As far as I can tell, there are four basic types of interaction in a two-player simultaneous game:
Pure cooperation -- you work together against the game. One could make this REALLY obvious by having the players share a life meter, score, etc.
Pure competition -- players are in direct opposition, and there is a Winner and a Loser (or a draw game, maybe).
What I call "coopetition" -- there are two possible outcomes: either one player wins and the other loses, OR both players lose (but a shared victory is not possible). This forces the players to work together to avoid the we-both-lose outcome, but they also have to work against each other to avoid the I-lose-you-win outcome. Extremely rare in the field, but one of my favorite types of game forms.
And then there's this fourth, open-ended style where players can cooperate or compete at their discretion. The most obvious examples are games like Joust and Wizard of Wor where players can kill each other (and the game offers some reward for doing so), but players can also keep out of each other's way and just work as a team to get to the highest level they can. What do you call this style of gameplay? The best I can come up with is "co-option-tition" or "co-optional", both of which sound really lame to me.