One of these arguments is the difference between complicated and complex. While most people would use these words interchangeably, I have particular definitions for both. I don't know how widely spread these definitions are, but I like them, and they are a useful tool when I am trying to explain or analyze a game.
To me, complicated means that the game is hard to understand. This is what most people would think both complicated and complex mean, but I only use this definition for complicated. This definition often has negative connotations. If a game is hard to understand, it becomes more challenging for players to become invested, as they don't know what is going on or how to interact with it. League of Legends is a complicated game. While you can understand vaguely what is going on- two teams fighting, trying to destroy the other team's base, without significant study or experience, you can't follow combat, strategies or play the game well. Without knowing the skills of the champions, playing against them (and in some cases with them) becomes a matter of random chance.
Complex indicates that a game is mathematically complicated. It is easily confused with being complicated, because if complexity is added in the wrong way, it ends up being significantly more confusing for the player. It doesn't mean that you have to do advanced math, or that the game needs to be hard to predict. Labeling a game with complexity means that you're options are varied and extensive, not random. Chess is a un-complicated game. Without advanced rules, there are six pieces that you have to care about and a couple of rules regarding how the game ends. However, it is an incredibly complex game. After one move, there are twenty possible board positions. After both players have gone once, you're looking at four hundred different possible positions. After both have gone twice, there are seventy-one thousand, eight-hundred fifty-two possible positions. Other, more complicated games can be much more complex because of the larger number of positions and/or pieces available. Starcraft 2, for example, can have over four-hundred pieces in a two player game, and these pieces aren't limited to gridlike movement. However, this makes the game more complicated as well as being complex.
I can't think of a situation where a designer wants to make their game more complicated. I can understand letting it become more complicated in a trade-off for making the game more complex, as increasing complexity leads towards increasing the value of strategic thinking. This in turn makes the player choices have more of an effect on the player because they know that their choices are either making them win or lose, rather than their reflexes or pre-planned plot devices. Add complexity where you can, but be wary of making a game more complicated without making it more complex.
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