If human culture were to develop again from scratch there would probably still be games, but most of them would be different. The space of possible games is very large, and I don't know any examples of a game that was developed independently in multiple cultures. I would expect, however, that the probability of playing a game would be dependent on the complexity of the rules and the quality of the game those rules define. Some complexity in rules is generally required for an interesting game: tic-tac-toe has simple rules but after a few games people reliably learn how to force a draw, while backgammon has more complex rules and experts can play interesting games.
While it's hard to pin down exact levels of complexity and interest in games, I think go is the simplest known game that a person could reasonably devote their whole life to playing.
(Star Trek Diplomacy is right out.)
Update 2013-08-13: to investigate the complexity question I wrote some programs.
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