When I tell people about the football/baseball/soccer/basketball game they are supposed to have today, I say

We have a game today.

which I think is strange. It is strange that it sounds idiomatic and even grammatically correct.

But this is the specific game we have today at some specific place at some specific time. And there is only one game that I am talking about. So why is it not 'we have the game today?'

Side thought: Also should I say 'the specific game we have today' or 'a specific game we have today'? I think both are acceptable, but have slightly different meaning. Can't explain the difference though.