Now playing

30 April, 2008 at 2:37 pm (actual play) (, , )

I’m playing in two games right now; Thulen‘s victorian game (he hasn’t written down any actual play yet; I hope he will) a Nobilis game set in Star Wars universe, GM’d by Ukkostaja.

Good and poor play

I’ll be judging my own play and that of the others; by “good” play, I mean creating and grabbing story opportunities, doing things, being entertaining an so forth. By “poor” play, I mean being a wallflower, turtling, not interacting with other participants, and so forth.

Victoriana

Cast of players: Me, wgaztari, ksym, Ari (with no web presence that I am aware of). Ari is new to the group, but has roleplayed before. His integration was quick and painless.

There’s been two sessions of actual play and one of character generation and prelude-thingies. I am slowly getting used to the game. Next session will probably be genuinely enjoyable; the end of the previous was good, but beginning meh; the first one was indifferent. This no fault of anyone else; wgaztari played exceedingly well in the first session and okay during the second; Ari is overall a good player, and ksym has been doing well, though his character is quite different from the others; he knows magic and uses it, and is the only one who does.

I seem to always play pretty poorly for the first session or two. Maybe it is anxiety with the group, maybe with the playing style, maybe with something else.

There’s been robberies, a smidge of intrigue, meeting worthy opponents, but not yet truly engaging them. Oh, and the end of the last session was when ksym’s character threw my char with a sword. Hit or miss; not determined yet.

SW Nobilis edition

Cast of players: Me, Thulen, possibly pickaboo, and even opusinsania paid a visit (I made a guest appearance as a crazyish NPC in his game just the other week; didn’t go very well, but was enlightening). Estimated play time remaining: Session to three.

For some background: Nobilis is a roleplaying game in which players are Nobles, something like gods or demigods. It is diceless, but has a definitive economy with miracle points used to power all sorts of things. Like Amber diceless, but more structured and characters are more exotic. Mine is a Noble of light, with a special ability called elemental form: Can turn into light at will. Convenient.

Thunderer’s game mastering style seems to be that there is a definitive plot that will happen, more-or-less as planned, but the way it happens is up to players. (Also, if we build a mess ourselves, that’s a bonus.) Basically: I don’t try to get any enjoyment from story-crafting; using the rather formidable powers that nobles have is the point of the game, as far as I am concerned. The story happening around that is a perk/hindrance, depending if it adds new stuff to play with or involves boring puzzle solving or following clues to wherever. (Actually, I just suck at puzzle solving and following clues, and hence find them boring, and hence suck at them, …)

Thunderer (Ukkostaja; direct translation) also built a cheat sheet for the price of using miracles, given the miracle level and character’s relevant attribute: PDF.

Interesting is the fact/impression that I got used to Ukkostaja’s game more quickly than to Thulen’s, despite knowing the latter better. It may be the smaller group of players. I also think that having formed a clear picture about the way Ukkostaja runs the game helps a lot. This picture might or might not be accurate, but it is clear. I am not yet quite sure what I should actually be doing and enjoying in Thalin’s game.

In other news

Eric has an idea for something like a custom roleplaying search thingy. Maybe something like Mahalo. I’ll be helping along in whatever ways I can, since the idea seems interesting enough.

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Summer

22 April, 2008 at 3:08 pm (meta)

Within a month the following shall happen:

  • I will take on a summer job that involves cleaning factories with heavy machinery.
  • I will not study in university before autumn.
  • I will live in Hämeenkyrö (with mother, stepfather, sister and dog) or Tampere (with father), neither of which is exactly close to Jyväskylä.
  • I will not roleplay as much, if at all. Maybe this shall be the time to try some more roleplay via the internet, either with friends or in Roleplaygateway.
  • I may play board games or card games, especially if there is no or too little roleplay.
  • I will not spend as much time on the internet. Particularly I will probably not read forums or use stumbleupon. Also, I will blog less if at all. This all is quite probable (P({this all}) = 7/10), but not certain, where “this all” means this bullet point.
  • I will read more. Interesting material: Fantasy (preferably not direct D&D derivative and preferably not Eddings), philosophy (ontology, philosophy of mind, epistemology, whatever else is interesting), logic, mathematics (particularly game theory is of interest), mythology and old poetry, rpg books.
  • Take a dog, walk somewhere, anywhere. Good fun. Run once a while to keep it interesting. Great for thinking and designing rpg settings and calming the mind. Grab a friend if one is nearby. Remember to take water.

