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The Mathematical Basis of Game Design

by Rinku

What are games? Are all games mathematical? What is the relationship between measurement and games? Here is a philosophical approach to these questions. I have succeeded in condensing it down to a manageable 16 paragraphs.

I wish to thank Ishalltriumph for discussing this definition with me, via livejournal comments, in its early stages of formation, and in specific, for pointing out to me that the game (process) form has art and non-art entities, just like the word (concept) form has art and non-art entities -- fiction-literature and nonfiction. I later expanded this to all art forms, noting that all of the other forms of art also have art and non-art entities: for example, there is a difference between visual art on the one hand and smears of paint and photography and stick-figure drawings on the other hand. He also helped in pointed out that a rule-system cannot be recognized as a re-creation of reality without some visual, conceptual, or musical 'teammate' -- ie, that if you took a game and reduced it to *only* the rule system, without a game board or game pieces or graphics designating the game tokens and game world (ie, Chess with squares and triangles instead of kings and knights), it would lose something essential to art: the re-creation aspect. This would lead to one not being able to identify exactly what the game's theme (what process it re-creates in game form) actually is (Chess with squares and circles would no longer have the theme of warfare).

I also wish to thank the contributing members of the New Romantics email group: while they did not actually help in this idea's formulation, they did provide valuable criticism of expected objections to this theory, which doubtless helped me formulate the exact wording in this article so that their misinterpretations were taken into account.

I also wish to thank Komera Waddi for finding examples of the battle theme in painting and in statue, and for recommending a summary paragraph.

1. All games -- games qua art, and games without artistic status -- both share something: the specific form of an interactive mathematical-logical causal rule-system, with built-in values. Any game you can think of -- whether board game, a sport, a card game, a videogame, a children's game, etc. -- shares this definition.

2. A game qua art, like all art, it performs a necessary psycho-epistemological (that is, pertaining to nature of human consciousness) function: it bring's abstractions to the concrete level, letting one apprehend ideas directly, as if they were concretes. Just as Goldilocks and the Three Bears is a selective re-creation of a certain concept (the idea of too little, too much, and just right) in the form of a story, so Chess is a selective re-creation of a certain concept (the idea of war between armies) in the form of a game. The difference is that Golidlocks and the Three Bears does it through -language-, and Chess does it through *causal rules*. But what is essential is re-concretization: the going from things in reality (concrete) to a mental symbol of a certain category of things in reality (ideas) and from there creating a new concrete with the essential characteristics of that classification (concrete).

3. A single theme may take different form in different art forms: the theme of "War" for example, may take the form Richard Wagner's The Ride of the Valkyries (music form), Tolstoy's War and Peace (novel form), Delacroix's Liberty Leading the People (visual art form), First Emperor Qin Shihuang's Terra-cotta Warriors and Horses, or Blizzard Game's Warcraft II (game form, that is, "interactive mathematical-logical causal rule-system, with built-in values" form). These all take different forms, but are all united by the theme of warfare; in fact, you could not group them all under one concept if you did not yet know the concept of warfare. Existentially, they vary by a great deal, it is only in what they are reconcretizations of, what they symbolize, that they can be classified together. Therefore, games qua art have this additional characteristic in common with all other art forms which allows them to be grouped under the concept of Art. What remains for this article to do is define what the game form precisely is, as opposed to the other forms art can take.

4. Please note that I am not saying all games are art, just as I wouldn't say that everything that has 500 pages and is in paperback novel form is art, or that everything that is hung on a wall in a frame is art, or that everything that is seen in a movie theater is art. Unless it has the characteristic of being a selective re-creation of reality according to the author's metaphysical value judgements, and therefore serving an essential epistemological role in consciousness, it isn't art (this definition of art was first discovered by Ayn Rand, for more on this if you are unfamiliar with this philosopher, see my forthcoming article on the Objectivist Aesthetics).

5. However, this is not an article about the artistic status of games. I covered that in part last issue and may elaborate on that topic in future articles. This article is only concerned with what all games share, whether art or not. And what all games share is that they are interactive mathematical-logical causal rule system, with built-in values. The hope is that in understanding with clarity what games are in essence, game designers will have an easier time designing their architecture. So let us leave the defining characteristic of art for a second and unravel the defining characteristic of the game form.

6. Now let me expand on my mentioned defintion of a game as "interactive mathematical-logical causal rule-system, with built-in values". I shall go over each of those terms in turn, in this order: (1) mathematical, (2) logical, (3) causal, (4) rule-system, (5) built-in values.

7. First, the mathematics: there is no such thing as a game without numbers (or measurements). From the most basic to the most advanced symphonic games, there is some type of counting, score-keeping, and breaking things down into units of measurement. In RPGs, there are stats, HP, MP, and other numbers, all representing the status of the game. In the game Hide-and-Go-Seek, there is counting, and the Boolean (binary) status of 'found not found'. In Chess, there is the location (in a pair of ordered numbers) of each piece on the board, and the Boolean status of whose turn it is. In most videogames that you can think of, there is some type of 'location system' (held in numbers) and 'health meter' (held in numbers), and many also have a 'score' (held in numbers).

8. The assertion here is that it would be *impossible* to have a game without measurement; and the science of measurement is mathematics. So games qua games can be described as essentially *mathematical* entities, which is starkly different from the forms of vision and touch (for the visual and plastic arts), hearing (for music), and conception (for literature-poetry). Note that each form of art uses a uniquely different faculty of humanity, and the faculty used here is our ability to think in numbers. No animal can play games exactly because they can't think mathematically.

