Professor Carse writes in the first chapter, "There are at least two kinds of games. One could be called finite, the other infinite. A finite game is played for the purpose of winning, an infinite game for the purpose of continuing the play." link
Finite games have a definite beginning and ending. They are
played with the goal of winning. A finite game is resolved within the
context of its rules, with a winner of the contest being declared and
receiving a victory. The rules exist to ensure the game is finite.
Examples are debates, sports, receiving a degree from an educational
institution, belonging to a society, or engaging in war. Beginning to
participate in a finite game requires conscious thought, and is
voluntary; continued participation in a round of the game is
involuntary. Even exiting the game early must be provided for by the
rules. This may be likened to a zero sum game (though not all finite games are literally zero sum, in that the sum of positive outcomes can vary).
Infinite games, on the other hand, do not have a knowable
beginning or ending. They are played with the goal of continuing play
and sometimes with a purpose of bringing more players into the game. An
infinite game continues play, for the sake of play. If the game is
approaching resolution because of the rules of play, the rules must be
changed to allow continued play. The rules exist to ensure the game is
infinite. The only known example is life.
Beginning to participate in an infinite game may be involuntary, in
that it doesn't require conscious thought. Continuing participation in
the current round of game-play is voluntary. "It is an invariable
principle of all play, finite and infinite, that whoever plays, plays
freely" (p. 4).