The more exact date of my departure is between three and four weeks from today.

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Mathematical proofs

21 April, 2008 at 7:44 pm (mathematics) (, , , , )

In which I shall relate a few ways of building mathematical proofs. Should be useful for several kinds of problem solving.

Basic structure

The basic structure of mathematical problem solving is simple: There are certain assumptions (or axioms or definitions) and a certain desired outcome. The assumptions are used to get the desired outcome.

For example (disregard if not interested; this is a toy example, and the substance continues after it), Bolzano’s theorem states that if there is a continuous function f such that f(a) < 0 and f(b) > 0 (f is a mapping from real line to real line, a < b), then there exists c between a and b such that f(c) = 0.

So, assume that one has a continuous function, such as g(x) = x^2, that has g(0) = 0 and g(2) = 4. Can Blozano’s theorem be used to prove that there exists z between 0 and 2 such that g(z) = 1? As Bolzano’s theorem only says that if the function gets positive and negative values, it also gets zero, it can’t be directly applied. The trick is to define h(x) = g(x) -1. Now h(0) = -1, h (2) = 3, and hence the there is z between 0 and 2 such that h(z) = 0, and hence g(z) = 1. Tacitly I assumed that continuous function minus a constant is still continuous, which would also have to be proven.

Indirect proof a.k.a. proof by contradiction

Also known as reductio ad absurdum, the idea of an indirect proof is that one assumes that what is being proven is actually false and from that follows a contradiction. The point of indirect proofs is that they give a free assumption to play with, essentially. That is: If one assumes A and B and must prove C, an indirect proof would mean assuming A, B, and “not C”, and find any contradiction.

It is easy to make a mistake in formulating the antithesis (“not C”): Take the definition of a continuous function, which says that for all x, for all positive e there exists a positive d so that if the absolute value of z-x is less than d, then absolute value of f(z)-f(x) is less than e. If something is not continuous, it means that there exists and x for which there exists a positive e such that for all positive d |z-x| < d and |f(z)-f(x)| is at least as large as e. This is a fairly simple concept (continuous function on the real line), which just happens to look scary.

An indirect proof can use any other tool in the box; it gives a free assumption and is never actually harmful, though often useless.

Constructive proof

A proof can be said to be constructive when there is an item that must be shown to exist and this is done by actually constructing the item in question. Constructive proofs are often cumbersome (if rigorous) and longer than uncontructive ones. The reason for appreciating constructive proofs is that they are concrete: Something is actually built. It makes understanding the proof a lot easier.

For example, a function is (Riemann-)integrable if one can put rectangles below and above it such that as leaner rectangles are used, the approximations grow arbitrarily accurate. The problem is that if the function is not a very simple one, building the rectangles is difficult. Hence people instead learn a bunch of rules and tables so that they don’t need to and can instead do easy calculus. Or, in case of people who study physics, handwave it all away with an infinite number of infinitesimals. (One can treat them rigorously, but…)

One can also name unconstructive proofs, if one wants to. For example, everything that uses the axiom of choice is unconstructive (I know of no exceptions and have hard time figuring out how to create one, but someone will hopefully come and tell me I am wrong). Some hardcore mathematicians only accept constructive proofs; they are consequently known as constructivists, and are a rare breed. The scope of what they can prove is greatly limited.

Proof by exhaustion

Proof by exhaustion means dealing with every special case in order, one-by-one. Proof by exhaustion is often long, ponderous, boring, and avoided at all costs by most mathematicians. The most famous example if the four colour theorem, which essentially says that given any map, one can colour the nations that share borders using different colours and only use four colours in the process (this means that they must actually have a small bit of common border; single point is not sufficient). It was proven by a computer that essentially went through all the interesting situations. Five- and six-colour theorems can be proven in the conventional way with relative ease.

