Finite games like the one invented by the children described by the OP have end states, they have winners, and the goal within them are always framed from that perspective. While infinite games have no winners, have no end, and the goals are often framed around ensuring that the game never ends.
It's easy to think of finite games in our lives, they are everywhere in our society, and the OP does a good job of pointing out some of the less obvious cases. Infinite games are less obvious, the one that I found most illuminating was "language", a game where people actively collude to extend the game to the end of time, and has no winner.
Anyay, it's an interesting read, but a little bit mumbo-jumbo by the middle of the book. I would recommend it.
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