The most common and most likely answer to this crossword clue is the 6 letter word TIPCAT
I've seen this crossword clue in The Times
T
I
P
C
A
T
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 = tipcat.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.
What does TIPCAT mean?
TIPCAT noun- An old game in which a small piece of wood pointed at both ends is tipped, or struck with a stick or bat, to make it travel through the air as far as possible.
- The wooden piece used in this game.