Much has been made about Joshua Smithley's prediction of a 390k vote-by-mail (VBM) firewall for Kamala - it originally seemed to be framed as the margin at which VP Harris' supporters can start to feel confident in PA, but seems to have since moved to being framed as the "break even" point - and has further since been suggested by Smithley that it will be "revised" up.

As far as I could tell, he did not indicate at all how he actually came up with that number, so it is hard to really say if it is justified or not. I decided to do some simple modeling to see if it is.

Methodology

We will take the "break even" interpretation: we seek to model various scenarios for total ballots requested, total ballots returned for each party, how the returns break for each party (i.e. some D's return as R votes, etc), how the rest of the population turns out, etc, and use the modeled results to determine the election day margin required by Mr. Trump to tie (not statistically, literally) VP Harris on election.

To do so, we will take priors over a variety of parameters. Because I have limited knowledge of these things, I used uniform-random priors with fairly wide ranges to capture a very diverse range of outcomes; however the code (linked at the bottom) is incredibly simple to edit, so feel free to update the priors.

1. The total voting age population of Pennsylvania ~ U(1e7, 1.1e7)
2. The total number of VBM ballots requested ~ U(1.8, 2.2)
3. The fraction of VBM ballots requested by D-registered citizens ~ U(0.6, 0.75)
4. The fraction of the remaining VBM ballots requested by R-registered citizens ~ U(0.8, 0.9)
5. [Remaining ballots are I-registered citizens]
6. The fraction of democrat-registered ballots returned (for any party) ~ U(0.6, 0.8)
7. The fraction of republican-registered ballots returned (for any party) ~ U(0.55-0.75)
8. The fraction of I ballots returned (for any party) ~ U(0.5, 0.7)
9. [note that I assumed a slightly higher D return rate]
10. The fraction of returned-democratic ballots which are votes for Harris ~ U(0.8, 0.9)
11. The fraction of remaining returned-democratic ballots which are votes for Trump ~ U(0.5, 0.9)
12. [remaining returned democratic ballots are votes for third-party]
13. The fraction of returned-republican ballots which are votes for Trump ~ U(0.8, 0.9)
14. The fraction of remaining returned-republican ballots which are votes for Harris ~ U(0.2, 0.9)
15. [Remaining returned republican ballots are votes for third-party]
16. The fraction of returned-independent ballots which are votes for Harris ~ U(0.2, 0.9)
17. The fraction of remaining returned-independent ballots which are votes for Trump ~ U(0.2, 0.9)
18. [Remaining returned independent ballots are votes for third-party]
19. [We now have enough information to deterministically compute the D VBM net total lead in votes]
20. Election day turnout as fraction of population that did not request a VBM ballot ~ U(0.6, 0.8)
21. The fraction of election day voters who vote third party ~ U(0.0, 0.05)
22. [This means we now know the exact number of voteres who are voting either D or R on election day, and can compute the election day margin Trump would need to hit to reach a perfect tie]

We perform the sampling above 40,000 times and determine the returned ballots net lead for the Dems, the actual vbm lead for the dems, and the election day margin trump would need to achieve to tie. One motivation for doing it this way is that we don't need to take any priors on how the election day ballots split (except for the small one on third party votes cast).

Results

With all that out of the way, let's take a look at what these priors yield:

https://imgur.com/rdjy9n3

The priors result naturally in Harris building a lead from about 360k to 530k via VBM (in terms of actual votes! note returned ballots!) and Trump needing around a 6%-9% victory in terms of the *election day* vote to break even with Kamala. In the scatter plot however, we can see an extremely clear correlation between the Democratic vbm actual-vote margin and the election day margin needed by Reps to break even. For every 100k actual votes that democrats add to their VBM lead, it forces republicans to increase their election day victory margin by 1.71%. A 390k lead corresponds to about a 6.6% margin on election day give or take a a percent or so.

However, keep in mind... the number that the firewall refers to is actually the returned ballots, not the actual vbm vote tallies... let's look at those plots:

https://imgur.com/V5N02Hn

In almost all scenarios, the dems naturally end up with 390k returned ballots vis-a-vis R returned ballots, suggesting my priors might be a bit aggressive, however, we see that the margin correlation, though still strong, is quite a bit more uncertain - every 100k votes added to the *returned* D-ballot lead only equates to forcing the R candidate to add an additional 1.28% to their election day margin of victory to tie - and 390k corresponds to forcing the R candidate to just a 5.1% lead on election day, but it could be as low as 3% or as high as 6.5% or so.

Interpretation

To me, this seems to be (a) already a bit aggressive in the leads it builds for Harris through VBM, and (b) pretty feasible margins for Trump to hit on election day. So it seems reasonable to think that if the Dems have a 390k lead in returned ballots, the race could be a tossup - but they really need to build up more than that to force a higher election day margin for Trump.

Code - try it yourself in a Jupyter notebook and tweak the priors!

Obviously I set a variety of priors here - you might have better numbers! Feel free to plug them in yourself and run the notebook to get new results.

https://colab.research.google.com/drive/1lNJp4L3EeNxQbZuH5ERYC1gyAV9i0D6i?usp=sharing

Edit

If anyone has twitter, please tweet this at Smithley, curious what he would use as inputs for the priors!