Integral of motions for RPGs
02 Sep 2025
The concept of the integral of motion of a system is interesting. If you have something made of many interconnected parts that influence each other movements by impacts and/or forces of attraction and repulsion it’s a mess, BUT you can try to get off the mess by finding something that, despite of all interactions, stays pretty much the same.
In a more “physical” system, usually, total energy is a good candidate as something that does not change during whatever you are trying to model.
In other systems it could be something else but, since mathematics people are stubborn, it’s not unusual to still call it that way.
The concept of “something that stays the same during whatever is happening”, anyway, is called integral of motion and it’s usually a nice idea to find it if you want to study a system. And a Roleplaying Game ruleset is a system. So I did.
Wait, is it interesting?
No.
Because it’s data, and data, by itself, is boring.
Correlations in between data are far more interesting, because they try to point out that there can be something meaningful there and there actually is an interesting correlation. It’s taken from far away, it’s quite obvious and I need to double-check everything, but it’s quite there.
If you look at the various game systems, many of them have pretty specific integrals of motions wired in their system. They are usually constraints laid to limit the power that a single player has to intervene on the agency of other ones. More interestingly, many games do not have these constraints, but players put them in anyway. They serve a purpose: since the rules fix these limits, whatever bad things happen to the character comes straight from the rules and not from the player.
Take Daggerheart: the GM must spend Fear, an in-game currency, to make bad stuff happen, it’s capped, so the GM must spend it, and when it’s over some monsters cannot activate their worst abilities.
Take Band of Blades: the game has quite the structure, I played two full campaigns (both really enjoyable) but in both cases the GM put an extra structure on missions, turning them in the filling of three clocks, to match a three-act story structure that the game doesn’t have.
The interesting part is that there is a correlation in between how much structure a game has and how much successful it is. With some outliers, you can see that the less a game has a precise integral of motion, the more is played.
D&D, Call of Cthulhu, Vampire: the Masquerade, Pathfinder all of them do not really tell you “during a session you should follow this flow”: the flow is pretty much in the hand of one of the players* at the table and there is not really a limit on how much duress the party can meet. Nothing stops that player to say “ok, your newbie adventurers get out of the inn and they see the Tarrasque emerging from the ground, devastating the temple of the Five Thinders. Roll initiative” - and that is fine. I mean, when I said that it’s been the beginning of a great game night :)
Even removing D&D, which is an outlier per se, the correlation still stands. My gut response to this is that these systems stand on the trust players have on each other and, being built upon that, the final experience is improved.
That matches another thought of mine, that D&D books are not a game, but more of a framework that allows you to bring on the table your D&D.
I need more data and/or experiments to sustain or disprove the thesis. Everything in time.
( * ) Uh, I almost forgot this one. Yes, D&D’s 2024PHB uses player excluding the Dungeon Master, but that’s part of the lexicon. For the colloquial use of that term, the DM is still playing a game & is usually the one with some social duties, all delegable to others. So, yes, players, not capitalized nor italic, still includes the DM ;)