The Advanced Math of Choice
Hold onto your wizard hats folks, interactive storytelling can get pretty crazy
By MARTIN REZNY
Since somebody seemed to like my article about choice in games, here is the promised continuation. But first a little bit of context, necessary because I may actually be about to present an equivalent of a unicorn in today’s world — a (somewhat) original idea that would have been obvious if ancients had PCs.
I don’t know about you, but I always felt like I don’t belong in this time period, like not at all. I always tend to either reinvent or reverse engineer things that were normal so long ago that nobody today seems to even know what they were anymore, or do things that will become a cool thing only years later.
I used to compose music electronically when I had to carry it around on floppy discs. I used to think, wouldn’t it be great if you had a fantasy RPG in an open world, mixing potions from ingredients you gather, creating custom spells from variations of individual effects, and other later realized game elements.
When I saw Doctor Who on TV, my thinking was “Finally, a normal person I can identify with.” When I saw Mr. Nobody, for the first time I have witnessed a depiction of how I’ve always thought about relationships — predicting a web of possible future trajectories and trying to navigate them. So yes, I am weird.
And I think it’s about time I came out of the closet as a wizard, non-ironically. It’s necessary because even though this is relatively math-heavy as far as storytelling goes, it is built around certain ideas about the world and nature of meaning that have gone out of fashion academically long before computers. Well, the current ones, at least. Greeks had computers. Check out this laptop:
To sum up the basic idea from the previous article, balancing a system of choices in a game so that it can satisfy both the player and the requirements of math can most effectively and easily be accomplished if you stick to a certain arbitrary integer number and its multiples, so that there is symmetry to it all.
How it works can be easily visualized, if you take a hint from the ancient Greeks and instead of algebra start thinking about math as geometry. Also, math and ideas were not really separate concepts to them, especially for the likes of Pythagoras, who had similarly developed for instance the tone scale.
I’m using this example because it is important to understand that to many of the ancient Greek thinkers and tinkerers, meanings were not arbitrary. They were big fans of rigorous logic and believed mathematically representable ideas to be real. This was not just theory, it had practical, if abstract, goals.
Much like music was improved by the the tonal theory, an applied piece of holistic qualitative math coupled with acoustics, so a similar approach can be used to improve games on a qualitative level — allow for more meaningful experience of choice, more elegant coupling of story with game mechanics.
WARNING: The long boring math part of the article is beginning!
Okay, now to the basics of the geometry of thought. You always start with a circle. An infinite amount of points equidistant from each other, as perfect a representation of the whole of all possible meanings as can exist in reality. You can draw points on the circle, connecting them with lines is optional.
Angular distance between those points on the circle’s circumference expresses the degree of difference, meaning that the most different ideas lie literally opposite each other (180 degrees), and conversely, the closer two points are to each other, the more similar ideas they represent. So far, a piece of cake.
Earlier I said that the number you choose, the number of points, is arbitrary, but it’s actually a little bit more complicated. As you may begin to see now, it really matters if you choose an even or an odd number, because the easiest tool to create a system of choice are polar opposites, points 180 degrees apart.
There are no points 180 degrees apart in equilateral shapes with odd number of edges, only in even numbered equilateral shapes. It’s a problem because the less symmetrical a shape is, the less balanced it is, and if you just remove one of the points of an equilateral shape, you’re making your system incomplete.
An example of incompleteness. Did you know that in most fantasy RPGs, Elder Scrolls series included, there’s a missing class? Think about it for a second. You have a warrior applying might (concrete, will), a wizard applying magicka (abstract, will), and a thief applying cunning (abstract, tech).
What’s missing? And to the credit of Bethesda, it is a part of their world, just literally missing. The Dwemer, the dwarves, the engineers (concrete, tech). It’s the four elements again. The fourth part of the whole is missing, mine and craft, an obvious logical necessity, and it doesn’t have a class of its own.
At this point, it might just seem like an abstract intellectual exercise to you, but that’s precisely what it isn’t. Without understanding any of this, players will still be thinking, gee, wouldn’t that be nice to play as Dwemers. That’s why, eventually, mining and crafting were demanded and later added.
This is about what’s logically inevitable. It may only speak of the ways in which humans think or conceptualize things, not of the physical world, but you’d be making a game, a story, not trying to make a physics experiment, so it only concerns you completely. But back to the oddity of odd numbers.
In the ancient (not an exaggeration) lore of this sort of approach to math, odd numbers were understood as being qualitatively different, chaotic, lucky or mysterious, and this is why. They are that way if you try to base a qualitative system on them. Three is change, five is strife, seven is luck, and nine mystery.
