Non-Linearity In Games

Games can be of two types: linear, or non-linear. The definition of linear is "of, relating to, resembling, or having a graph that is a line and especially a straight line." Thus, a linear game is one that follows a straight line, having no deviation from that line, and the player is not allowed to stray from that line.

What is the role of linearity in games? 90% of the time, the story determines the role of linearity in games. Strategy games, which usually have little or no story, are very NON-LINEAR, giving you many options of how you play the game. RPG games usually require you to hop around a world not forcing you into a path, which makes the game half-linear. And adventure games and FPS games usually restrict you to the story, making those games completely LINEAR.

Completely linear games are usually frowned upon because they limit the player's choices and freedoms, creating for a more boring playing style. After all, the more choices the player has to victory, the more interesting the gameplay will be. But then comes along a game like Half-Life 2 which is completely linear, yet ultimately amazing in every aspect. If HL-2 was a non-linear game, it would ruin the atmosphere and the story aspect of the game. It would detract from the cinematic atmosphere if you could run around that world and preform your chores in any order you choose. So, how do you incorporate non-linearity into a game (giving the player more options and choices) without ruining the story or atmosphere?

Half-Life 1 and many other games have developed a certain formula for balancing linearity and non-linearity, without ruining the story. Half-Life 1 would throw you into various "arenas" where you had to solve puzzles and fight enemies to get to the next level/arena. Once in the arenas, you usually had to find the exit and get to the next level, however you did that. The arenas gave you freedom, but each arena was in a linear path toward the end.
So, you would say that Half-Life 1 has a linear level structure, but non-linear aspects to finishing each level (some of the time).
And to add to the non-linear formula, the player is given multiple weapons, multiple ways to kill enemies, and multiple ways to solve puzzles.

If Half-Life was completely non-linear, that would ruin most of the story progression and "cool moments" in it. So instead, they have this formula, which one person put it as "...sections of non-linearity wrapped up in a path of linearity."

Zelda uses this formula in a way too. You are required to beat each dungeon in a successive order, but each dungeon can be beaten (somewhat) non-linearly (is that a word?). And then in between dungeons, you can explore the world if you want.

My rule of thumb would be this: if you can incorporate non-linearity into a game, without detracting from the atmosphere, then do it. But by all means, if it will detract from the atmosphere (which it most often does), keep the game linear in those scenes.