Mathematical induction

Normal induction works as follows: The sun has risen every morning that I remember. Hence, it will probably rise tomorrow morning, too. Pretty sensible, though it often goes wrong in nasty ways (every black-skinned person I have met thus far…).

Mathematical induction is used to prove things that apply to all natural numbers (1, 2, 3, …; 0 may or may not be included), or to anything that can be numbered by them, such as the propositions of propositional logic.

For example, the sum 1 + 2 + 3 + … + n = (n+1)*n/2. (E.g. 1 + 2+ 3 + 4 +5 = 6*5/2 = 3*5 = 15).

The first step is to check that the equation works when n = 1. This is often trivial. Particularly: 1 = 2*1/2 is indeed true.

The second step is to assume that the equation works for some natural number k, which is not specified. That is: 1+ 2 + … + k = k*(k+1)/2. This step is not particularly strenous. That is: Assume the equation holds for n = k.

The third step, which is the actual substance of the proof, is to see that the equation now holds for n = k+1. In this specific example, it must be shown that 1 + 2 + … + k + (k+1) =  (k+1)*(k+2)/2. The assumption made in step two will be useful here. (k+1)(k+2)/2 = (k+1)*k/2  +  (k+1)*2/2. By the assumption in the second step the first term equals  1 + 2 + … + k; that is, the formula looks like 1 + 2 + 3 + … + k + (k+1)*2/2, where the last term is simply k+1. Hence, 1 + … + k + k+1 = (k+2)*(k+1)/2.

The first step established that the equation holds for n = 1. The second and third established that if it works for n = k, it also holds for n = k+1. That is: Because it works in the first case, it must also work in the second case, and hence also in the third case, and so forth. Hence it works for all n, as it well should.

Proof by handwaving

The most powerful tool wielded in a seminar of lecture, proof by handwaving involves drawing pretty pictures, writing down some equations, and appealing to the common sense of whoever is subjected to the explanation and the vigorous hand movements. Phrases such as “Clearly”, “One can easily show that”, “It can be proven that”, “Gauss has proven that”, “this is left as an exercise for the reader/student” are often used to great effect. Proof by handwaving can even be used to prove false statements, which makes it the strongest method catalogued herein, even if not the most rigorous.

For real life examples, see such achievements in accurate film-craft as everything by Michael Moore and the “documentaries” titled “The great global warming swindle” and “Expelled! No intelligence allowed” (or something to that effect). Even if they are correct in some parts, such facts are established by vigorous handwaving and propaganda, and hence can’t really by trusted. (Disclaimer: I have seen part of the global warming propaganda and a film or two by Moore.)

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Between blocking and resolving

18 April, 2008 at 10:09 pm (game mastering, rpg theory) (, , , )

This post will be about techniques for accepting and influencing the inputs of other participants when roleplaying. Inspirations: Improvisation for roleplayers by Graham and an rpg.net thread by R00kie. Observant readers can see why I use exactly six different categories. I am sure they can be merged and more can be added if such are searched for.

Context

Say, I am running a random dungeoncrawl. A player character has discovered a secret passage to what seems to be a room full of treasure and wants to and grab some. The secret passage is quite crambed.

How can I react, as a GM? The process of picking which way I actually react may be a matter of rules (e.g. failed roll, no treasure; successful spell digs a tunnel), predetermined facts (the character is fat, no treasure; the character can pass through stone, easy entrance), on-the-fly setting creation (there’s a forcefield between you and the treasure; a minor earthquake opens up the passage), or by other means. That’s not the main focus here (not that I’ll keep quiet about it).

Blocking

“You can’t get through.” Blocking means that the input of the other participant is, for whatever reason, by whatever means, mande insignificant. As a general heuristic, one should avoid blocking. It slows everything down and disrupts flow of the game. Blocking is the way into boring failures when dice are not favouring the players.

here are some expections: The first is an idea that is totally out of synch with the rest of the game. A gritty and serious dungeoncrawl and someone is yelling Superman to widen up the entrance a bit. (Another common reaction is treating it as in-game sillyness. I’m not seeing the benefits of that, but won’t start yelling badwrongfun, either.)

