Possible Futures

There’s a marriage waiting between the sense of meaning that applies to video games and the trees of future states you’ll find in game theory.  Meaning takes outcomes into account even if they never actually happen; these trees, likewise, branch out into every conceivable outcome.  Modification of future possibilities, even possibilities that will never be realized, will change the interpretation of the immediate situation.  Players sense this and experience it as meaning.

Decision trees

Let’s start by admitting that, for the present at least, we only need an informal sort of decision tree.  This is the kind of structure the player will actually work through mentally; a really complete enumeration of all possible futures is a useful construct for working out proofs, but it’s utterly intractable in games as they are actually played.  The number of possible moves each turn, and the number of turns, is simply monumental: Take Brogue.  With a full inventory and in a wide open space, you can wait, search, apply or equip or remove any one of your items, throw any of your items at any of the 400 or so dungeon cells around you.  Many of those applications, such as of a scroll of enchantment, will prompt for another item or for a direction to zap in.  In such a situation, the player has at least ten thousand possible moves.  Over the typical course of a twenty thousand turn ascension, that’s an astronomical profusion of futures to account for.

Thankfully, however, most of that variety is degenerate.  Many of those moves will actually be impossible (not least because they get you killed), and many others will be functionally indistinguishable.  There’s an incredible degree of self-similarity in the futures of these trees; sitting around waiting, in the absence of any enemies, takes the game to a new state that is exactly the same except that all child trees are one turn shorter — the rogue is one turn closer to starving.  The same is true, for instance, of the various ways of pathing across a room.  The exact sequence of steps really won’t matter much, and you’ll find yourself facing the same future as before.

This self-similarity turns out to be the key to making headway, which is rather unusual: In most situations, introducing the fractal nature of a problem serves as a distraction.  Look, look, it’s so complicated, but isn’t it pretty?  Here, instead, what we want to notice is that what had seemed dreadfully complicated (ten-thousand to the twenty-thousandth power) really mostly comprises the same simple patterns over and over again.  The differences between futures can largely be broken down into situation and resources, and this, again, is exactly what players do.  These are the tactical and the strategic subgames.

Situation and resources

My situation might be bad, but it’s just one path between my earlier state and a target state, and once I get there it doesn’t matter how I got there.  It doesn’t matter whether my health dropped to 50% or 10%, so long as I didn’t die.  It doesn’t matter whether I triggered a fire trap, so long as I didn’t burn a scroll.  It doesn’t matter whether I went north and then east or east and then north, so long as I end up where I’m going.  Situation is the part that gets scrubbed away by the passage of turns.  I might only have two possible moves this turn, because I’m in a corridor, but I’ll have more as soon as I step into a room.  My situation changes quickly.

Resources, on the other hand, describe the differences between sub-trees that propagate uniformly from parent to child.  Any turn on which I have a Potion of Healing is a turn on which I can use a Potion of Healing, but after that turn I have no more Potions of Healing to use: The shape of my options each turn changes the moment I actually use it.  Likewise, an item on the floor that I haven’t picked up is an item I can walk over to and grab whenever I please, until I’ve actually done it.  Once I’ve picked it up I can’t pick it up again.  Anything that functions in this way, from finite monster spawns to inflammable terrain to walls that can only be tunneled through once (and are forever open after), functions as a resource.

The two, it should be obvious, commute.  On a long enough timescale most resources look a lot like situations; on a short enough timescale, most situations look like resources.  How far ahead you’re willing to look determines which is which, and so, again, the nature of the heuristic becomes clear: we’ve constructed one tree of futures at a fine scale and called it our tactical situation (even though we’re not likely to distinguish every one of our ten-thousand possible moves each turn, and we’re really only considering a few dozen of them), and we’ve constructed another one by recognizing self-similarity and labeling each distinct feature of each potential future with a named resource.  A tree that looks like this is one that has a healing potion; a tree that looks like this is one where the rogue is starving; a tree that looks like this is one where the player has a powerful suit of armor.  We take each of these, treat them as a single state, and get (what else?) another tree out of it.  This is the tree we actually think about for the strategic subgame; do I pursue this resource or that one, expend this one or that?  The ugly turn-by-turn of it is abstracted away.

And the ugly turn-by-turn of it goes about attaching a sense of meaning to resources.  The general sense we get of when a particular item is useful and which items we’d like to put in our kit together, itself yields a deeply rewarding sense of drive and of mastery.  Just watch the lengths that actual players of roguelikes will go to to ascend with a more peculiar build than ever before.

Randomness and uncertainty can be incorporated just as they are in game theory: Any time the dice will be thrown, you construct a turn that will be taken not by a player but by the dice, and you annotate the edges with their probabilities.  Curiously, the optimal strategy in the face of probability tends towards two extremes, depending on whether the player can mitigate the risk of absolute failure: in the one state, expected value dominates, and rolls are treated in terms of their averages; in the other, the player considers only the worst case.  At full health, for instance, a player will treat a 20% chance of hitting as doing a fixed 20% of the damage.  When that 20% chance might kill, however, a good player will treat that as inevitable.  (A bad player, on the other hand, will systematically succumb to wishful thinking, and bet on the 80%.  This is one of several hard skills that roguelike players must acquire — but that’s for another post.)

Smoothness of time and space

The heuristic we use to counter the combinatoric explosion of a roguelike’s future can be adapted with no real trouble to games with smooth time and movement.  Indeed, smoothness is no obstacle at all: Even without relying on discrete frames of motion, we can identify arbitrary key frames between which no decisions take place.  The length of time between them does not have to be uniform, and we don’t (for reasons to be developed) need to consider reaction times.  This is entirely reasonable, in practice.  No, the true obstacle is the matter of skill.

A player is certain to decide which of two platforms to attempt a jump from, for instance, on the basis of which is more amenable to success.  The player has firsthand knowledge of skill, and can treat performance as a random roll.  A thoroughly unskilled player, aware of these shortcomings, is likely to harvest resources — to grind — in order to overcome a situational weakness by resource sufficiency.  As a child playing Mario World, I would get two feathers and a blue yoshi before entering any difficult level.  Most people do the same.

Players actually do experience these exercises of skill as gambles, and I would suggest that it is because of this equation of skill with gambling that gambling tickles our reward anticipation as strongly as it does.  Before people came along, nothing could exploit this property of our learning mechanism, just as nothing could add refined sugar to a 32-oz drink.

Bringing it back around

The interesting thing is that the process by which a player identifies possible futures is itself subject to failure, and can itself be assigned a probability.  The application of our situation/resource heuristic is difficult.  Remembering present possibilities is hard enough (and it’s one of the cornerstones of roguelike difficulty).  Anticipating future consequences is another beast altogether. The savvy player, rather than fret over this, assigns a probability to the failure to consider all outcomes, just like the platformer player did when considering a jump.

A naïve player will also internalize actual randomness as an expression of skill, and experience a psychological reward accordingly.  Designers find it easy to exploit this failure.  Traditional power curves stand on several psychological legs, one of which is this inability to distinguish skill from external conditioning, in a peculiar misreading of cause and effect.  (The same effect is also, quite probably, responsible for our trancelike enjoyment of television and film, but that’s for later.)

Players plan for their own fallibility.  This can also help us understand the psychology of hoarders; cognizant of their total inability to plan for a game with unknown rules, they preserve every possible resource for later use in repairing the damage their ignorance will probably inflict.  Good players plan for fallibility, too, and attempt to distinguish between what they could have anticipated and what they never could have.  Good designers respect that.

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