The most common and most likely answer to this crossword clue is the 3 letter word NIM
I've seen this crossword clue in The Daily Telegraph
N
I
M
Buddy explains!

The straight part
We know that "game played with matches" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case game played with matches = nim.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 NIM mean?
NIM noun- A game in which players take turns removing objects from heaps.
- To take or seize.
- To filch, steal.
- To walk with short, quick strides; trip along.