This is one of the chief attractions of a role-playing game in which you are playing a role. In fact, the games might more appropriately be called role-building games. You are building a role and by playing it out you give it substance. Cooperating with somebody else somewhat diminishes the individual in a game such as this. It is a bit more blatantly obvious in role-playing games, but it is an element present in any multiplayer game.
People cooperate only briefly and only to confront and solve immediate problems.
This gives multi-player games a great deal of flexibility. They can go on and on because no one person is going to gain a long-lasting advantage.