They describe a leaderboard built from server members' shared gaming activity, while actual rollout remains unconfirmed.
When It Appears
“a game” can appear on a server view that ranks shared gaming activity and shows a timed competition when Discord displays the singular game unit used when the count is one.
What It Means
When “a game” is shown, the current step is presenting the singular game unit used when the count is one.
A user who saw “a game” in Discord is likely searching to confirm that it means the singular game unit used when the count is one, where it appears, and which state it describes.