I'm a big fan of games of all kinds, and one of my favourite styles is the good ol' table-top role-playing game. I do so like throwing dice across a table, describing and imagining scenes of glorious, wondrous fantasy. It's a form of beautiful escapism that I don't think any other kind of entertainment quite captures, where you can immerse yourself in the imaginary world of the game at exactly the level you want, so no-one is ever "doing it wrong" and everybody round the table is comfortable and having fun.
However, like many 'cult' hobbies, role-playing has something of a divide in it in terms of style. While everyone is in agreement that there is no wrong way to play these games, you can bet your bottom dollar that everyone has a right way.
The divide, curiously, can be drawn through the name of the genre: Role-Playing Game. Specifically, the line separates the last word from the rest of it - creating the "Role-Players" and the "Gamers". You can see this in practice when people talk about themselves in the hobby; many will tell you that they are a role-player, while just as many will brand themselves a gamer. This is really just harmless, but it speaks of the range of subjective methods to something that they are all doing.
I'll note here that while I speak of two groups of people, they are by no means extremes to which everyone in this hobby subscribes. The issue is of course much greyer than that, so see them as the two parts of a player's perception of the hobby, with one having maybe a bit more precedence than the other in individuals.
First, I shall speak of the gamers. I identify more as a gamer, so this is where my train of thought starts. RPGs started as a bizarre twist on wargames (something which is unquestionably a gamer thing), so were more about the rules with the concept of deeper personal narrative being this whole new unexplored thing. Everything that happened was based on equations of numbers and dice, with the referee having the final say. The rules dictated what went on, so it was very much a game like its boardgame and wargame cousins. Other than a bigger and more complex manual (and of course a totally different premise), there was little to differentiate D&D from Monopoly in terms of what market they occupied.
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