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In this interview with Jacqueline Policastro, I read

President Trump:

No, not trying to suppress it. I want to have the vote. I want to actually have the vote. I want to have the real vote. I don't want an election to be stolen from either party. I don't want it to be stolen. But when they give the post office virtually no notice, and they say, "We're going to give millions of ballots out and you have to go and deliver them and do whatever you have to do." I mean, how can anything run like this? The post office didn't know about it. But just take a look at what's happened in New York, Carolyn Maloney. It's a mess. Take a look at Paterson, take a look at many other places. It's a mess. And it's going to be a very big embarrassment for our country.

My first thought was that one extra piece of mail for each voter doesn't sound like it would strain the system. I searched and found ONE DAY IN THE LIFE OF THE U.S. POSTAL SERVICE, which says

Each day the Postal Service processes and delivers 181.9 million pieces of First-Class Mail.

I am unsure of the number of registered voters, or how many of them would use mail-in. Can anyone definitively say if mail-in voting would strain or break the postal system?

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    I am concerned that this question is asking about a hypothetical event that hasn't happened yet. There is no empirical data. There are only predictions and opinions. What would be considered a "good" answer, in your mind? – Oddthinking Aug 6 at 10:44
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    Totally agree with @Oddthinking . Consider this quick Fermi Approximation: Every human in the US gets about 10-20 pieces of mail a week. If everyone eligible to vote does so by mail, that is about 1/4 piece of mail per human, all in the same week. Obviously and evidently it's not going to be a big deal. The only worthwhile answer, would, simply, tighten up the figures on this Fermi Approximation. And indeed it should be very easy to get very precise figures for those two figures. – Fattie Aug 6 at 12:28
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    Here's another excellent Fermi Approximation. At Xmas time, a vast amount of excess pieces are handled. (I would Fermi Estimate, on the order of 10x extra pieces per human above the normal, in a week.) Given that they're never lost at item during Xmas rushes, it does seem hard to argue there would be a problem with the 1/4 extra piece per human for voting. – Fattie Aug 6 at 12:34
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    It seems mail-in voting was a disaster for the New York primary last month: no way to tell when ballots were mailed, so no way to tell which ballots should be counted. On the other hand, the system in some other states has been working fine for years ... ballots must be received by election day, not merely postmarked. – GEdgar Aug 6 at 15:14
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    I agree that a conclusive answer would be very difficult, but a good summary of relevant publicly available evidence on either side of the claim would still be very helpful. The NY Times and CNN have stories that look helpful as a starting point. – Brian Z Aug 6 at 16:24
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TL;DR: USPS is currently facing major financial issues due to an unsustainable model and struggles to work as it is. Additional load from mail-in ballots would definitely add to the problem.


First thing: how many voters can we expect?

According to PennState University:

Around 138 million Americans voted in the 2016 presidential election. From Business Insider.

However, those 138 million Americans only make up 58.1% of our voting-eligible population (those American citizens over 18). From United States Elections Project.

In Pennsylvania, it is estimated that 61.3% of voting-eligible population voted, around 5,965,000 people. From United States Elections Project.

While these numbers seem high, our voter percentage is not higher than the 2008 (61.6%) or 2012 (58.6%) election turnout. From United States Elections Project.

It is difficult to ascertain how going to an (almost?) entirely mail-based system would change those numbers, but assuming no change compared to previous elections, it would be around 130-150m mails.

As noted in the OP, the USPS (United States Postal Services) deals with 181.9m mails per day on average, so it would definitely require spreading the load over several days, but it appears to be manageable at first glance.

However, USPS is currently facing major issues.

Trump mentions "New York, Carolyn Maloney". While he does not clearly mention what the exact problem he is referencing to, QNS reported in April about USPS have serious financial issues and asking for help from the stimulus bill:

“I want to commend the brave men and women of the Postal Service for all they are doing in the midst of this pandemic,” Maloney said. “The Postal Service is holding on for dear life, and unless Congress and the White House provide meaningful relief in the next stimulus bill, the Postal Service could cease to exist.”

More recently, USPS also released a statement indicating that they still can't sustain their current model:

We are currently unable to balance our costs with available funding sources to fulfill both our universal service mission and other legal obligations. Because of this, the Postal Service has experienced over a decade of financial losses, with no end in sight, and we face an impending liquidity crisis.

As well as indicating that they are facing massive penalties due to overtime in processing, transportation and mail delivery:

[...] The Postal Service Inspector General issued a report entitled “U.S. Postal Service’s Processing Network Optimization and Service Impacts.” In that report our Inspector General indicated that the Postal Service spent $1.1 billion in mail processing overtime and penalty overtime, $280 million in late and extra transportation, and $2.9 billion in delivery overtime and penalty overtime costs in FY 2019. Yet, even after incurring these additional costs, the Postal Service has not seen material improvement in our service performance scores.

While mails meant for election have a special tag meant to expedite their processing, many ballots still end up missing:

Between 2012 and 2018, 28.3 million mail-in ballots remain unaccounted for, according to data from the federal Election Assistance Commission. The missing ballots amount to nearly one in five of all absentee ballots and ballots mailed to voters residing in states that do elections exclusively by mail. [...] Although there is no evidence that the millions of missing ballots were used fraudulently, the Public Interest Legal Foundation, which compiled the public data provided from the Election Assistance Commission, says that the sheer volume of them raises serious doubts about election security.

In light of the problems USPS is currently facing, additional load from a transition to mail-in ballots would severely strain a system that is in fact already struggling.

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    You have quotes from various people suggesting that the USPS has a chronic problem, but not that it won't be able to handle an acute issue. You haven't discussed how the election period will be funded and staffed. You haven't quoted anyone saying the election will be the straw that breaks the camel's back. Perhaps an election contains such efficiencies that it will actually save them? That wouldn't be inconsistent with your sources. – Oddthinking Aug 6 at 10:50
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    @Oddthinking Reference added for election-specific mails. Turns out there is a tag for election ballots, but it doesn't appear to be enough based on what I found. – Asmael Aug 6 at 11:24
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    This question (admirably) points out that Certain People claim the USPS is "facing problems". While it is admirable to know that fact: (1) Since the beginning of the universe, the main purpose of the USPS has been to constantly, relentlessly complain that it is swamped, understaffed, and underfunded. (2) Note that the complaints mentioned are not specific, in any way, whatsoever - at all. (If the complaints said something with quantities such as "We can handle an extra 2.35m pieces in a given week, but beyond that we cannot" or – Fattie Aug 6 at 12:31
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    .. cont .. "After 3.767m extra pieces we start throwing away mail" those complaints would be of value. As they are, they are of absolutely no value other than (interestingly) noting that (completely vague) complaints have been made. The first part of this answer ("number of mail in votes") is perfect and outstanding. A second useful piece of information would be "how many pieces do they normally handle in a week". And a 3d useful datum would be "what are their estimates for how much extra crunch they can handle." – Fattie Aug 6 at 12:32
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    My bad: with the edits I missed the final outstanding paragraph which has great quantatative information regarding "ballots remain unaccounted for ...". That's key info from @Asmael , good one – Fattie Aug 6 at 12:35

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