9. (There are some things children call 'games' which entail no mathematics, such as the game of 'Cowboys and Indians' or 'playing dolls' or even just 'pretend', but I would not consider them games in this sense, but rather role-playing or pretending activities. They are certainly playing something, but they aren't playing anything essential similar to what we usually think of as games. In other words, a game does not simply mean 'that which is fun to do', that is known by a different concept: entertainment.)

10. Second, logic: there is no such thing as a game with self-contradictory rules, and no such thing as a game that uses "non-Aristotolean" logic, such as the so-called super-logic of dialectic. The rules must not contradict eachother. You can't have two rules covering the exact same circumstance at the same time. There must never be confusion about what the rules tell you is supposed to happen. Otherwise it would be impossible to play the game -- you wouldn't know how things worked in it if the rules were sometimes A and sometimes B. That is why games like Baseball have umpires, and games like Dungeons & Dragons have Dungeon Masters -- upon any confusion between players about the rules, that person has the final say on what the rules mean. This isn't really a problem with computer games, because if a computer program is contradictory, the compiler will spit it back out and the program won't work (unfortunately, no such device exists for looking for contradictions in language).

11. The reason I combine mathematical and logical into mathematical-logical is to distinguish it from its twin, linguistic-logical, which is noncontradiction of concepts which do not involve numerical measurement, for example, most nouns and verbs and adjectives and adverbs and other parts of speech. Of course, the logic of mathematics is not essentially different from the logic of language, both are non-contradictory, but the point is that one can have a game with illogical text language (dialogue between characters with grammatical constructions and concepts that don't make sense and contradict, for example), but one cannot have a game with illogical mathematics. So it's essential that a game have a logical rule system, but *not* essential that it have logical language (although that would of course be desirable). Novels, in the true sense of the word (ie, not in the 'stream of consciousness' James Joyce / Virginia Woolf variety) need non-contradictory language, and aren't novels without non-contradictory language, and likewise, games need non-contradictory rules, and aren't games without non-contradictory rules.

12. Third, causal: all rules are causal, of course, so this is just for emphasis -- but the emphasis is perhaps needed for fullest clarity. A rule is an if-then statement, that is, anything which says "if such and such conditions are fulfilled, then such and such is the resulting change". A rule in Chess is: if the king is killed, then the game is lost for that player. A rule in Tag is: if someone who is 'it' touches someone who isn't 'it', then the person touched becomes 'it' and the old 'it' is no longer 'it'. Some rules in The Legend of Zelda is: if Link has 5 hearts, then he can take the White Sword from the old man at the waterfall; Octorocks are entities such that they shoot rocks at link; Link's shield is an entity such that it can deflect rocks; when Link loses all his hearts, the game is over. This is a very important point to understand about games -- all complex rules can be broken down into if-then statements, and what is an if-then statement? Something describing cause and effect.

13. Fourth, rule-system: a game's rule system is not a collection of units, but a system, and in that way it works like an organism; each part having a purpose. If you remove any organ at random from a human, what you have left will be either a human who is decreased in efficiency or a corpse; each organ is defined by its role in the system. Similarly, you can't just remove rules at random and look at them apart from the whole; and each rule of the game is defined by its role in the total rule-system. A collection of rules, by itself, isn't enough, they need to work as a *system*.

14. Fifth, built-in values: if a game had no 'object', ie, no 'goal', then it wouldn't be a game. The object of Chess is: take the king of the other player. The object of Tag is: do not be 'it'. The object of The Legend of Zelda is: find all the triforce pieces, defeat Gannon, and save the Princess Zelda. All of these are *values*, things that the player must act to gain or keep. So this is another very important point to understand about games -- all games have *built-in* values, the game itself tells you what you must act to accomplish. Even non-art games, like newspaper crossword puzzles, have built-in values: the built-in value here is to fill out the crossword puzzle, as much as possible, preferably to fill it out completely.

15. Another last important point to remember is in regards to not losing the essential characteristics of art: a "mathematical-logical causal rule-system, with built-in values", by itself, *cannot* be recognized as a re-concretization of a process of reality without some form of visual, linguistic, or auditory cue that directly relates it to that process or implies that process. These cues would have to have also been created by the author specifically for the game. To illustrate this rather simply, think what would happen if we replace Chess pieces with squares, circles, triangles, and other shapes. It then ceases to be art, and becomes a sport or a contest, like dice, soccer, golf, checkers, or poker. The basic difference here is that Chess with pieces designed to look like kings, queens, and knights serves an epistemological function -- examining a process of reality via a re-creation into an essentialized rule system -- and "chess" with pieces made of squares, circles, and triangles serves no such function. It may still be as much fun, and the rule system may be exactly the same, but it has now lost its psycho-epistemological function as art. So in this manner, a game can never be art alone, it always has to be art in combination with something which allows you to see the rule system -- some set of visual, plastic, audial, and/or conceptual tokens. This isn't really that odd, because you also cannot have a conceptual re-creation of reality without some sensory (either visual, audial, or tactile) symbols to hold the concept (let alone the whole novel) in reality -- note that a novel needs some visual symbols (written words are visual objects), or in the audio-book format it needs audial symbols (spoken words). But this is particularly important to recognize in the case of games because a rule has to be symbolized *somehow*, a rule system isn't a re-creation of reality unless it is held in reality by something. In the current generation, games are held in reality in the form of programs (computer code) and shown via graphics.

16. Conclusion: The definition of games as "interactive mathematical-logical causal rule-system, with built-in values" appears to hold as the essential characteristic a wide range of phenomena called games, both art-type and non-art-type. It is proposed that the game form is the form of the basic human capacity to count and measure, i.e., games : mathematics :: literature : language :: paintings : vision :: music : hearing. Just as understanding language fully makes for superior literature, understanding mathematics fully makes for superior a game architecture.


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