This is a topic near and dear to my heart! I’ve written up two different approaches on my blog and have one other that I’m currently using (I’ll summarize all three below). You might also check out Blades in the Dark for a fairly simple system: each faction is on a Tier (scope of influence), and within its Tier has a Hold (how effective at acting within that tier). Factions try to seize turf, which conveys various benefits, and each faction has a Status with other factions (how friendly they are). If that sounds appealing or the right amount of complexity, check it out.
tldr for 3 methods below
Power Dice: Each faction has a different die, bigger dice are more powerful. Roll and interpret high as good and low as bad. This is what I use right now.
Factions as NPCs with Reaction Rolls: Based off of Courtney Campbell’s system in On the NPC. Each faction has a bond with other factions, which represents how likely they are to get along, and then you roll reaction rolls to see what actions they take against each other, which affects bond positively or negatively. Great if reaction rolls are natural to you, maybe fiddly if they aren’t seared into your mind.
Faction Stats and Reputation: Each faction has a stat for Force, Cunning, and Wealth, and the PCs have reputation with the faction. When they want something from the faction, the stat defines what kind of things the faction can do, and the reputation defines how likely they are to provide it. Maybe a touch fiddly, especially for “off-screen” factions.
Now, here are the approaches I’ve tried:
This is my current system and the most lightweight of these three. Each faction has a die size assigned to it which represents how powerful it is - d4s are the equivalent of street gangs or small businesses, d20s are the mighty rulers of the setting. When two factions come into conflict, roll a die for each, and whoever gets higher makes progress/beats the other, a bigger difference in score implies a more decisive win. Combine with clear ideas of what each faction wants/does not want and some short and long term projects. If you want to be more granular, combine with countdown clocks like @ItsPizzaTom mentioned, and each “win” advances the clock, maybe two spots for a really big difference in rolls.
Want something even faster? Grab a die for every faction, roll em all at once, and interpret higher as better, to get a feel for how everybody is doing.
I use a relationship map showing each faction inside an icon that shows their die size and with color coded lines connecting allies and enemies. If I want to see if factions are growing or failing, I treat them like exploding and/or usage dice - a maxed out roll might give them a chance to go up a die size, while a 1 might mean they go down a die size.
I’ve toyed around with having a different die for a handful of resources/stats, but that seems to get away from the freeform oracular value of this system.
Faction as NPC with Reaction Rolls and Bonds
I based this one on the system Courtney Campbell lays out in his book On the NPC and I detailed the basic idea in this blog post, and further built on it with inter-faction combat here. I never got to use it all that much in game, so it likely has some warts. I think it would be most helpful in a situation where all of the factions want roughly the same thing (say, territory in a megadungeon or a share of crime in a city), and less useful with lots of different factions that all want different things.
The basic summary is that each faction has a “bond” with other factions, a morale or strength score, and for each faction “turn” rolls a reaction roll. You use the bond to modify the reaction roll, and that determines what action each faction takes towards the others. When factions come into conflict, roll 1d20 and add their morale/strength, and the winner harms the loser.
If you really like reaction rolls and/or normal distributions, and if you have a fairly limited number of factions, this one might be good for you. If you have more than a handful of factions, tracking each faction’s bond with every other faction will likely become crazy, so you’ll want some combination of a default bond (probably zero) and/or a way to limit which other factions each faction cares about (maybe only their neighbors/rivals/allies).
Faction Resources and Reputation
I built this system for my “Heresies Without Number” game - as you might guess, it’s doing Dark Heresy with Stars Without Number. I wanted a way to maintain the benefits of treasure as XP in a campaign based primarily on gathering information and where money isn’t really a constraint. My solution was to make “treasure” be information/access wanted by various factions, which is rewarded by reputation, which allows you to get the things that faction has to provide. It’s definitely the most fiddly of the three I’ve worked out, and it’s the closest to Kevin Crawford’s systems, so it might not be what you’re looking for. The good news is that like Crawford’s systems, once set up it creates a lot of “living world” adventure material for you with only a bit of fiddliness. So this is most likely helpful if the “faction game” is meant to be the primary motivator for adventure, and you want the PCs to have lots of interaction with it and how it’s going.
Hope some of these come in handy!