The second situation that may merit blocking is when something has been established as futile, yet someone keeps trying. You really, really can’t get past that forcefield by hitting it with a club. Really. No, not even when raging. This is usually a case of communication failing between the participants and should be handled as such.

Shifting

“No, you don’t get inside the chamber, and further the dragon hears you.” The idea with what I’ll name shifting is that the previous outcome is not achieved and something else surpasses it in importance. This is what I used a lot in the Burning Wheel game (in context of someone failing a test). Basically, shifting is one interesting way to handle severe failures and setbacks. Not only does the attempted action or contribution to the fiction work, but also something else comes and grabs the attention.

Shift is something one might do when the game is running too slowly and some character screws something up, or other suitable situation occurs. More generally: Use a shift to change the pacing or other aspects of the game significantly. Like, “The orcs overwhelm you. You are standing there with a spear to your throat. The demon who leads the orcs walks through their ranks to face you.”, where an encounter with orcs does not lead to (immediate) character death and a potential BBEG (big bad evil guy) is introduced. Hectic combat is replaced with some probably hectic in-character dialogue and potential deals with demons. (Now I want to run that game. Damn.)

Opening

“You can’t get through, but there is this jar you just could tip over to make some noise (presuming that there is another entrance to the treasure trove and guardians in the place).” Opening still prevents the original intent from happening, but offers some other viable action or cause of play. Note the “offers”. Shift forces one instead of opening up new possibilities. Openings tend to slow down the game a bit, as people like to evaluate different options they have.

If running a game where the characters are not sticking together, open up opportunities for one player and move on to the next one. I wish I had figured this out and explicitly written down way before this. Using shifts in the same way may work as (mini-)cliffhangers, but killing the momentum is at least as likely.

Complicating

“You get through the passage, but a several guardian skeletons rise from the thrones they were sitting in.” Complication means that whatever was attempted actually worked, but so did something unanticipated and usually unwanted. Complication, like shift, changes the nature of the conflict, but tends to keep the goal fairly intact, which shift is likely to not do.

Complications are easy to introduce when some action is almost a success (or partial success or whatever), or when some minor mistake is done is some way. Complications often slow down the overall speed of events, but their effect on tension varies; use them as a pacing tool when something important is happening too fast to be enjoyable. “Your finger of death kills the BBEG, who barely manages to snap a wand in two. Red haze fills the room.” The action of the player is still relevant, but the climatic battle is just about to start.

Building

“You get to the treasure vault and of all the treasure a particular golden ring catches your attention.” Building means that whatever the other participant wanted to add to the game is now part of the fiction, and something that enhances the effect also happens. Interesting successes are situations where something is built. Building means that the goal, if any, is achieved, and yet something interesting happens. Run from the bandits and discover a hermit living the woods.

It is usually possible to ignore the new hooks that entered play, but it is considered bad form in some groups. It is essentially a way of blocking: “No, I don’t even touch or look at the ring.” In other groups the same behaviour might be called smart play.

Resolving

Where block is a clear No, resolution is a clear Yes. It is a closure, an end. A time to move towards other points of interest, or to end the game entirely. The trick is using resolution if and only if it is appropriate: Too rarely and the game bloats with new options, making it a huge mess full of unsolved events (I do this.); too often and the game will look episodical with only tenous connections between the different sessions or other instances of play. (If you enjoy episodic play, reduce the duration of the episodes until you are no longer interested, like so that every encounter is very much a discreet unit of play with little connection to anything resembling a setting or story.)

Larger scale

Assume that a given instance of play can be divided into scenes, each of which is fairly continuous with regards to characters, location and time. Ignore the scenes that are not interesting (for your particular definition of interesting), if any such exist.

