Dice probabilities

20 November, 2007 at 8:43 pm (dice, game design, mathematics, rpg theory)

First, a broad approach on the subject can be found here. I’ll tackle the specific case of rolling a bunch of dice and summing the result, but it’ll be an exact formula. I won’t prove it, because I can’t, as of yet. It will be a proof by induction and require rereading Generatingfunctionology by Herbert S. Wilf with thought.

A die is defined by a probability distribution. An n-sided die, where n is a positive integer, has propability of 1/n of giving each integer on the closed interval [1, n] when rolled. A die with n sides can be written as a polynomial (x + x^2 + x^3 + … +x^n). A general polynomial is of the form (a0*x^0 + a1*x^1 + a2*x^2 + … + an*x^n). To convert it into a propability distribution of a dice roll, take ak/(sum of ai for all i that are in the interval [0, n]) as the chance of the result being k.

An example: The polynomial that corresponds to d6 is (1*x + 1*x^2 + … + 1*x^6). The chance of rolling 2 is 1/(1+1+1+1+1+1) = 1/6.

If two arbitrary distributions are already known, they can be combined by multiplying the relevant polynomials. This corresponds to the sum of the two results signified by the two original distributions.

E.g. The distribution for 2d6 is that of d6 + d6. Hence it can be determined by multiplying the polynomials of d6. (x + x^2 + … + x^6)*(x + … + x^6) = (x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 5*x^8 + 4*x^9 + 3*x^10 + 2*x^11 + x^12). The chance of rolling 6, 7 or 8 is (5+6+5)/(36) = 8/18 = 4/9.

Constant a can be represented as x^a. So, the distribution of d3+2 is (x^2)*(x + x^2 + x^3) = (x^3 + x^4 + x^5). Chance of rolling a five is 1/3.

These can be a bit cumbersome to count. It is possible to do with a handy table, which does look a bit more complicated than it really is.

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A random burst of ontology and epistemology, part 1

20 November, 2007 at 7:37 pm (philosophy) (, , , )

Skepticism is a fun way to think. Global skepticism is a philosophical stance according which we can know pretty much nothing. Everyone who is not a skepticist tries, of course, to show that the skepticist is wrong. I have not seen a tight proof along those lines, as of yet. I think it is pretty futile to try, due to skepticism being right. There is precisely one thing we can know with certainty, though reasoning must be assumed to work (if it doesn’t, this entire exercise if futile, but so are any and all responses and claims of futility, so I find myself justified in assuming that reasoning does work).

The argument goes thusly: I think, therefore a thought exists.

Do note that “I” does not necessarily exist. Or the thought might be momentary; time and other measures of change may be illusions.

The skeptic can’t really touch that argument. Neither is it particularly strong argument. The existence of a world in which we live would be nice to know, for example, and would be a lot stronger claim. I just can’t figure out a way to prove it without nontrivial assumptions. This is why I will assume it and the capability of know things about it. There are further justifications for that assumption, gratefully.

It is a fact that I perceive something around me. I will call the immediate source (as opposed to the ultimate or final source, if there is such) of these perceptions a world or a reality or some word that is practically synonym thereof.

If there actually is no world, I will lose little by assuming it, because I can’t perceive whatever else there may exist (otherwise it would, by definition, be part of world, which is a contradiction). If there exists a totally irrational and random world (defined as one about which useful knowledge can’t be gained), I likewise lose little, because no matter what I assume or don’t assume, there is nothing useful I can know about it. At least trying to find patterns keeps me well amused. If a world about which something useful can be known exists, it is smart to assume so, because it is true. A world which works so that all human assumptions about it are false is contradictory, because one could assume that all human assumptions about it all false, which leads a and not a, where “a” means “all human assumptions about it are false”.

Hence, it is justified to assume that there is a world about which one can know things.

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Defining roleplaying

20 November, 2007 at 6:56 pm (definition, rpg theory) (, , , )

A perfect definition of roleplaying has the following features:

1. It includes everything that is roleplaying.

2. It includes nothing that is not roleplaying.

3. It uses no jargon.

4. It does not refer to “roleplaying games”, or provides a noncircular and perfect definition for them, too. Ditto story-games and other closely related concepts.

My definition fails on the second criterion, IMO. I still consider it a fairly good one, if not the shortest possible.

An activity is roleplaying if and only if all the following criteria apply to it.

1. It is social in that there are at least two persons whose activity qualifies by the other two criteria. (If an alien, a sufficiently advanced AI, or a supernatural but intelligent entitity happen to roleplay, they qualify, too.)

2. The aforementioned participants make decisions for characters. (Thinking in character is sufficient but not necessary.)

3. The aforementioned decisions matter in the fiction of the game in a way that the players care about.

That’s it. Most, if not all, forms of miniature/war/figure gaming are included, which is the major flaw of the definition. Or just an observation on the similarity of such games and rpgs. Whichever way one wants to take it.

