Example!
The safavids are currently undergoing an Institution Quest. I rolled the years waaay back on 227 (There are, like, five dice rolls? My phone freaked so we're going to pretend that that was on purpose in an attempt to use them as examples that failed because
I rolled three fours).
Fluff: The Shah is going for a combination of increased literacy and massively funding aeronautics programs to improve
his rightful mastery of the sky the Persian understanding of the Airships so much of their economy is based around. As such this Institution Quest is tied to the Aeronautics and Rocketry field.
My first External Factors roll is a 5, Tradition. That doesn't particularly affect anything at the moment, as I've got 9 Magic, so I'm getting 1 point from my first year of the Quest. So now we go on to the Modifiers. I don't have spare MP and don't want an Econ hit, and both of my heroes are doing things at the moment, so I'm going to commit an artifact (The Ring of Solomon, good for infrastructure development everywhere) for an additional points. If I make a fluff post by Friday that'll be 3 points for me, and really that's more than enough for my purposes.
So, let's say my Econ goes down one next turn and I'm too busy to make more effortposts but otherwise everything stays the same. As such my next turn gets a +0 (Funding) because I rolled a two, for a total of 2 points since the Ring remains committed, but the turn after that I get
That's a total of 3+3+2, 8 points. That means I'm one over the threshold so if I wanted to change nothing else, I'd get +3 Institution Points and +3 years of Aeronautics Tech (Then affected by the relevant multiplier, which would probably be 1915-1930 in '88 so, well, it just adds three years).
There are a few ways this could go squiffy, someone could invade (-1 from troops on shore) or I could make a sacrifice, voluntarily slowing down my institutional development to appease my Elitist faction, losing a point (Going to 7) but gaining some loyalty with the Dari, or I could have rolled a 1 instead, netting a -1 penalty for my 18 million population. If two or more of those happened I could even drop a tier to 4-6, but for our purposes let's pretend things go as outlined above.
+3 Institution Points increase me to Scholarly 6/5, which brings me to the next tier (Systematic 0/5) and gives me a free Boost in '89 under my nifty new Systematic tech progression. If I'd gone to 7/5 I'd start at Systematic 1/5.
What if I don't want to focus on a tech and just increase my institution?
That's fine but you don't get a bonus for it.
..... What happens if you have no clear Ruling Faction
That's probably Pan-Bharatists for you right now, though for previous Mughal emperors that would be either the Nizams or the Rajputs. Basically 'who do you derive legitimacy and the ability to actually rule from'. For the Shah that's the Shahsevan, for El Presidente it's his military junta faction, for Naglfar it's now the Kalmarists, etc.
, and the most loyal faction is tied for most powerful with another faction that is neither most loyal nor most disloyal?
+1.