It is convenient in these discussions to see the GM as the author of a story and the player's as the audience. Already we see that they are not flawless analogies, but they work well enough to keep them up. However, there is a third aspect of tabletop gaming to consider, and that is the system that people use.
As I mentioned in the beginning, role playing needs some amount of randomness to represent the uncertainty of life. They need ways to determine what characters can and can't do, and how well they can do a given task. There are more systems than I am aware of, but I want to give a few examples.
Dungeons & Dragons is a D20 system. That means that all the chance in life is determined by a 20-sided die. When a character tries to do something they are skilled at, they get to add a modifier number to the result that the die gives (a negative modifier can also be given to skills that people would struggle to do). The total number is put against a certain Difficulty Check, and if the result exceeds the DC, the action succeeds. Players generally do not know the Difficulty Check they need, which adds a lot of mystery and excitement when the GM announces what happens as a result of the roll.
Call of Cthulhu is a D100 system. This is very similar to a D20 system in that a 100-sided die (or two 10-sided dice) is used to determine the success of all actions. A major difference between the two is that the D100 is checked a character's personal skill ability. For example, if a person wanted to break into a building, they could try to pick the lock on the door. If the character has a 30% lock picking skill, then they would have to roll 1-30 on their D100 in order to succeed. This can make a very different experience for characters in that they always know if they succeeded or not, which can take away from the excitement. However, it makes for a simpler system as far as math goes. And if you compare Call of Cthulhu, which is all about normal humans investigating paranormal happenings, it is a much more sensible system than Dungeons & Dragons, which is all about heroes of legend acquiring increasingly potent items and abilities, and therefore focuses a lot on modifiers that affect any given circumstance.
World of Darkness is a 10-Again system. In this case, characters have a pool of 10-sided dice that they roll when they attempt to do something. The number of dice they roll depends on a number of factors, including how skilled they are at the task. Any result of 8 or more is a success and an action needs five successes to work. If any die lands on 10, then they reroll the die, keeping the original success of that ten. This system doesn't care about modifiers to roll results as much as it cares about how many dice you get to roll. It becomes a great study in statistics in that regard.
The reason I explain at least the basics of these systems is that we must understand that the game system is a framework. Sometimes the story we tell depends on the game system we're using. Sometimes the game system we use depends on the story we're telling.
The closest analogy to this is picking the right format for your story. If you have an idea, how long is it? Novel? Novella? Short story? Flash fiction? Poem? Does it need visuals to go with it? Should it be a comic book or graphic novel? Does it need audio as well to turn it into a movie? What about a stage play?
Choosing a medium to deliver your story in the most effective way is an important step (or picking a story that works well with the medium you wish to work in). The same goes for your chosen game system. They are frameworks that you build your story in and around.
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