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Comments on a bachelor’s thesis on Gadamer and roleplay

20 November, 2007 at 5:58 pm (academic rpg theory, philosophy) (, , , , , )

Opusinsania wrote a bit something about Gadamer’s understanding of play and roleplaying. It is in Finnish, though. He may be willing to share the writing if asked, or not.

http://opusinsania.livejournal.com/3417.html

A very brief summary: First part of the thesis is about Gadamer’s philosophy and such. The interested should utilise Google, Wikipedia and the local university for additional and actually accurate information. What follows is my on-the-fly translation and commentary of the relevant bits. You have been warned.

Gadamer consider both play and art to be fundamentally the same thing, at least in that both of them interact with the person experiencing them and the interaction becomes primary, in a way, and separate from the rest of the world. I am reminded of Christopher I. Lehrich‘s article Ritual Discourse in Role-Playing Games, hosted on the Forge. Back to Gadamer. In his philosophy, a play (or game or art) attains perfection when it becomes a structure, or form. In theater this happens when an actor stops acting and becomes what she was trying to act. A similarity with (some definitions of) immersion and gaming are clear. Gadamer further says that the players will focus solely on the content of the game, which reminds me of flow, which, consequently, has sometimes been connected with immersion (discussion on rpg.net some years back). In spite of this, play can’t happen without a player.

Turku school manifesto by Mike Pohjola was quoted: “Role-playing is immersion (“eläytyminen”) to an outside consciousness (“a character”) and interacting with its surroundings.” A definition of immersion by Pohjola: A process in which the actor’s consciousness melds totally into the character’s consciousness. This is seen as the goal of play by Turku school. Pohjola has said that perfect immersion is impossible to attain, but can be approached (see also: limes in math). One way of understanding this is to think that the player’s consciousness is negated. Pohjola has not shown public support for the alternative (that immersion is a relation to another ontologically independent entity). A third alternative, not from Pohjola: That roleplaying is taking a social mask, stolen from ludology, and immersion has little role. There are intermediate positions.

Another important concept is diegesis, defined as everything that is true within the game. If you are into Forge theory, shared imagined space (SIS) is not that different a concept.

The purpose of gaming can be defined as immersion, the simple joy of gaming (or flow) or social reasons and other concerns outside the game. Aside from Pohjola, every definition of rp considers the social dimension to be essential.

The meat of the thesis (IMO, at least) is the contrasting of Pohjola’s and Gadamer’s understanding of roleplay (play in Gadamer’s case). Gadamer requires interaction to call something play. This can be interaction with, say, a deck of cards (solitaire) or other players. The degree of interaction required is unclear. This can be seen as supporting Pohjola in that one can play by himself (I certainly do), but it is still possible to define roleplaying so that it requires social interaction (which I find smart). Some pervasive games (which are at best fuzzily separate from normal world) may complicate the matters a bit, but I am not sufficiently interested in them to say anything further and they were not mentioned in the thesis.

Both Pohjola and Gadamer consider play to take precedence over normal life and create some sort of self-contained reality. Pohjola took the concept from Hakim Bey.

According to the thesis, a significant difference between Pohjola and Gadamer is related to Gadamer’s claim that game becomes primary and player secondary, while in Pohjola’s model the player consciously becomes the character and then the immersion happens within the player, so the player is still central. In Gadamer’s model the player no longer exists when the games achieves perfection.

I disagree with the point (badly) referenced above. In both models, the player disappears. In Gadamer’s this happens as the game reaches perfection, but the player is still necessary, as there can be no play without someone playing, as was claimed earlier in the thesis. In Pohjola’s model, the player disappears as he achieves perfect immersion (since this is impossible, mere sufficient approximity to this stage can be assumed, but it doesn’t change things significantly, IMO), but is still necessary, due to the other consciousness being the same actual person. Or, if this is not the case, the game would partially happen outside the observable reality of humans, and is thus quite irrelevant. Essentially, I disagree with the point that there is a fundamental difference between those two points of view, but can’t actually say why, probably due to not understanding the argument well enough and due to it being hard to prove negatives.

Opusinsania, if you don’t want this text to be public or here at all, say so and I will act accordingly. If you do decide to publish the paper somewhere, I’d suggest The International Journal of Role Playing.

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About theory

20 November, 2007 at 3:30 pm (meta) (, , )

I am a theoretical person.

For me, the value of theory is in the new patterns of thought it provides one with. For this purpose, math is a way to think. This particularly means that theory can be useful without a clear practical application or even a tie with the real world. Academic theory is generally of value due to there being some standards of publication. In the field of roleplaying theory, hobby theories are of value because the people who develop them usually are both passionate about and familiar with the subject. This is not always true of academic rpg theory.

Anyway. My approach to theory: Semantic debates are boring. The actual meat of the arguments and the form the arguments take are interesting. If you really want to argue semantics with me, do go ahead, but don’t be surprised if I refrain from engaging. If I misuse a word or a concept, informing me of that is appreciated, though.

There will be a theoretical point of view on practically everything, but little unexplained and unlinked jargon. Scientific referencing is a bit too much to expect, though.

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