Another “Difficulty” Experiment…

[Kobal2fr]

I don’t have a great understanding as to the mechanism involved, but can accept that any faction would react in exactly the same way given the same set of inputs. Except that, I think (cannot prove), that a lot of the decision made involve a random factor that will give some variety to the outputs of the decisions made.
You have already listed a few other variables that can alter the balance of power, I could give you a dozen more.

Well I've worked with the AI files, and the only things they take into account are not random at all but rather a combination of set factors (like target's total power, theirs and ours numbers ratio in region X, how much of target's military power is "tied down" in other wars and in garrisons etc...), the only thing random about the AI was included in version 1.2 and it's about alliances (namely : that allies make a random check at the beginning of each turn, if they "pass" then they won't consider their allies as possible targets no matter the other factors, if they "fail" then their allies will be factored in the global "where to defend, and who to invade next, how and when ?" thought-process like they were in 1.0 and 1.1. Essentialy this gives AI factions a 50% chance per turn to be irrationally trusting of their allies.).

What gives the illusion of randomness is that most of these factors are not readily apparent, or connected, to the player. The HRE could invade Metz because Hungary has declared war to the Turks for example. Or because Arhus has built Large Stone Walls. What little randomness there is lies in the result of autocalc'ed battles that can dramatically switch the previous balance of factors in one roll of the dice.

This is true, but it doesn’t have to be.
For example.
If the faction King purse was determined from a random number at the beginning of the campaign then each campaign would have a different set of superpowers. And /or the amount could be adjusted, proportional to the number of territories, during the game to reward expansion. But only for the AI.

That's also what I came up with when trying to figure a middle-ground between "one faction to rule them all and in darkness bind them, the rest is just background noise" and "petty kingdoms über alles" : factions taking turns being the Top Dog, behind the scenes, at the start of every campaign. Or, to put in another way, one mechanical and predictable over-faction chosen at random. Or more than one, you could give 10k King's Purse to 3 or 4 to make it a bit more random.
But that's the only thing I can think of, and it's not even that good a solution :/

Currently the philosophy of the AI seem to be ‘Lets make it a bit harder for the human to win’ rather than each AI faction trying to achieve the victory conditions for itself.
Trying to win should be the goal of all the players human or computer.

Oh I do believe they all try to beat each other senseless and "win", they only go out of their way to be nuisances to the player on VH. The problem is that since they are all equally "strong" and use the same overarching strategy, in effect in trying to win for themselves, they just all lock horns while the player nibbles at their butts.
Another issue is that the AI is very, very conservative in its defensive decisions, and very confident in it's offensive ones. It will try to keep in every province an army that can beat its strongest neighbour should they invade. Meaning that it'll tie down a LOT more forces than the average player, who'll think nothing of leaving a city crewed by 3 Town Militias because "the army can come back if need be, but it probably won't need to and it's more profitable to take another province right now". By comparison, the AI doesn't gamble, and doesn't take risks, ever.

Which is why the player will *always* be faster than them in seizing rebel regions in the early game, for example.

OTOH, it will think nothing of attacking with 1,000001:1 odds, because that's better than 1:1 and thus victory is possible, failing to realize that "possible" is not exactly the same as "likely" .

Play a passive game as Spain to see exactly what I mean : Portugal starts with a large-ish army in Pamplona, more than enough to seize Zaragoza in one go.
Yet if led by the AI, Portugal will attack it piecemeal with just as many siegers as there are defenders in Zara, because "Spain could attack us", and thus they keep most of their forces in reserve back in Pamplona to cover this eventuality. The end result is that it takes them 10s of turns to eventually capture Zaragoza because they're beaten back time and time again by the defenders, and because the Spanish player/AI builds-up troops to take Valencia/to defend itself against the Moors/to spend cash ; meaning Pamplona needs even MORE men tied down to keep it "safe".
Whereas a human player will have taken Zaragoza, built it up and mustered enough defense in both Pamplona AND Zaragoza to free the original army to attack somewhere else in the same time.

You are probably right but why do the bother to land a force of only 200 and leave it there for 10 turns, in the first place.

I have no idea, because I really can't figure out how naval invasions work at all, and they don't seem (to me, at least) to follow the standard triggers and logic. For example, absolutely nothing had changed in the AI files between 1.0 and 1.1 - yet 1.1 made naval invasions commonplace, somehow. And I've yet to understand that somehow

[Sentinel]

Interesting description of the game mechanics. It helps me to understand some of the thing I have seen.

What I have seen is
Taking an infantry unit out of a stack.
Spying on a settlement.
Assignation attempt.
Sabotage attempt.
Trying a heretic.
Gifting money
Merchant acquisition
Auto calculating a naval battle.
Auto calculating a siege.

All use random numbers to determine the results and there are many more. If you load a game and carry out all the above in exactly that order you will get a certain set of results. If you reload and do it again you will get exactly the same results. But if you reload and skip the first action all the subsequent actions will be seeded with a different set of random numbers, which can make a big difference to the results. You may find that this time the siege is lost rather than won. When these differences are then fed into other calculations that don’t use random numbers the outcome from these decision can also change.
The AI is poor at introducing variables, but it can rely on the human to do that for it.