hardball vs baseball game

hardball and baseball game both are nouns.

Word	Noun	Adjective	Verb	Adverb
hardball	Yes	No	No	No
baseball game	Yes	No	No	No

As nouns, baseball game is a hypernym of hardball; that is, baseball game is a word with a broader meaning than hardball:

hardball: baseball as distinguished from softball
baseball game: a ball game played with a bat and ball between two teams of nine players; teams take turns at bat trying to score runs

Other hypernyms of hardball include ball, baseball.

hardball (noun)	baseball game (noun)
baseball as distinguished from softball	a ball game played with a bat and ball between two teams of nine players; teams take turns at bat trying to score runs
a no-nonsense attitude in business or politics