The question is: How are the scenes linked together? This all is after-the-play reflection or even analysis, though may work as a preparation strategy, too. Particularly: How do scenes end and how do they start? My gut reaction is that if a lot of scenes end in resolution, the need for contrived plot hooks and the like is increased to keep the character engaged. Compare: Kill Kranach the raider lord, gather reward from the sheriff, enjoy the reward, spot a plot hook, grab it, go rescue a puppy from a cave. Alternatively: Kill Kranach and rescue the tiny girl at the same time. The girl asks you to go find her dog, which got lost in the nearby dark cave.

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Rules as toys

15 April, 2008 at 7:28 pm (roleplaying, roleplaying-games) ()

Levi Kornelsen wrote a post about using rpg rules as a toy when playing. I find the idea intriguing, having done something similar as a gamemaster of several homebrew games.

The idea is that playing with the rules (as opposed to by the rules) is part of the fun: Assume a game does not have mass combat system. A large combat occurs with a player character leading one side. How to handle it? Or maybe the game doesn’t have good rules for climbing on a dragon and stabbing it to death while riding it. Or whatever.

My gut reaction is that d20 would be particularly suited for this kind of play, because it has many moving parts. Climbing on a dragon and then scewering it could mean bypassing natural armour, getting automatic critical threats (for a particularly weak spot), doing it extra con damage, having it treated as flat-footed against your char, or a myriad of other possibilities.

Combine this idea with Ryan Stoughton’s little pdf named Raising the Stakes (direct link), which is a free download on the E6-wiki. The relevant part of the PDF is that players can suggest extra effects for their rolls, but they also have to suggest the heightened negative consequences for failing the roll. GM can accept or decline the suggestion. Like: “I raise the dragon being flat-footed against me falling prone right next to it”, in context of aforementioned climbing. GM accepts or declines.

This will not work when some participants are not familiar enough with the rules. Further, a common standard for the kind of fiction to create is necessary: If some players are going for extremely gritty style and others high-action anime combats, there will be some disagreements ahead (this is true in general, but even more so in case where this variant is used).

It would be possible to even build a little d20 variant around this. Say, generic classes that get one ability/feat/whatever per level. Stunts would work as per raising stakes above, the feats/class abilities would be essentially stunts the character has mastered (and can use at will without explicit GM permission).

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Fluxx and Uno; system and memory

12 April, 2008 at 9:14 pm (game design, game element) (, , , )

My sister and a friend of hers visited my humble apartment. We played more than a bit of Fluxx. Here’s some reflection on that and on the numerous Uno games I have played along my short life.

Accessibility

Uno is extremely easy for even young people. Fluxx requires fair amount of skill with English (even my father had problems, surprisingly). I think they both are casual games. There’s a few factors that affect this.

Unpredictability

Fluxx is the epitome of a chaotic game. This chaos is amplified when there are several people. With two or three people, your play actually has a visible effect on your next turn; with six, it does not have too much of an effect. There’s some, but not enough to count on.

Uno has few variants that are played among the circles where I have played it: First is to draw one or two cards when you can’t play any, second is to draw up to three cards until you can play one (and immediately play that). The third is to draw cards until you can play at least one of them and then play that. The first are least unpredictable, the third most. Also, the number of players has a large effect: Usually it is possible to meaningfully affect the next player or maybe two, again depending on the rules used (stacking “draw”-cards either affect only the next player or each player gets one effect until all cards are used). As in Fluxx, the state of game can vary significantly between the turns of an individual player. This is almost the norm when there are many players.

Why do I think unpredictability is good? First, it reduces stress; you can always blame the luck and will often be correct. In addition, both games can take new players in midplay and not make a significant splash. Further, one can take a pause from the game when the others are doing their turns and often not a lot has been missed.

No death spiral

Fluxx tends towards equality among the players due to the numerous hand limits and keeper limits, as well as rules reset. Further, winning the game is always possible by shifting the goal, stealing or scrambling keepers (and the changing the goal), or just picking the correct keepers and playing them. If you’ve got no keepers, hope someone will put a limit on them. Fluxx doesn’t so much balance itself as it screws everyone equally and always keeps victory a possiblity.

