Does army size affect speed of siege?
Is it better to split my units to siege multiple counties at once, or does a stack wear down the defenders quicker?
Solution 1:
From the CKII wiki:
For a siege:
If the attackers do have enough troops, every 12 days a tick will count and progress will be made on the siege. The amount of progress that is made per tick depends on the fortification level of the holding and how severely the attackers outnumber the defenders.
This means the number of troops do matter, and it's more than just a case of if troops > x: get bonus . The exact numbers aren't on the wiki, however, and are probably only known to Paradox.
When assaulting, it is recommended to have 10x the number of troops involved in the attack as the defenders have in their garrison. This will minimize the attacker's casualties, as the defenders are overrun and unable to maintain any level of adequate defense.
This is also pretty obvious from trial and error. If you siege 400 troops with 2000 troops, you will lose about half your army while taking the province. If you siege the same 400 troops with 10k troops, you will lose about 100 troops, from my experience. (Obviously this varies a bit depending on fort level too)
Solution 2:
A large stack will wear down the defenders quicker during a siege, but not much quicker. One of the multipliers to siege speed is the "Outnumbered" bonus, and it gets large very, very slowly. I took a few notes in my last invasion, and this is the practical upshot: your Outnumbered bonus increases linearly with the multiplier of how badly you outnumber your enemy. Each extra multiple you have of their garrison will give you just under 0.1 more multiplier. (It's 0.09 for most of my data, but a couple of data points showed 0.10 or 0.07. Maybe there's a martial-skill component?)
Some sample data points:
Outnumbered by Outnumbered bonus multiplier
3.37 to 1 1.239
4.56 to 1 1.32
6.14 to 1 1.462
10.2 to 1 1.824
129 to 1 (!) 9.91
(!) The enemy had a fort level 6.5, which was offsetting almost all of this bonus. I was only costing him around 25% of his morale every 12 days. I assaulted him for minimal losses.
If you bring 5000 troops to the castle with a garrison of 1000, you'll get a multiplier of just under 1.4. Double your troops? Your multiplier becomes a bit under 1.9. You could instead siege 9 castles that size (with a small buffer for each army), in about double the time it would take for your 10,000-man army to take one castle.
The practical upshot: If you're trying to take an enemy's castles through siege (not assault), you're better off splitting your army into the smallest possible units that can take their holdings, with some buffer to handle the various siege events that can kill your troops. This will take a long time before you get your first holding, but over the course of a campaign, will give you all of your enemy's holdings faster. Fortunately, this also means that your sieging armies will fit inside the county's supply limit, too!
Unfortunately, you're still going to need to worry about his other armies marching around and attacking your holdings or your invading armies. And you might want to have somewhat-larger armies just so you can siege the cloister, city, and castle without having to fiddle with your army size.
Solution 3:
The answer is, as almost always, it depends. I'll try to do some research on numbers later, but I'm not sure where to look.
It boils down to this: there are bonuses to the amount of morale that the garrisoned troops lose each period (I think this varies, but 12 days was one I recently saw) during the siege. One of those is called "Outnumbered", and it increases with the amount of troops sieging the county. I think the benefit has diminishing returns, so there's going to be an ideal siege size for a given amount of garrisoned troops.
There are other factors to consider though. For instance, what's the supply limit in the county under siege? Is the opposing army a big threat? Do you plan on assaulting the holding to end the siege faster?