Does United do equipment swap based on low load?
 

I noticed that this week's SFO-PVG flight changed from 773 to 772 last minute twice in a row. The flight was mostly empty, so I think it makes sense to use a smaller/cheaper airplane.
However is this just a coincidence or they actually optimize based on load? What do they do with the larger 773? It's unlikely they can suddenly sell all the extra space on the same day.
FWIW, the original 773 flew SFO-HNL today, but 773 is the default airplane on that route anyway.

UA shows you the exact tail number for incoming flights, so you can track that to see how the plane is being re-allocated.
Last minute swaps are very common though it is hard to say whether load is a primary or secondary factor. Certainly if a swap is necessary then UA will consider any plane type that can accommodate the load. Of course, there is a ripple effect when changing the capacity on a flight so that adds a constraint that makes it impossible to optimize a single flight in isolation.

SFO PVG is likely a straight out and back which makes a swap easy if loads in both directions are low (and each cabin’s capacity permits). But UA would still need a spare 772 to be available for the entire duration.

I don't think United does close-in cancels or equipment swaps because they want to save a buck or two on low-load factor flights. Rather, operational factors tend to drive these decisions.

That said, where loads might come into play, is if United has an operational disruption elsewhere, and they're looking to minimize impact. Let's say SFO-LHR is running at 100% LF and the aircraft scheduled to fly it has a mechanical issue. Instead of canceling SFO-LHR, United may look to "steal" an aircraft from another route with a similar operational profile but a much lower load factor, and cancel that flight instead. United now has fewer passengers to rebook. It might not always happen this way, but it can be a relevant factor.

Many years ago, during a Flyertalk DO at the United HQ in Chicago, they demonstrated some software that helps in exactly these types of scenarios. They can play "what if" and find out when a designated flight cancels, how many passengers are disrupted, how long each would be delayed in their eventual arrival. They can even break it out by M+ elite level. ie 3 Global Services passengers -- 1 delayed 4 hours, 1 delayed 8 hours, etc.
They can then run the same on other flights, and compare levels of disruption caused.

It was very cool back then and I can only imagine how sophisticated it is now.

Besides passenger loads, cargo loads are a factor.

Due to low pax loads only? I don't think so.

If they need the scheduled plane somewhere else due to IRROPS? For sure.

Yeah, a given flight may be low load, but what about the next flight on that same frame? If they downgauge, how does it impact those downstream flights? In other words, the nearly-empty flight to an outstation may be overbooked on the next leg, so downgauging could cause more issues.

Somewhat related, I actually flew ORF-IAD earlier this year where everyone was sure the flight was going to be canceled, because the inbound IAD-ORF had delayed overnight, and by then everyone had either rebooked to the now-oversold flights the next day or just driven (3.5 hours). GAs and check-in agents said there didn't seem to be a reason for the flight to even operate and were sure it would cancel. I stuck with it, and the inbound came in light with maybe half load... and then the return had exactly one pax (me). I asked the GA why they hadn't moved some of the pax from the oversold flights to mine, and they said it had to do with the fact mine was still dated the previous day, and the computer wouldn't let them--it was technically just a 14 hour delayed flight, not an extra section run the next day.

but not last minute swaps. A swap for a/c size is done well before schedules are published.

With United? Is that true?

Yes though I should have specified that it is very common only within the same aircraft type (just different tail numbers).

Ok. That is a bit different from the scenario raised by the OP then.

Perhaps I was not clear., because of cargo commitments, downgauges may be constrained regardless of light passenger loads.

Yes. There are always disruptions from weather, delays, maintenance, etc. and airplanes (and gates) have to be swapped to minimize the impact of those disruptions. That's true for any airline. If the airline is very small, then there's often no other airplane to do a swap.

even so, that does not happen. excess cargo will simply go on the next flight/day.