Uno self-balancing in a very elegant way: The more cards one has, the faster one can get rid of them by playing many at the same time. Also, as one gets more cards, the chance of drawing cards that match them in symbol only increases. (People rarely forgetting to say Uno is also something of a balancing mechanism, though very weak one).

The lack of death spiral means that skilled players don’t seem to dominate, because they could be toppled at any moment.

General observations

Take any system where participants have turns, take some action during a turn, then wait for the next one (examples: Heroes of might and magic n, most rpg combats, ADOM, roleplay with a split party). How much does a single turn matter?

Number of actions

Obviously, the number of actions one can take are very important. If one can somehow get more actions or deprive opponents of theirs, such abilities often are extremely valuable. For example: Haste in D&D 3rd, reflexes in Burning Wheel. If the number of actions or action points that are used when doing anything can be altered, one would do well to start with a fair number of them. If everyone starts with single action, getting another is worth very much. If everyone has 10 action points, getting 1 extra is very nice, but won’t as easily break things. Getting 5 or more does break things.

Whiff chance

There may be a chance that the actions one takes simply have no effect. High whiff chance is undesirable, because it tends to be frustrating (I want to hear a counter-example for this one). Further: With a significant whiff chance, the system becomes more chaotic; a given amount of play may give no results or be hugely effective, depending on luck. Obviously low number of actions and high whiff are a bad combination.

Power

One has actions and does not whiff. What happens? In all examples I can recall right now the power of different actions (choices) is different. This may be balanced by different costs (in actions or other resources), different whiff chances (magic missile always hits), or other factors.

Memory

Memory may not be quite as obvious a factor as the others. In a system with long memory the effects of the choices one makes linger for long. They may change or weaken but one can easily see that a particular effect is there due to a particular choice made. System with short memory obfuscates these relations: The status of the system changes rabidly or radically. Or maybe there is a large number of choices made, so that the effects of single one are effectively buried. Or maybe there is a strong attractor the system tends towards, so choices tend to be lost as the attractor is approached again.

In roleplaying context: Traditionally, system has long memory with regards to character generation. Choices there count for a lot (hence the flames around point-buy vs. rolling and the tendency to let people do minor changes after actually playing a little). Character death is another event that games tend to remember for long.

Gamers try to avoid effects that are harmful and have long memory: D&D examples are level drains and ability drain/damage (prior to plentiful restorative magics). In the Mountain Witch the wounds that have duration for “rest of the game” tend to be nasty (this one is from experience), even if that duration is rarely more than three sessions. Generally speaking, permanently disfiguring a character is something that many gamers really dislike (exceptions abound). In some games, losing items is more harmful than character being wounded, because healing is fast and wounds matter little in the long run.

An interesting rules element

Whenever a player loses a conflict/roll (as suits the game and situation in play) any participant can suggest a permanent consequence, or at least one with long memory. If the player does not want that, damage time, for whatever values of damage the system recognises.

Example the first: A troll subdued the would-be trollslayer. Options: Take the harm (given the circumstances, may very well be death unless there is help coming) or take a semi-permanent nasty effect, such as a trollslayer cast into the river from which he is rescued with only his clothes on (or so the villagers insist), or the troll consuming the slayer’s right hand and leaving the slayer to die, not liking the taste.

Example the second: Negotiations with the high king. A failed diplomacy check. Options: Beaten up and thrown away from the castle, a humble apology (and charisma damage due to the humilation and loss of confidence), the ire and later assassins of the high king, losing some allies from the local nobility, being branded an outlaw, …

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Game design =/= rpg design

8 April, 2008 at 6:37 pm (definition, game design) (, , , )

During brief discussion with Phil I verbalised the idea of good game design not being the same things as good rpg design. This is obvious when discussing, say, Chess. I argue that it is also true when discussing roleplaying games, given the way I define good game design.

The definitions have my bias clearly articulated; they are there for all to see. If you have different base assumptions or definitions, your conclusions may also be different.

Definitions

Game design is building a (semi-formal) system where players can make mechanical choices that have mechanical consequences. Good game design makes this process of decision-making interesting: There are few null choices that have no effect and the best choice is often enough very hard or impossible to see, if it even exists and is unique.

