d, it's not even clear that every game
of Sylver Coinage must terminate
(i.e. that one player must lose
at some stage). Perhaps the two players can just continue naming huge
integers, each inexpressible as sum
s of previous move
That's a highly undesirable property in a game. When I play, I want to know that the game will be over at some stage. For instance, chess and Go both have this problem potentially. In chess, the 50 move rule prevents it (triple repetition doesn't quite cut it, as it only allows a player to draw, but doesn't force the draw). In Go, the Ko rule(s) (and some refereeing decisions) are supposed to prevent nontermination. Does Sylver Coinage also require some such rule? What could it possibly be?
It doesn't. See a proof that every game of Sylver Coinage ends.