Their three-bullet-point definition of a game is:
- A closed, formal system, that
- Engages players in structured conflict, and
- Resolves in an unequal outcome.
The last point is problematic; it doesn't appear to account for single player games, because in order for the outcome to be unequal there have to be multiple outcomes (for different players) to be compared together. I think a better phrasing might be "Can be resolved into multiple, unequal outcomes," because that would allow for only one of those outcomes being selected - the fact that the player wins doesn't change the fact that they could win *or* lose (two unequal outcomes).
It's led me to wonder, though. A game with only one outcome isn't a game (is it? Certainly not one worth playing). And yet, there are plenty of games out there which only have a single outcome, such as Tetris: the game continues until you lose. (I think we must assume that all games are played to resolution; "the player gets bored and leaves" is not a valid outcome).
Perhaps it's necessary to redefine the outcome by accepting that the player loses the game eventually, and considering it an event rather than an outcome - so we examine the properties of the system when that event occurs, and that determines the outcome. Bingo - the player beats the top score, the player does not beat the top score. The player achieve rank #5 in the scores table. The player beats their previous best. And on and on - a massive number of outcomes to choose from, with much inequality.
So, is it possible to extend that to all play-till-you-drop games? To say that getting a "game over" isn't an outcome, but rather an event that triggers the concrete selection of an outcome? (The player can have achieved slot #5 on the high scores table at any point in the game, but that won't actually be the outcome until they get a game over). I'm thinking it is. If anyone's got any counterexamples, I'd like to hear about them.
I think it covers cooperative games as well - irritatingly I can't think of any examples of coop games where players aren't "versus the AI" - but you can just condense all cooperative players into a "team", and view that team as a single player. And then it becomes single-player reasoning.
So, all games have multiple outcomes. Consider also that an outcome which will never be reached should be discarded as an impossible outcome.
If we assume that our players are rational, we can say that given a fixed situation with fixed circumstances and fixed unequal options, the player will always pick the same option (because they evaluate the situation the same way and always pick the best option, because they're rational). The situation is deterministic - the option they select is a function of the situation and circumstances. Same situation and circumstances, same option. If we switch the word "option" for "outcome," we can quickly see that deterministic situations allow for only one outcome.
Games have multiple outcomes; therefore, games cannot be deterministic.