Rpg design is building a fiction and a system that describes how the choices the players make affect the fiction. Good rpg design makes the process of play interesting: There are actual choices to be made, they are about something the player cares about, and there are several roughly as lucrative alternative ways of making many choices (in this paragraph several can be arbitrarily large, but not too small).

Good rpg/game design does not imply that the game itself is good, because there are numerous other factories related to that. As such, if one is only interested in how much enjoyment can be derived from a (roleplaying) game, fixating too much on the quality of the (rp)g is not advised. There is correlation: On average, well-designed stuff is more enjoyable.

Do note that the other kinds of design are immensely important (and not part of the above definitions): Designing the game so that it has a suitable social footprint (the time, effort and commitment gaming takes), building the game so that it encourages the creation of certain kinds of fiction, building functional character sheets, elegance and other usability issues, and doubtless other factors. I may someday extend this post to explicitly include some or all of those things. This is not that day.

The thesis

My thesis is that good game design and good rpg design, as defined above, are not very tightly linked. One can have an rpg that is well-designed game but not very interesting fiction-wise; likewise, a well-designed rpg need not have interesting mechanical elements.

What I am not saying is that the two design issues are orthogonal; they certainly affect each other. I am also not saying that they are independent; the quality of one factor tends to influence the other for the positive, because it is common to link certain fictional and system-level effects together.

Examples in the abstract

Assume a game with very complicated (and intense and fun) combat system. Assume the output of the system is the amount of hit points the participants have at the end of the combat. All other variables that change only affect the single combat encounter and any used resources are recovered with a moment of rest or such. This combat system is (one can assume) good an instance of game design, because it has many (mechanical) choices that are interesting. It is not good rpg design, because none of those juicy choices are persistent; all that remains is the number of hit points one is left with. To be honest, there are other potential choices one can make: Which opponent to kill, how much of one’s abilities to reveal, for example, but they are pretty minor and would work with almost all combat systems.

A game where each (player) character has a number of memories (some of which are utilitarian, some have emotional value, some both) and the character can sacrifice them to demons in order to get wishes or other benefits could be well-designed, rpg-design-wise; if the character sacrifices too much, that character can no longer enjoy from the achieved victories; if too little, something bad will happen. OTOH, sacrificing the utilitarian memories (where was the artifact hidden again?) can have much the same effect as sacrificing nothing: Failure at preventing the bad things. Game-design would only make this interesting if the memories with emotional value gave some sort of benefit; otherwise they are like spell points.

On D&D 4th

From what I have read, 4e is focused on encounters and the designer are doing game design. What about rpg design? No idea. Experience for achieving certain story points could do that, but I am more than slightly doubtful. This does not mean that “there will be no roleplay in D&D 4th”. The system just will probably not do all that much to promote the kind of roleplay I am looking for.

Bonus: Proof by antithesis

Assume that all good rpg design is always good game design. See the two example above. They are non-trivial counter-examples to the antithesis and hence the antithesis is wrong, from which it follows that the thesis is true. QED.

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Some probability theory

4 April, 2008 at 9:56 pm (dice, mathematics) (, , , , )

I’ll explain what I see as the point behind some elementary concepts of probability theory. The primary source is Introduction to Probability theory by Geiss, Christel and Stefan.

The basis

The idea behind probability theory is to treat uncertainty with mathematical rigour. The first necessary component is a (non-empty) set of events; for example, if rolling a d4, this would be {1, 2, 3, 4}. When arriving to traffic lights, {red, yellow, green} is a passable set of events (at least as the traffic light are around here). The lifetime of an electronic device could have the set {1, 2, 3, 4, …}; that is, the set of natural numbers, which indicates how many days the device functions. If there are news in the radio every half an hour, the time one has to wait for the next news after turning on the radio creates the real line between 0 and 30 minutes; in math, [0, 30[ (0 is included, 30 is not).

Sigma-algebra

Sigma-algebra is defined by a certain set of properties, which I’ll list a bit later. The idea behind sigma-algebra is to list the collections of events one might want to measure the probability of.

