What Are Infinite Games and Why Does It Matter?

James Carse, in his wonderful book, Finite and Infinite Games, suggests:

There are at least two kinds of games.
One could be called finite, the other infinite.

The finite game is played for the purpose of winning,
an infinite game for the purpose of continuing the play,
...and bringing as many persons as possible into the play.

Finite players play within boundaries;
infinite players play with boundaries.