There are three types of games.
First: Players create avatars in a virtual world. These avatars compete against each other (levels, equipment).
Second: Players improve their skills while playing the game (chess, first person shooters, soccer). They compete directly against each other.
Third: Players spend money while playing. He who spends most wins.
Question: what kind of game do you like most?
Fake question: what kind of game do you like least? ;)
PS: When I say "compete" I also mean the competition for prestige inside a team. This doesn't necessarily mean PvP.