I have woefully neglected mentioning and linking to Markus Montola’s chaos model of roleplaying. The way I actually use the model, which is somewhat different from what Montola has written, is as follows. Ignore if you actually know something about chaos or are math-phobic. For the record, I don’t anything about chaos.
Chaotic system is one which has a given starting state and then changes from that recursively, where each iteration can be determined but predicting the end result of multiple iterations becomes increasingly difficult due to the interaction of multiple factors all part of the system. Technically, there should no random factors involved, but I don’t think they actually chang the model at all, assuming that the possible effects of randomness are possible due of the state that is used as base for current iteration and might not be possible, or at least not as likely, given some other state of the system. Even with no randomness, there must be some state that might happen as the result of given iteration that could not be the result of some other starting situation.
I think the model can be applied to roleplaying on two levels: The diegetic level, which is a fancy way of saying the level of fiction , where a every situation is a different end and beginning of a new iteration, or the social level where the players actually function and play happens. It could be argued that only the social level is important, but at least I also find it interesting to investigate how the diegetic situation changes and which factors change it to specific directions. There would be no roleplaying without the fiction.
Attractor is a certain path (an ordered list) of situations. The social situation or gameplay tends towards attractors. On social level, there being a rules expert in a given group is an attractor. It is likely that someone will take on that particular group, and if only a single session is observed, it is likely that there is only one such person. On diegetic level, the player characters fighting the undead hordes could be an attractor, given suitable adventure and PCs.
Bifurcation point is a situation where the game (there doubtless are social equivalents, but I haven’t thought about them as much and can’t come up with a suitable example) can take at least two directions; that is, one attractor is chosen, others disregarded. For example, in a totally original plot twist the GM decided that the big bad evil guy (BBEG) is one PC’s father, who asks the relevant PC to join the dark side and rule the world with him. This is very much a bifurcation point, because the PC and the BBEG might start working together towards world domination after a family reunion or the PC and the father might fight and there might be much angst. Either could happen. Other factors, like group mentality (everyone must be a good guy) or character portrayal (BBEG is very evil and nasty and more-or-less literally drips darkness and goo) can affect the situation, or even not make it a bifurcation point at all by making the response a given.
The next post outlines three playing styles, applies the terminology introduced in this post and provides more examples as a side effect.