Complexity and Noise in Games

Puzzles have solutions, games have strategies. A solution is a complete sequence of moves or actions that leads to victory. A strategy is a general plan, in which the exact actions or moves may vary. This shows a property of games – they are not predictable in practice, or at least, they should not be. They may or may not be predictable in principle – rolling a dice or drawing a card from a shuffled deck produces outcomes that are, both in principle and in practice, unpredictable. Other players are also, in principle, unpredictable. Many digital games make use of random number generators that are really no different from dice rolls, and these too, are in principle unpredictable.

The distinction between in principle and in practice unpredictability is an important one, because on some level it relates to the very functioning of games as a medium. Games are about prediction. Every decision a player makes is based on predictions about how the system works, and how it will respond to the player’s actions.

A players’ ability to predict a system is not a fixed quantity – as a player builds their knowledge and skill, they are increasing their ability to predict future states of the system. Thus, practical unpredictability decreases with experience and skill. In principle unpredictability on the other hand, can be thought of as noise in the system. It cannot be reduced with knowledge, and so it acts as a limiter on the granularity of predictions that can be made about the system. Or an alternate framing – any given system that contains noise will have a certain signal-to-noise ratio, and this means that certain signals (useful information about the system) are too low strength to be discerned from the noise. While noise may provide some level of excitement or tension on a thematic or experiential level, its effect on the players’ attitude towards the system will be that a certain band of signals, from the smallest details up to the level of the noise, is simply ignored. An extra task has also been given to the player – that of sorting signal from noise, and of determining where the lower limit of discernible signals lies. Whether or not this is a good thing I leave to the reader.

One seemingly positive aspect of noise in a game system is that it forces players to understand and operate on a higher level of abstraction. In this way it behaves a little like chaos. Here is the distinction that I want to make – the two sources of unpredictability are complexity and noise. Complexity is the measure of the unpredictability of a deterministic system. Complexity is, in principle, predictable, noise is not.

At this point I think it may be helpful to establish a vocabulary :

Complexity
Simply, how difficult it is to predict a non-random system. Practical unpredictability can be overcome through knowledge (heuristics) or calculation (sequentially simulating future states). A more complex system is more difficult to predict, that is, requires more knowledge or calculation to accurately predict. This can be due to any combination of the following factors –

Chaos
Small variations in earlier states of a chaotic system lead to larger changes in later states. Systems can be more or less chaotic. Chaos arises because the elements within a system interact with each other, and are in some way dependent on the state of other elements in the system. Every subsequent state of a chaotic system is highly contingent on the details of previous states. [See here for a more thorough exploration]

Scale 
The number of elements in a system.

Detail
The amount of information within a game state or game element. Describes individual states more cleanly than whole systems, as the amount of information in a system can change over time, but we could say that the amount of information within the average state of a given system defines how detailed that system is. Can also be used to describe how intricate the rules governing the system are. Detailed game rules are often referred to as “complex” colloquially, but I think “complicated” is a better term. A game system can be complicated but not complex, by having many rules but being simple to predict once the rules are known.

Notably, scale and detail are quite trivial to quantify, but chaos is not. We could imagine that there exists some kind of formula for calculating complexity – each element in a system has (d) bits of information, the system contains (s) elements, and some amount of chaos (c). Any given state of the system contains d * s bits of information, and the complexity is given by (d * s)^c . This is only to illustrate the idea, presumably any consistent and useful formula would be much more complicated.

Complexity, or practical unpredictability, comes from some combination of scale, detail, and chaos. Any given state can also be difficult to predict due to range – how distant it is in the future. For example, the system could be highly predictable, but the player is trying to predict a future state ten thousand moves away- that would be hard to predict primarily due to range. If the system is highly chaotic, such that every sequential state is highly dependent on the previous state, then it could be difficult to predict a future state even just a few moves away. Or there could simply be a lot of detail, for example each piece on a game board could move very differently- this could also make a future state hard to predict. Noise, on the other hand, simply means that the future state being predicted can only be narrowed down to some range of possibilities, no extra effort or knowledge can narrow it down any further. In truth, any system that is only in practice unpredictable, through complexity, if it is unpredictable enough to be interesting, has a similar property – the player will be predicting future states with limited accuracy, and so they will be predicting some range of states rather than any single state. However, in a game with no randomness and no hidden information, the width and accuracy of this range will be limited only by knowledge, memory, or time. If the player ever does achieve perfect prediction of the game, then they have “solved” it, and the game is no longer of any value to them. So an “evergreen” or “lifestyle” game must have sufficient complexity to resist solution over years of regular play.