Taking the d4 as an example, the largest possible sigma-algebra is the set of subsets or power set of {1, 2, 3, 4}, which means a set that contains {} (the empty set) {1} {2} {3} {4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} { 1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 2, 3, 4} for a total of 16 sets (16 = 2^4 which is not a coincidence, and it also is the reason for using d4; d6 would have involved 64 subsets).

What if one is only interested in the end result being even or odd? The following sets also form a sigma-algebra: {}, {1, 3}, {2, 4}, {1, 2, 3, 4}.

The properties of a sigma-algebra

Let E be a non-empty set. The sigma-algebra of A always contains the empty set and E. The idea is that the propability of nothing happening and that of something happening are always known. In addition, if any subset A of E is part of sigma(E), the complement of A (negation of A) is also part of sigma(E). The idea is that if the probability of A is measurable, the probability of “not A” must also be measurable. Further, if any (finite or countable) group of subsets of E are part of sigma(E), so is their union, which means that for any group of measurable events, one can measure the chance of at least one of them happening.

From these follow a lot of things; see the PDF for more detail on the process and results.

Probability measure

Probability measure intuitively assign weight to every set that lives in the sigma-algebra. To take the d4 again, the weight added to every outcome is 1/4 (0.25 or 0,25 or 25% for those who fear fractions) if the die is fair (and is rolled fairly). If the die is weighted, the probability of {4} could be 1/2 (50% aka 0,5 aka 0.5), while that of every other number could be 1/6 (about 17% or 0,17 or 0.17).

The rules that probability measures must conform to in order to be called probability measures are as follows: The probability that something happens is 1, which is written P(E) = 1. If A and B are disjoint subsets of E (they are entirely distinct; for example, even and odd number or the number 1 and cats), the probability that something out of A or B happens equals the probability that something out of A happens plus the probability that something out of B happens. In symbols, P(A or B) = P(A) + P(B) for disjoint A, B. This applies to all numerable groups of subsets. The third rule is that the probability that something out of A does not happen equals one minus the probability that something out of A does happen, which is equivalent to P(not A) = 1 – P(A). It follows that the probability of nothing happening is zero.

The connection to sigma-algebra

In addition to every probability measure requiring a sigma-algebra to even be defined, there is another connection between the rules that defined them. Every sigma-algebra of E always includes the empty set and E; likewise, the probabilities for both of these are always defined. Likewise, if A is part of sigma(E), “not A” also lives there. Contrast to the fact that if the probability of A is known, so is that of not A. The final part of the connection is that summing up probabilities and taking a union of subsets work in similar way (there is an easy way of making any numerable group of sets disjoint; take the first set as is, take the second but remove any overlap with the first, take the third and remove any parts that overlap with the first or the second part and so forth).

The existence of this connection is obvious; there is no sense in building the definitions in a way that does not produce these connections. Still, they are useful guidelines for remembering the definitions, since it is sufficient to only remember one of them and the other can be deduced from it.

Elaboration on d4

I won’t build any more theory, since it is well-presented in the lecture notes and this is long enough as is. Go read those if you are really interested and already know some mathematics. The notation that follows is occasionally a bit clumsy, but there are reasons for it. Anything in square brackets indicates a set.

The measurable space ({1, 2, 3, 4}, {{}, {1, 3}, {2, 4}, {1, 2, 3, 4}}) can be used to determine the probabilities of getting an even or odd number with a d4. First, assuming a fair die, the relevant probability measure is defined by P({1, 3}) = 1/2 (it follows that P({2, 4}) = 1/2). The probability of rolling for example 3 is not well-defined, because {3} is not part of the sigma-algebra in use. One can think of this as a friend telling that the result was even or odd, but not what the exact number rolled was. Using the loaded die introduced earlier, the relevant probability measure would be characterised by P({1, 3}) = 2/6 = 1/3 from which it follows that P({2, 4}) = 4/6 = 2/3.

Note that with the measurable space given one could as well flip a coin; it would have two options, though they would be heads and tails, not numbers, but they could be mapped to the real line to give numeric results.

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