The most common and most likely answer to this crossword clue is the 7 letter word REVERSI
I've seen this crossword clue in Financial Times
R
E
V
E
R
S
I
More clues leading to the result REVERSI
Buddy explains!

The straight part
We know that "Game" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case game = reversi.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.