I dreamed last night that I was giving them (being a big nerd), so here goes:
Pace Wittgenstein, we can give a definition of what it is to be a game. To be a game, a set of actions must contain routine or ritual components which have an indeterminate outcome that depends either on chance, on skill, or both.
There is no need to specify the number of players, the fact that some games are played with balls, others with words, etc.
There may be concepts that can't be pinned down through definitions, but this isn't one of them. (It's certainly true that we don't LEARN most concepts in this way, but that doesn't mean definitions can be given afterwards. Harder concepts to do this for are basic sensory concepts, like 'red,' and perhaps qualitative concepts.)