Not arbitrary, not superstition. Pure logic. It’s hard to use ideal typologies (modern sciency term for this kind of ideograms) based around an odd number of points for static description of something, because you cannot have a balanced system of mutually symmetrical axes in odd-sided shapes.
Put simply, either it’s all based on balanced scales of polar opposites, or it’s never going to be complete or balanced. The best you can hope for is what the Wizards of the Coast have managed to create in Magic: The Gathering. A world of perpetual conflict due to imbalance of powers. See, five is strife.
The only logically functional alternative is a cheat, a system of something like a two and a half instead of two or three. Like in the example of StarCraft, where you have only one scale with two opposite ends (few expensive units versus many cheap units), and you put the Terrans in between. Balanceable.
That’s why instead, the odd numbers have traditionally been used by classic thinkers to describe cycles of change in time — each point a step in a loop. The famous example is Hegelian triad (thesis, antithesis, synthesis). In games, this can be used for meaningful automated storytelling, or to balance stuff in time.
And now for the final idea that I may have conceived of myself, not just reinvented, but invented. All of this, all that the ancients or the classics have done in qualitative math, has been limited to numbers below mid-twenties, usually twelve (as the most divisible small number, it still has its proponents).
The “magic” of a number like twelve rests purely in its utility, because within a twelve-sided shape, you can embed multiple symmetries or subsystems based on both odd- and even-numbered systems (1, 2, 3, 4, 6, and 12). Packs the most meaning in the smallest space. And it matters, because of our brains.
See, we have a difficulty processing large numbers in our heads, at least most of us do, and you don’t want to confuse people with your game, story, choice system, religion, whatever. Even if you have designed your game around the number twelve, most of the time, you’d only be showing 2–6 options at once.
It would still be useful, though, because that could allow for mathematical balancing between different minigames, aspects of a game, or even separate games. Oh transgaming, another idea I had ages ago. But go team Eve Online, it’s just an inescapable logical conclusion, necessary next step. Where was I…
Size of numbers. It matters. Especially if you want to create anything like an automated generator of story elements, or if you want to make even hundreds of meaningfully different aspects of your game still balanceable. Normally, and this is still advisable, you have to use intuition resulting in imperfection.
But beyond certain measures of stylistic imperfection or perfect imbalance, it would be hard to have a game based around a choice system of something like twelve fundamentally different flavors of story and elements of gameplay and create for example hundreds of unique characters and make it all intelligible.
For that, you can use something I decided to call contextual resolution, and it works thusly. Imagine you choose four as your base number and you want to make classes for four different game mechanics, but not just four singular types, you want also a system of subclasses, or a way to structure the skill tree.
Key word is multiples. If you have four styles of play, say smash, counter, sneak, and gear up, you can make each class in four flavors, like a smasher who can super smash, counterstrike, sneak attack, and weapon-craft. By dividing further, you increase the meaning resolution, like adding pixels.
Just make sure that you clearly and fully understand what differentiates your choices, which is possible if they’re permutations of polar opposites on symmetrical axes. You need to understand if you’re doubling the same thing, mixing the similar, combining the dissimilar, or merging absolute opposites.
If you had say three character roles to create, you could split each into three mixed classes, and each of those into three more, even more finely mixed classes, and so on for as long as any meaningful shades of difference exist. If the roles are based on equidistant abstract ideas, this should not be difficult.
Following the StarCraft example of a swarm versus a team versus a superbeing on appropriate power/cost levels, you could have three subclasses of units in each of these categories (based on power/cost ratio), and three even lower subclasses in each subclass (based on power/cost ratio), and on and on.
As for the “contextual” part of contextual resolution, if you devise a numerical system like this based on principal differences, you should have no trouble transferring it between worlds, genres, or game elements, or in other words, between different contexts. Let’s analyze it on the simplest possible example.
Take smash. Smash can be a character with such personality, a personification of smash (yes, you’re thinking of him now). Smash can be a weapon, smash can be a spell, a buffing or cursing item, smash can even be a place or concept in theoretical physics. As Plato would see it, it is a real idea in and of itself.
What would always be very similar would be its mathematical and practical representation in a game system — exactly that in a given context which tends to result in maximum damage being dealt straightforwardly while being open to retaliation. Meaningfulness is all about logical consistency, balance is too.
To sum up, thinking about your system of options or choices in a game in this numerical fashion will make it possible to achieve mathematical balance, as well as meaningfulness of the whole and every partial aspect of it, as long as you’re mindful of your own principal logic and how it applies in given context.
It is not an exact science, I admit, but the goal is not objective perfection, it is providing a better tool to enable finer and more deliberate artistic expression. Music also cannot be mathematically solved, but that doesn’t invalidate the utility of a tone scale. Or at the very least, this is how ancient philosophers would make video games.
