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Archive for field theory of games

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On the application of commonalities among games to the learning curves of games

Moshe Callen
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ἄνδρα μοι ἔννεπε, μοῦσα, πολύτροπον, ὃς μάλα πολλὰ/ πλάγχθη, ἐπεὶ Τροίης ἱερὸν πτολίεθρον ἔπερσεν./...
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1. A qualitative approach to the theoretical study of games

Game theory is the mathematical study of "games" in a very abstracted sense, but intrinsically the subject deals with mathematically abstracted representations of games. Obviously the field is highly productive, but it lends itself only to addressing those questions which can be precisely mathematically formulated. Especially but not exclusively when dealing with non-abstracts, this limitation can be severe because a number of interesting questions do not readily lend themselves to mathematical representation. Other questions could in principle be mathematically formulated but doing so for many actual games becomes highly unwieldy. Therefore I propose an alternate approach to studying games which is supplementary to mathematical game theory, namely a qualitative approach which seeks to describe patterns in actual extant games by direct observation.

The analogy I will make is the contrast between linguistics and philology. The former attempts to take a scientific approach to the study of language whereas the latter describes patterns in the extant corpus of languages without being predictive. In the analogy, game theory would stand in lieu of linguistics and what I am talking about would then be comparable to philology. Thus for example, instead of optimizing utility curves to determine ideal strategies, one would seek to delineate a set of templates into which all known games fit and qualitatively describe those features.

2. The nature of the approach and its practical application

I have previously described what I call the field theory of games, but I did not then have in mind a practical application. Now I do. To be blunt, we are gamers.We like to play games-- lots of games-- and we play them to win. Sure, having fun and social interaction (blah, blah) are important but ultimately a game is most fun if everyone plays to win and most people enjoy winning more than losing. Yet with so many games out there in the world to be played, one cannot learn all of them individually if one wants to play well. This aspect of gaming is where what I term game-field theory comes in. One does not have to learn all games separately if one can learn the underlying structure of games. Field theory attempts to describe the template into which all games fit and how it varies from game to game.

To clarify what I'm talking about, one may consider the example of the Traditional Games: Tafl games which I discussed here in my list of reviews cum strategy articles in a series. This family of games is characterized by:
1. custodial capture,
2. asymmetric goals in which the defending player needs to move the king-piece to the edge or corner of the board while the attacking player (who has precisely twice as many pieces as the defending player, excluding the king-piece) must capture the king-piece, with complete immobilization of the defending pieces counting as capture,
3. orthogonal movement on a rectangular grid and
4. a rotationally symmetric initial set-up of pieces with defenders surrounding the king-piece which occupies the exact center of the board.
The actual size of the board and number of pieces does not matter in a fundamental sense. Even the pattern of the initial set-up may not be an essential element, although it must clearly influence the early game. Once one learns the basic principles of strategy, the specific variations do not matter. Even the inclusion of the guards in Alea evangelii effectively only increases the number of pieces which need be captured. In other words, by learning the key elements one in practice learns to play all games of the tafl family of games.

Another example from more modern games is the Risk family of games which I discuss via reviews cum strategy articles in this list. The commonality in all games of this family is the combat mechanism. Board topology, player goals, the nature and function of cards and die modifiers all vary widely. Overall strategy will differ widely, but the tactics remain essentially the same in terms of combat. A similar observation could be made about the A&A games. Where this becomes more interesting is when one extends the idea to games which are not of the same family.

3. A practical example: on dice-based combat systems

I'm going to start with an example which pointedly could be treated mathematically, and I'm going to discuss it qualitatively: dice-based combat systems. The useful qualitative questions largely set the stage wherein mathematically oriented questions become useful and meaningful. The point is that no question of either-or in approach exists; both approaches are useful and overlap. The only novel notion here is that a systematic qualitative approach to the study of games can also be useful.

While a wide variety of dice-based combat systems exist, all such systems can be classed as either comparative or independent. Within those categories exist two similar sub-categories based on the types of probabilities involved, whether uniform or Gaussian distributions. For example, games like Axis & Allies/Britannia, Conquest of the Empire and Battle Cry use an independent system in that one rolls dice singly (in principle) to see if a unit either hits an enemy unit or is hit by an enemy unit without comparison to what the other player involved rolls. Similarly, the combat system used in many wargames such as Successors (Third/Fourth edition) is also independent. The vast majority of such games use a single die with perhaps die modifiers and/or a CRT, but one could imagine a similar mechanic in which each player rolled a pair of dice; that would be a Gaussian independent combat system. In contrast, games like Risk or Struggle of Empires are comparative in that the value of a result is relative to what the opponent rolled in the same conflict, and the probabilities involved are Gaussian due to dice combinations. Yet any wargame in which players roll a single die but only one side takes casualties would use a comparative combat system with uniform probability.

One useful application we see immediately is a delineation of the possibilities for game designers. Yet for players, one also sees that one does not have to start from scratch each time one learns a new game in terms of tactics; one can carry over one's experience from similar games. The idea is to know when and to what extent one can do so.
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Sun Feb 20, 2011 1:34 am
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