The first type of game is called a finite game. A finite game is defined as any game which is played for the purpose of winning. Things that we usually call "games", such as Yahtzee and WWF wrestling, are in this category, along with things that we don't usually call games, but which are indeed played to win (at least, most of the time), such as war, college scholarships, and politics.
The second type of game is called an infinite game. An infinite game is a game that's played for the purpose of continuing the play. If someone starts to "lose" an infinite game, the other players will conspire to keep him in the game, either by helping him somehow, or by changing the rules.
The concept of changing the rules is what makes this idea really interesting to me. In a finite game, the rules are static, because rules are what determines who wins. If you change the rules, it becomes unclear who the winner is, and thus it's no longer a finite game (imagine you're playing Monopoly, and one of the other players declares a new rule: that he can take money directly from the bank whenever he wants. If this guy "wins", has he actually won?). In an infinite game, you make up the rules as you go along, and you change them whenever they become inconvenient. If the point is just to play, and not to win, then rules are only worthwhile as long as they make the game better, longer, or more fun.
Music is an infinite game. Although some people are fond of the misconception that music has "rules", such as time signatures and keys, even a limited investigation of the scope and history of music will reveal the shortcomings of this theory (free CD to anyone who can tell me the time signature of this Angola Prison Spiritual). Music theory is a way to describe music, and a language with which to communicate about it. Like any language, it's imperfect, and creates some distinctions which are lousy with exceptions.
I will say that a given song starts with a set of rules, but that those rules can be changed by the composer or the players whenever they agree to change them. Modulation is an easy example of this - if "play in the key of G" is a rule for a given song, the players or composer may choose to change that rule in order to make the song better. Dig the intro to my favorite version of "Honeysuckle Rose" if you have any doubts.
We human beings have a nasty habit of trying to turn infinite games into finite ones. Just think about the Guinness Book of World Records. Building giant replicas of things was definitely not a contest before that. The ugly business of trying to turn music into a finite game started longer ago that I can possibly determine, and is still going strong, with contests like the Grammys and American Idol leading the whole ridiculous pack.
I'm only saying all this to remind myself, and anyone else who's guilty of the same habit, that music is an infinite game. There are no rules, there are no winners, and we play it for only one purpose: to play.