count curves. Like so:

If I assume a final vote tally for each candidate I get count curves like the ones shown next. The final numbers are the percentage of the vote and I have arbitrarily chosen A's, B's, and C's such that both receive just less than 50% of the vote but there is a clear winner. Red Candidate wins by a nose with 48.2% of the vote to 46.9% of the vote for Candidate Blue. The counting rates are identical for each candidate since the votes are assumed completely shuffled and reported simultaneously by precinct. All in all it seems to represent the real world pretty well.

In one state however we got a late addition to the vote counting which created an interesting twist and has led me here to this post. Specifically the "jump." One state found a thumb drive with  votes that were not tallied for only one of the candidates, and the subsequent count curve had a jump in it like so:

Now Candidate Blue wins. Candidate Red's count tally never varied so if a model holds for the one it should hold for the other. This implies that one of two scenarios is true. The first is that the original count curve is accurate and the votes were fraudulently added. The second is that the vote tallies for Candidate Blue from a large number of precincts were withheld and the actual count curve results in meeting the final tally as is shown. The two green lines represent the two scenarios.

So, what do you believe?