Turning the knobs
If we approach this topic somewhat naively, and think about tweaking these values – scale, detail and chaos, can we make any useful generalisations? Increasing detail increases the knowledge burden placed on the player – there are more rules governing the system that they will need to understand. This increases the amount of layer 0 (non-emergent) information that the game system contains. This information must be absorbed by the player before they can really be said to understand the game. When a player lacks some knowledge about the game rules, the behaviour of the system may be more than just surprising, it will often appear incoherent. While for some digital games, discovering the underlying rules governing the system is the whole point, this is undesirable for a strategy game, where the point is to try to make accurate predictions about future game states. For a strategy game, increased detail means a greater upfront cost to players – they must spend longer learning the rules before they can meaningfully interact with them.

Increasing scale would not have this effect, instead it would simply require more calculation from the player to find the optimal move. Usually we don’t want the player to ever find the optimal move, so we either have to get the complexity value ((detail * scale) ^ chaos) to be very high, beyond complete human ability, or try to limit the effectiveness of calculation. Only limited time or hidden information can truly limit the effectiveness of calculation, but that is beyond the scope of this article. Suffice to say, that too small a scale allows for mathing out the solution, as we see with noughts and crosses (tic-tac-toe for you yanks). On the other hand, scaling up a system with low chaos will tend to reduce the importance of the finer details, allowing for degenerate strategies to become viable. Degenerate strategies are high level strategies that work sufficiently well to allow the player to ignore lower levels of emergence.

This is only a cursory examination of these topics – the relationships between detail, scale and chaos are themselves complex, but hopefully the ideas presented here at least help you to build some rough intuitions.

Complexity is Other People
I skipped over this topic earlier, but what about other players? I noted that they are in principle unpredictable, but they are not the same thing as noise. In some contexts it is possible to improve ones ability to predict other players. While this prediction can never get to 100% accuracy, which is technically possible for complexity, it can get above chance. Players are, at the very least, weighted noise generators, and in some contexts they are noisy pattern generators. This may be best illustrated with a hypothetical. Suppose you are playing rock-paper-scissors against another person. This other person is not in the same room, and you play via a computer, so you cannot try to read their expression or body language. You have no information to employ against them except for the moves that they have chosen in previous rounds.

If we were to change the game so that a win with paper is worth five points instead of one, then you would expect your opponent will be aiming to play paper more often than rock or scissors. Players produce weighted noise when there are unequal payoffs for the different options that they have available, as they will naturally be seeking to maximise the expected value of their actions. But weighted noise is still just noise – a weighted number generator has the same range of outcomes as an unweighted one, only the probability of each is different.

To do better than break even against your opponent you will need more information to go on. As I said, in this hypothetical your only interaction with the opponent is the moves that you each make, and so there is only one kind of non-probabilistic information that can be encoded in these moves – patterns. A pattern is a common sequence of moves. Perhaps you notice that your opponent always plays paper after they play rock. Of course, this is very simplistic, a more likely scenario is that your opponent plays paper more often than scissors after they win with rock. Their patterns are both noisy (probabilistic), and contingent on your own plays.

Really, human opponents are whole systems in themselves that players can try to understand and predict. They do not reside within the game system, and so their inputs are, like noise, the outputs of an external system entering into the game system. Dice rolls and card shuffling are the outcomes of physical systems entering the game system, random number generation is the outcome of some computation entering the game system.

To my mind, only fighting games have ever successfully made use of opponents as pattern generators. This is because they are fast paced, and so players have to rely on instinct and unconscious behaviours, and because they allow enough repetitions of certain scenarios for patterns to form and be observed. Due to the fast pace, players do not have time to randomise their actions in order to avoid creating exploitable patterns, as they would be able to do in a turn-based game. The faster the game, the more players have to rely on heuristics and instinct, which are prone to producing patterns, rather than methodical, conscious calculation, by which one can systematically avoid creating patterns.

As stated, patterns are the only in-game source of information on which to base predictions about a specific opponent. But in most competitive multi-player games, players are not trying to predict each other precisely, so much as they are making predictions based on past game sessions. They have a model of how the average player in the current meta game would behave, and they make predictions based on that. This is not a bad thing, it is not all that different from how a player predicts the game system itself – they have a mental model which they refine through experience. It is much easier to make a competitive multi-player game that can be played for ten years than a single-player game. The meta-game can evolve over time, so even if the game systems themselves offer nothing new to learn, players can at least learn new information about other relevant systems – their opponents.